Michel Janssen, Jürgen Renn, Tilman Sauer, John D. Norton, and John Stachel A Commentary on the Notes on Gravity in the Zurich Notebook Table of Contents 1. Introduction 1.1 The Four Heuristic Requirements 1.1.1 The Relativity Principle 1.1.2 The Equivalence Principle 1.1.3 The Correspondence Principle 1.1.4 The Conservation Principle 1.2 The Two Strategies 2. First Exploration of a Metric Theory of Gravitation (39L–41L) 2.1 Introduction 2.2 The Three Building-Blocks: Gauss, Minkowski, Einstein (39L) 2.3 Finding a Metric Formulation of the Static Field Equation (39R) 2.4 Searching for a Generalization of the Laplacian in the Metric Formalism (40L–R) 2.5 Transforming the Ellipsoid Equation as a Model for Transforming the Line Element (40R–41L) 3. Energy-Momentum Balance Between Matter and Gravitational Field (5R) 4. Exploration of the Beltrami Invariants and the Core Operator (6L–13R, 41L–R) 4.1 Introduction (6L–13R, 41L–R) 4.2 Experimenting with the Beltrami Invariants (6L–7L) 4.3 Investigating the Core Operator (7L–8R) 4.3.1 Covariance of the Core Operator under Non-autonomous Transformations (7L–R) 4.3.2 Generalizing the Constituent Parts of the Core Operator: Divergence and Exterior Derivative Operators (7R–8R) 4.4 Trying to Extract Field Equations and a Gravitational Stress-Energy Tensor from the Beltrami Invariants (8R–9R) 4.5 Exploring the Covariance of the Core Operator under Hertz Transformations (10L–12R, 41L–R) 4.5.1 Deriving the Conditions for Infinitesimal Hertz Transformations (10L–R) 4.5.2 Checking Whether Rotation in Minkowski Spacetime Is a Hertz Transformation (11L) 4.5.3 Checking Whether Acceleration in Minkowski Spacetime Is a Hertz Transformation (11L) 4.5.4 Trying to Find Hertz Transformations under which the Core Operator Transforms as a Tensor (11R) 4.5.5 Checking Whether Rotation in Minkowski Space- Time Is a Hertz Transformation Under Which the Core Operator Transforms as a Tensor (12L, 11L) 4.5.6 Deriving the Exact Form of the Rotation Metric (12L–R) 4.5.7 Trying to Find Infinitesimal Unimodular Transformations Corresponding to Uniform Acceleration (12R, 41L–R) 4.5.8 Geodesic Motion along a Surface (41R) 4.6 Emergence of the Entwurf Strategy (13L–R) 4.6.1 Bracketing the Generalization to Non-linear Transformations: Provisional Restriction to Linear Unimodular Transformations (13L) 4.6.2 Trying to Find Correction Terms to the Core Operator to Guarantee Compatibility of the Field Equations With Energy-Momentum Conservation (13R) 5. Exploration of the Riemann Tensor (14L–25R, 42L–43L) 5.1 Introduction (14L–25R, 42L–43L) 5.2 General Survey (14L–25R, 42L–43L) 5.3 First Attempts at Constructing Field Equations out of the Riemann Tensor (14L–18R) 5.3.1 Building a Two-Index Object by Contraction: the Ricci Tensor (14L) 5.3.2 Extracting a Two-Index Object from the Curvature Scalar (14R–16R) 5.3.3 Comparing Tiκ and the Ricci Tensor (17L–18R) 5.4 Exploring the Ricci tensor in Harmonic Coordinates (19L–21R) 5.4.1 Extracting Field Equations from the Ricci Tensor Using Harmonic Coordinates (19L) 5.4.2 Discovering a Conflict between the Harmonic Coordinate Restriction, the Weak-Field Equations, and Energy-Momentum Conservation (19R) 5.4.3 Modifying the Weak-field Equations: the Linearized Einstein Tensor (20L, 21L) 5.4.4 Reexamining the Presuppositions Concerning the Static Field (20R, 21R) 5.4.5 Embedding the Stress Tensor for Static Gravitational Fields into the Metric Formalism (21R) 5.4.6 Synopsis of the Problems with the Harmonic Restriction and the Linearized Einstein Tensor (19L–21R) 5.5 Exploring the Ricci Tensor in Unimodular Coordinates (22L–24L, 42L–43L) 5.5.1 Extracting the November Tensor from the Ricci Tensor (22R) 5.5.2 Extracting Field Equations from the November Tensor Using the Hertz Restriction (22L–R) 5.5.3 Non-autonomous Transformations Leaving the Hertz Restriction Invariant (22L) 5.5.4 Extracting Field Equations from the November Tensor Using the ϑ-Restriction (23L–R) 5.5.5 Non-autonomous Transformations Leaving the ϑ-Expression Invariant (23R) 5.5.6 Solving the ϑ-Equation (42L–R) 5.5.7 Reconciling the ϑ-Metric and Rotation (I): Identifying Coriolis and Centrifugal Forces in the Geodesic Equation (42R, 43La) 5.5.8 Reconciling the ϑ-Metric and Rotation (II): Trying to Construct a Contravariant Version of the ϑ-Expression (43La) 5.5.9 Reconciling the ϑ-Metric and Rotation (III): Identifying the Centrifugal Force in the Energy-Momentum Balance (24L) 5.5.10 Relating Attempts (I) and (III) to Reconcile the ϑ-Metric and Rotation: from the Energy-Momentum Balance to the Geodesic Equation (43Lb) 5.6 Transition to the Entwurf Strategy (24R–25R) 5.6.1 Constructing Field Equations from Energy- Momentum Conservation and Checking Them for Rotation (24R) 5.6.2 Trying to Recover the Physically Motivated Field Equations from the November Tensor (25L) 5.6.3 The ϑ^ -Restriction (25R, 23L) 5.6.4 Tinkering with the Field Equations to Make Sure That the Rotation Metric Is a Solution (25R) 5.6.5 Testing the Newly Concocted Field Equations for Compatibility with Energy-Momentum Conservation (25R) 5.7 Conclusion: Cutting the Gordian Knot (19L–25R) 6. Derivation of the Entwurf Equations (26L–R) Introduction The Zurich Notebook provides what appears to be a virtually complete record of Einstein’s search for gravitational field equations in the winter of 1912–1913. He had just started to explore a new theory of gravitation in which the ten components of the metric tensor take over the role of the gravitational potential in Newton’s theory.For discussion of how Einstein arrived at this theory, see “The First Two Acts” (in vol. 1 of this series). In terms of this metaphor, the search for field equations comprises the third act. It starts with the Zurich Notebook and ends with the four communications to Berlin Academy of November 1915 in which Einstein completed general relativity as we know it today with the formulation of the generally-covariant Einstein field equations (Einstein 1915a, b, c, d). The notes documenting Einstein’s search for field equations for this theory take up the better part of the notebook. They start on pp. 39L–41R and continue on pp. 5R–29L and pp. 42L–43L. Our text is a detailed running commentary on these notes.The commentary does not cover pp. 27L–29L, which deal with the behavior of matter in a given metric field. It provides line-by-line reconstructions of all calculations and discusses the purpose behind them.We follow the notation used in the notebook and clearly indicate whenever we use elements of modern notation (such as the Kronecker delta or a more compact notation for derivatives) in our reconstructions of Einstein’s calculations. Einstein’s notation should be easy to follow for readers familiar with the basics of the standard modern notation of general relativity. The one exception is Einstein’s idiosyncratic convention before (Einstein 1914b) of writing all indices downstairs and distinguishing between covariant and contravariant components (e.g., the components gμν and gμν of the metric) by using Latin letters for one ( gμν ) and Greek letters for the other ( μν ). In the notebook, to make matters worse, Einstein sometimes used Latin for contravariant and Greek for covariant components. The contravariant stress-energy tensor, e.g., is denoted by Tμν in the notebook while its covariant counterpart is denoted by Θμν (see, e.g., p. 5R and equation ()), which is just the reverse of the notation used in (Einstein and Grossmann 1913) and subsequent publications. The summation convention (repeated indices are summed over) was only introduced in (Einstein 1916a, 788). In the notebook, however, Einstein occasionally omitted summation signs, thus implicitly using it. When providing intermediate steps for calculations in the notebook we shall frequently and tacitly use the summation convention. The commentary is self-contained and can in principle be read independently of the notebook. It is designed, however, to be used in conjunction with the facsimile and the transcription of the notebook presented in vol. 1 of this series. The reader seeking guidance in reading a particular passage of the notebook can go directly to the section of the commentary in which that passage is analyzed. To help the reader find the relevant section, both the table of contents and the running heads of the text match (sub) (sub-)sections of the commentary to pages of the notebook. When looking up the annotation for a specific passage of the notebook, the reader is advised to consult the introduction to the (sub-)(sub-)section of the commentary dealing with that passage first. After covering the earliest research on gravity documented by the notebook (pp. 39L–41R) in sec.  and the derivation of the law for the energy-momentum balance between matter and gravitational field (p. 5R) in sec. , the commentary continues with its two longest sections, sec.  covering pp. 6L–13R and 41L–R and sec.  covering pp. 14L–25R and 42L–43L. These two sections have extensive introductions, providing an overview of the material cross-referenced both with the page numbers of the notebook and the numbers of the (sub-)subsections of the commentary where the material is covered in detail. Sec.  covers a stage in Einstein’s research during which, through trial and error, he found a body of results, strategies, and techniques that he drew on for the more systematic search for field equations during the next stage. Sec.  follows the fate of a series of candidate field equations as Einstein checked them against a list of criteria they would have to satisfy. All these candidates are extracted from the Riemann tensor, which makes its first appearance on p. 14L, the first of the pages covered by sec. . In the course of his investigations, Einstein already came across some of the field equations published in November 1915. One of the central tasks of the commentary is to analyze why these candidate field equations were rejected three years earlier. In the notebook Einstein eventually gave up trying to extract field equations from the Riemann tensor. Drawing on results and techniques found during the earlier stage instead, he developed a way of generating field equations guaranteed to meet what he deemed to be the most important of the requirements to be satisfied by such equations. In this way he found the field equations of severely limited covariance published in the spring of 1913 in a paper co-authored with Marcel Grossmann (Einstein and Grossmann 1913). The theory and the field equations presented in this paper are known, after the title of the paper, as the “Entwurf” (“outline”) theory and the “Entwurf” (field) equations. The derivation of the Entwurf field equations on pp. 26L–R of the notebook is covered in sec. 6, the short section that concludes the commentary. To get an overview of the contents of the gravitational part of the Zurich Notebook, the reader is advised to read the balance of this section as well as secs.  and –, the introductions to the two main sections of the commentary. The Four Heuristic Requirements For our analysis of the notebook it is important to distinguish two strategies employed by Einstein in his search for suitable gravitational field equations and four requirements serving as guideposts and touchstones in this search. In sec.  we introduce the two strategies; in this section we introduce the four heuristic requirements, which, for ease of reference, we have labeled relativity, equivalence, correspondence, and conservation.For further discussion, see “Pathways out of Classical Physics …” (in vol. 1 of this series) and (Renn and Sauer 1999). The Relativity Principle Throughout the period in which Einstein formulated general relativity as we know it today (i.e., the years 1912–1915), he was under the impression that the principle of relativity for uniform motion of special relativity can be generalized to arbitrary motion by extending the manifest Lorentz invariance of special relativity in the formulation of Minkowski (1908), Sommerfeld (1910a, b), Laue (1911b) and others to general covariance.For an analysis of Einstein’s conflation of the status of Lorentz invariance in special relativity and the status of general covariance in general relativity, see (Norton 1992, 1999). Only in the fall of 1916 did Einstein come to realize that general covariance does not automatically lead to relativity of arbitrary motion.This became clear to Einstein in discussions with Willem de Sitter. For further discussion, see the headnote, “The Einstein-De Sitter-Weyl-Klein Debate,” in (CPAE 8, 351–357). Kretschmann (1917) was the first to give a precise formulation of the difference between general covariance and general relativity.For discussion of Kretschmann’s paper and Einstein’s brief response to it in (Einstein 1918), see, e.g., (Norton 1993, sec. 5). Given his conflation of the two at the time, Einstein tried to implement the relativity principle in the notebook by constructing field equations of the broadest possible covariance. He was thus drawn to generally-covariant objects such as the Beltrami invariants (p. 6L) and the Riemann tensor (p. 14L). He found these in the mathematical literature with the help of Marcel Grossmann. Einstein would sacrifice some covariance to meet the other requirements the field equations had to satisfy. Contrary to what we know today, he assumed that both energy-momentum conservation and the recovery of Newtonian theory for weak static fields put constraints on the class of admissible coordinate transformations. Initially, his hope was that this class would still include transformations to frames of reference in arbitrary states of motion. In the course of the research documented in the notebook it became clear to him that this is not the case. By the end of the notebook (pp. 26–R), Einstein had settled for invariance under general linear transformations, hoping to extend the principle of relativity to non-uniform motion in a different way, suggested by the equivalence principle. The Equivalence Principle Einstein’s fundamental insight in developing general relativity was that there is an intimate connection between acceleration and gravity. As Einstein put it in 1918, the two are “of the exact same nature” (“wesensgleich,” Einstein 1918, 176). They are two sides of the same coin, which explains Galileo’s principle that all bodies fall alike in a given gravitational field, or, in Newtonian terms, that inertial mass is equal to gravitational mass. Einstein first explored the connection between acceleration and gravity in (Einstein 1907) and started calling it the equivalence principle in (Einstein 1912a). Einstein wanted to use this principle to extend the relativity of uniform motion to non-uniform motion and develop a new theory of gravity at the same time.For discussion of Einstein’s equivalence principle, see (Norton 1985). The general-relativity principle that Einstein formulated on the basis of the equivalence principle is of a somewhat peculiar nature. It is best illustrated with a couple of paradigmatic examples. Two observers in the vicinity of some massive body, one in free fall, one resisting the pull of gravity, can both claim to be at rest as long as they agree to disagree about whether or not there is a gravitational field present in their region of spacetime. The observer in free fall can legitimately claim that there is no gravitational field, that he is at rest, and that the other observer is accelerating upward. The observer resisting the pull of gravity is equally justified in claiming that there is a gravitational field, that she is at rest, and that the other observer is accelerating downward. They can make similar claims when they are both in some flat region of spacetime, one hovering freely, the other firing up the engines of her spacecraft. The first observer can claim that there is no gravitational field, that he is at rest, and that the other observer is accelerating upwards. The second observer can claim that a gravitational field came into existence the moment she turned on her engines, that she is at rest in this field, and that the other observer is accelerating downward in it. In neither of these two cases—close to some massive body or in some flat region of spacetime—are the situations of these two observers physically equivalent to one another. The equivalence, in fact, is between the observer in free fall in the first case and the one hovering in outer space in the second and between the observer resisting the pull of gravity in the first case and the one in the accelerating rocket in the second. In both cases it is not the motion of the two observers with respect to one another that is relative—in the sense of being determined only with reference to the observer making the call—but the presence or absence of a gravitational field.For further discussion, see (Janssen 2005, 63–66). Einstein implemented the equivalence principle by letting the metric field gμν represent both the gravitational field and the inertial structure of spacetime. For the equivalence principle to go through it is crucial that in all cases such as the ones considered above the metric field is a solution of the same field equations in the coordinate systems of both observers. This is automatically true if the field equations are generally covariant. But general covariance, while sufficient, is not necessary to meet this requirement. Whenever general covariance proved unattainable in the notebook (and similarly during the subsequent reign of the Entwurf theory with its field equations of severely restricted covariance), Einstein tried to meet the requirement in a different way, involving what he first called “non-autonomous” (“unselbständige”) transformationsEinstein to H. A. Lorentz, 14 August 1913 (CPAE 5, Doc. 467). The relevant passage is quoted in footnote . and later “justified” (“berechtigte”) transformations between “adapted” (“angepaßte”) coordinates.(Einstein and Grossmann 1914, 221; Einstein 1914b, 1070)In the case of ordinary, autonomous, transformations, the new coordinates are simply functions of the old ones. If the field equations are invariant under some autonomous transformation, any solution in the old coordinates is guaranteed to turn into a solution in the new ones under that transformation. In the case of non-autonomous transformations, the new coordinates are functions of the old coordinates and the metric field (expressed in terms of the old coordinates).For more detailed discussion of non-autonomous transformations, see the introduction to sec. . If the field equations are invariant under a non-autonomous transformation for some metric field that is a solution of the field equations in the old coordinates, only that particular solution is guaranteed to turn into a solution in the new coordinates. This suffices for the implementation of the equivalence principle. Given the difficulty of finding field equations invariant under a broad enough class of autonomous transformations, it need not surprise us that non-autonomous transformations play a prominent role in the notebook. Einstein was especially interested in non-autonomous transformations to uniformly rotating and uniformly accelerating frames of reference in flat spacetime, in which case the metric in the old coordinates is simply the standard diagonal Minkowski metric. The Correspondence Principle An obvious constraint on Einstein’s new gravitational theory was that Newton’s theory be recovered under the appropriate circumstances. We call this the correspondence principle. In the case of weak static fields, the 44-component of the metric (where x4 is the time coordinate multiplied by the velocity of light) is proportional to the gravitational potential in Newtonian theory. The correspondence principle thus requires that, in the case of weak static fields, the component of the field equations that determines g44 reduce to the Poisson equation for the Newtonian potential. Einstein expected g44 to be the only variable component of the metric in this case so that the spatial part of the metric would remain flat. This would allow him to connect his new theory both to Newtonian theory and to his own earlier theory for static fields in which the gravitational potential is represented by a variable speed of light (Einstein 1912a, b). It turns out that the correspondence principle does not require the metric for weak static fields to be spatially flat. Other components of the metric besides g44 can be variable without losing compatibility with Newtonian theory. This is because, regardless of the values of the other components, g44 is the only component to enter into the equations of motion for matter moving in the gravitational field in the relevant weak-field slow-motion approximation. Einstein only came to realize this in the course of his calculation of the perihelion motion of Mercury in November 1915 (Einstein 1915c).See (Norton 1984, 146–147), (Earman and Janssen 1993, 144–145), and “Untying the Knot …” sec. 7 (in this volume). Throughout the notebook he assumed that only g44 can be variable for weak static fields. At one point he contemplated relaxing this requirement but quickly convinced himself that this is not allowed.See p. 21R discussed in secs.  and . Despite this additional constraint on the form of the metric for weak static fields, Einstein found various field equations that satisfy the correspondence principle. On the face of it, it looks as if Einstein applied coordinate conditions to various field equations of broad covariance to establish that they reduce to the Poisson equation in the appropriate limit.See in particular pp. 19L and 22R discussed in sec.  and sec. , respectively. When considered in isolation, these two pages strongly suggest that Einstein applied coordinate conditions in the modern sense in the notebook, a conclusion that was indeed drawn in (Norton 1984). On closer examination, it turns out that Einstein actually used what we shall call coordinate restrictions. He took expressions of broad covariance and truncated them by imposing additional conditions on the metric to obtain candidate field equations that reduce to the Poisson equation in the case of weak static fields. Einstein did not see such truncated equations as representing candidate field equations of broader covariance in a limited class of coordinate systems for the purpose of comparing them with Newtonian theory, as we would nowadays, but as candidates for the fundamental field equations of the theory. The status of the conditions on the metric with which Einstein did the truncating is therefore very different from that of modern coordinate conditions. This is why we introduced the special term coordinate restrictions.For a particularly illuminating example of a coordinate restriction, see pp. 23L–R discussed in sec. . Coordinate restrictions played an important role in Einstein’s attempts to find non-autonomous coordinate transformations under which candidate field equations would be invariant. If such candidates had been extracted from expressions of broad covariance with the help of some coordinate restriction, their covariance was determined by the covariance of that coordinate restriction. The expressions involved in coordinate restrictions are much simpler than the field equations themselves and the covariance properties of the former are therefore much more tractable than those of the latter. Many pages of the notebook are thus given over to the investigation of the covariance properties of various coordinate restrictions.See, e.g., pp. 10L–11L (covered in secs. –), p. 22L (covered in ), and pp. 23R, 42L–R (covered in secs. –). Einstein routinely checked whether the coordinate restrictions he imposed allow non-autonomous transformations to uniformly rotating and uniformly accelerating frames of reference in Minkowski spacetime. The correspondence principle led Einstein to expect the left-hand side of the field equations to have the form of a sum of what we shall call a core operator, a term with second-order derivatives of the metric that for weak static fields reduces to the Laplacian acting on the metric, and various correction terms quadratic in first-order derivatives of the metric that vanish in a weak-field approximation. He either extracted equations of this form from equations of broad covariance with the help of coordinate restrictions or he determined what terms had to be added to the core operator on the basis of considerations of energy-momentum conservation. The Conservation Principle Einstein’s 1912 theory for static gravitational fields had taught him to pay close attention to what we shall call the conservation principle, the compatibility between the field equations and energy-momentum conservation. The field equations proposed in (Einstein 1912a) had turned out to be incompatible with energy-momentum conservation and Einstein had been forced to add a term to them in (Einstein 1912b).See footnote 119 for more details. The extra term, it turned out, gave the energy density of the gravitational field and entered the field equations on the same footing as the energy density of the field’s material sources. This, as Einstein realized, had to be the case given the equivalence of energy and inertial mass expressed in Emc2 , the equality of inertial and gravitational mass asserted by the equivalence principle, and the equality of active and passive gravitational mass. The field equations of the new metric theory would thus have to satisfy a similar requirement. Einstein accordingly sought to interpret the correction terms to the core operator mentioned above as the energy-momentum density of the gravitational field, occurring on a par with the stress-energy tensor of matter.Following the terminology in (Einstein and Grossmann 1913, e.g., p. 11 and p. 16), we shall use the term stress-energy tensor’ throughout the commentary rather than the more common ‘energy-momentum tensor’ or the more cumbersome ‘stress-energy-momentum (SEM) tensor’. In the notebook Einstein referred both to the “tensor of momentum and energy” (“Tensor der Bewegungsgröße u. Energie,” p. 5R) and to the “stress-energy tensor” [Sp[annungs]-Energie-Tensor] p. 20R). That the stress-energy tensor of matter should replace the mass of Newtonian theory as the source of the gravitational field Einstein had learned from the development of special-relativistic mechanics.In his 1912 manuscript on special relativity, Einstein called the extension of the central role of the stress-energy tensor in electrodynamics to all of physics “the most important recent advance in relativity theory” (“den wichtigsten neueren Fortschritt der Relativitätstheorie.” CPAE 4, Doc. 1, p. [63]). He gave credit to Minkowski, Abraham, Planck, and Laue for this development. The symmetry of the stress-energy tensor in its two indices encodes such physical knowledge as the inertia of energy and the conservation of angular momentum (Laue 1911a). This means that the differential operator acting on the metric set equal to the stress-energy tensor in the field equations must have that same symmetry. This requirement is indeed imposed, though not emphasized, in the notebook (cf. footnotes 219 and 220). The early years of special relativity had brought a transition from Galilean-invariant particle mechanics based on Newton’s second law to Lorentz-invariant continuum mechanics based on energy-momentum conservation, expressed by the vanishing of the four-divergence of the total stress-energy tensor of closed systems, ∂νTtotμν0 ( ∂ν shorthand for ∂∂xν ). The conservation principle requires that the gravitational field equations be compatible with this law in a weak-field approximation in which the energy-momentum of the gravitational field itself can be neglected. The weak-field field equations have the form □gμνκTμν ( □ is the d’Alembertian, ημν∂μ∂ν , with ημνdiag1,1,1,1 the standard diagonal Minkowski metric). Einstein imposed the coordinate restriction ∂νgμν0 in which case these field equations imply energy-momentum conservation in the weak-field approximation.See p. 19R discussed in sec. . This coordinate restriction, which Einstein used both to satisfy the conservation principle and to satisfy the correspondence principle, occurs so frequently in the notebook that we have given it a special name. We call it the Hertz restriction.Named after Paul Hertz, the recipient of an important letter in which Einstein discussed this restriction (Einstein to Paul Hertz, 22 August 1915 [CPAE 8, Doc. 111]). When the energy-momentum coming from the gravitational field cannot be neglected, the situation gets more complicated. On p. 5R Einstein derived an equation giving the exact energy-momentum balance between matter and gravitational field. In modern terms, this equation states that the covariant divergence of the stress-energy tensor of matter vanishes. It is the sum of two terms, the ordinary divergence of the stress-energy tensor of matter and an expression, once again containing this tensor, that can be interpreted as the gravitational force. The field equations set some second-order differential operator acting on the metric equal to the stress-energy tensor of matter. One of the most important tests to which Einstein submitted candidate field equations was to use them to eliminate the stress-energy tensor from the expression for the gravitational force in the energy-momentum balance and check whether the resulting expression can be written as the divergence of an expression that could be interpreted as the energy-momentum density of the gravitational field itself.See, e.g., p. 19R. Given the form of the stress-energy tensor of the electromagnetic field and of the stress tensor of the gravitational field in his static theory, Einstein expected and required this quantity to be quadratic in first-order derivatives of the metric. That way gravitational energy-momentum could indeed be neglected in the weak-field approximation. With the help of this quantity, Einstein could now rewrite the energy-momentum balance between matter and gravitational field as an ordinary divergence of the total energy-momentum density—of matter and of the gravitational field. The balance equation thus turns into a genuine conservation law. That a covariant divergence can be rewritten as an ordinary divergence in this manner immediately makes it clear that gravitational energy-momentum density cannot be represented by a generally-covariant tensor. It is what we now call a pseudo-tensor. Throughout the notebook, however, Einstein tacitly assumed that its transformation properties are the same as those of any other stress-energy tensor. He only recognized in early 1914 that this is not true.See (Einstein and Grossmann 1914). For discussion, see (Norton 1984, sec. 5), “Pathways out of Classical Physics …” (in vol. 1 of this series), and “What Did Einstein Know …” sec. 2 (in this volume). Einstein eventually turned this test of the conservation principle into a powerful method for generating field equations. It was this method that gave him the Entwurf field equations.See p. 13R, p. 24R, and p. 26L-R, discussed in secs. , , and , respectively. In summary, the conservation principle resulted in (at least) four related but distinct requirements that candidate field equations have to satisfy. First, the energy-momentum density of the gravitational field has to enter into the field equations in the same way as the energy-momentum density of matter. Secondly, the field equations should guarantee that the four-divergence of the stress-energy tensor of matter vanishes in the weak-field approximation. Thirdly, the field equations should allow the gravitational force density in the energy-momentum balance between matter and gravitational field to be written as the divergence of some gravitational stress-energy pseudo-tensor. Finally, this pseudo-tensor should be an expression quadratic in first-order derivatives of the metric. Given this rich harvest of requirements, the conservation principle was probably the most fruitful of the four heuristic principles that guided Einstein in his search for suitable field equations. The Two Strategies Einstein attacked the problem of finding suitable field equations for the metric field from two directions, clearing the hurdles he had himself erected with his four heuristic requirements— relativity, equivalence, correspondence, and conservation—in a different order. In what we call the ‘mathematical strategy,’ Einstein tackled relativity and equivalence first and then moved on to correspondence and conservation. In what we call the ‘physical strategy’ it is just the other way around. There he started with correspondence and conservation and then turned to relativity and equivalence.For more detailed discussion of these two strategies, see (Renn and Sauer 1999) and “Pathways out of Classical Physics …” (in vol. 1 of this series). The identification of these two complementary strategies not only turned out to be key to our reconstruction of many of Einstein’s arguments and calculations in the notebook, it also greatly enhanced our understanding of his work on general relativity during the subsequent period of 1913–1915.See “Untying the Knot …” (in this volume). In this section we briefly characterize these two strategies. The mathematical strategy was to use one of the generally-covariant quantities that can be found in the mathematical literature, such as the Beltrami invariants or the Riemann tensor, to construct a second-order differential operator acting on the metric (or its determinant) that is then set equal to the stress-energy tensor of matter (or its trace). If this can be done without compromising the general covariance of the initial quantity too much, such field equations will automatically meet the relativity and equivalence requirements. The problem that Einstein ran into was that the correspondence and conservation requirements, if they could be met at all, called for severe coordinate restrictions. Still, as explained in sec. , knowing that candidate field equations can be extracted from equations of broad covariance with the help of a coordinate restriction makes their covariance properties much more tractable. Their covariance is fully determined by the covariance of the coordinate restriction. Unfortunately, Einstein found again and again that the coordinate restrictions he needed to satisfy the correspondence and conservation requirements ruled out the kind of transformations to accelerating frames of reference needed to meet the relativity and equivalence requirements. The physical strategy was to model the field equations for the metric field on the Poisson equation of Newtonian gravitational theory and Maxwell’s equations for the electromagnetic field.For detailed analysis of Einstein’s use of this analogy, see “Pathways out of Classical Physics …” (in vol. 1 of this series). As explained in secs.  and , the correspondence and conservation requirements suggest that the field equations have a core operator, which for weak static fields reduces to the Laplacian acting on the metric, and a term representing gravitational energy-momentum density on the left-hand side and the stress-energy tensor of matter on the right-hand side. The conservation principle can be used to determine the exact form of the gravitational stress-energy pseudo-tensor. The physical strategy thus amounts to constructing candidate field equations guaranteed to meet the correspondence and conservation requirements. The problem is that their construction sheds little light on their covariance properties. Only their covariance under general linear transformations is assured. It thus remains completely unclear whether they satisfy the relativity and equivalence requirements. The best Einstein could do on this score was to check whether they allowed the Minkowski metric in various accelerated frames of reference so that they would at least be invariant under some non-autonomous non-linear transformations corresponding to acceleration. In the first half of the notes (pp. 39L–41R, 5R–13R), we see Einstein vacillate between the mathematical and the physical strategy. On p. 6L, for instance, the Beltrami invariants are introduced and used as input for the mathematical strategy. Two pages later, Einstein switched to the core operator and the physical strategy. On the following pages Einstein combined his two strategies trying in vain to connect field equations based on the core operator to the Beltrami invariants in an attempt to clarify their transformation properties. On p. 13R he made further progress along the lines of the physical strategy by introducing considerations of energy-momentum conservation. Then, on p. 14L, the Riemann tensor makes its first appearance in the notebook, and Einstein abruptly switched from the physical to the mathematical strategy. What follows is a concerted effort, taking up most of the second half of the notes (pp. 14L–23R, 42L–43L), to extract field equations from the Riemann tensor. Eventually (p. 24R–26R), Einstein went back to the physical strategy, continuing the line of reasoning begun on p. 13R. He briefly combined the two strategies again, trying to find (on p. 25L) a coordinate restriction with which to extract field equations found along the lines of the physical strategy (on p. 24R) from the Riemann tensor. He then decided to go exclusively with the physical strategy, which led him to the Entwurf field equations on pp. 26L–R. First Exploration of a Metric Theory of Gravitation (39L–41L) Introduction Einstein’s attempt to construct a new dynamical theory of gravitation starts from three basic elements: (1) the representation of the gravitational field by the ten components of the metric tensor; (2) the four-dimensional spacetime formalism of special relativity; and (3) his scalar theory of the static gravitational field. For Einstein, all three components were recent additions to his stock of knowledge. In Prague, in the spring of 1912, he brought his attempts to formulate a theory of gravitation for the special case of a static field to a satisfactory conclusion. Only after returning from Prague to Zurich in the summer of 1912 did he recognize the relevance of Gauss’ theory of surfaces to the gravitation problem. Gauss’ theory represents the metrical geometry of surfaces of variable curvature by a line element, the square root of a quadratic differential form invariant under arbitrary coordinate transformations. Einstein had become familiar with the four-dimensional formalism for special relativity developed by Minkowski, Sommerfeld, and Laue, and he realized that a four-dimensional extension of Gauss’ theory could provide a mathematical framework suitable for a new, dynamical theory of gravitation. The three components, out of which he hoped to build the new theory, each posed distinct but interrelated problems. Gauss’ theory for two-dimensional surfaces had to be extended to a four-dimensional space with indefinite signature. The flat Minkowski spacetime formalism had to be extended to a vector and tensor analysis valid for arbitrary coordinate systems in a non-flat spacetime. Einstein’s static gravitational theory was formulated in terms of a single (three)-scalar gravitational potential. The single partial differential equation governing this potential had to be generalized to a system of partial differential equations for the ten-component tensorial gravitational potential. On pp. 39L–41L of the notebook, Einstein explored a few simple ways of combining these three components to find the gravitational field equations for a static field in special coordinates. No clear candidates emerged from Einstein’s first foray into the problem. This may have signaled to him that a higher level of mathematical sophistication was called for. Even at the elementary level of these early calculations, however, one can see an alternation between physically and mathematically motivated approaches foreshadowing the two basic strategies that we distinguished in sec. . The Three Building-Blocks: Gauss, Minkowski, Einstein (39L) This page includes three groups of formulas, separated from each other by two horizontal lines. Each of these groups can be associated with one of the elements mentioned above. Einstein started by writing down the square of the four-dimensional line element ds2ΣGλμdxλdxμ . His use of a capital letter “ G ” indicates that this is the earliest occurrence of the metric tensor in the notebook. After p. 40R, Einstein switched to the common lower-case g . He then derived the transformational behavior of the metric tensor under four-dimensional coordinate transformations from the condition that the line element be invariant under such transformations. The transformations between the unprimed and primed coordinates are expressed by a matrix of coefficients and its inverseAs is indicated by a line through the first column, the transformation is given by x′1α11x1+α12x2+α13x3+α14x4 , etc. A similar matrix is found in (Laue 1911b, 57) and in Einstein’s 1912 manuscript on special relativity (CPAE 4, Doc. 1, secs. 15 and 16). From the invariance of the square of the line element under such a transformation, ∑∑Gλμdxλdxμ∑∑G′ρσdx′ρdx′σ ∑ρ∑σ∑η∑ζG′ρσαρηασζdxηdxζ, Einstein read off the transformation laws for the components of the metric Gλμ∑ρ∑σG′ρσαρλασμ , and “analogously” (“analog”) G′λμ∑ρ∑σGρσβρλβσμ . Next to these results, Einstein noted explicitly the equations for the coordinate transformation x′r∑sαrsxs . So apparently Einstein was considering linear transformations at this point.In the transformation matrix at the top of the page he may have had non-linear transformations in mind with non-constant α ’s: In the top row of that matrix Einstein seems to have written dx instead of simply x for the first three entries, and it may have been only later that he restricted himself to linear transformations. For the corresponding first-order partial derivative operators, he wrote ∂∂xs∑rαrs∂∂x′r . In the middle of the page, below the first horizontal line, Einstein considered a “special case for the Gλμ ” (“Spezialfall für die Gλμ ”), namely the case of a coordinate system in which the metric is diagonal G11G12G13G14100001000010000c2 . The spatial metric is flat and expressed in Cartesian coordinates, and the 44-component of the metric is identified with the square of the speed of light.A striking feature of this expression is the signature of the metric, which is in contrast to the appearance of the metric for this special case on pp. 6R and 21R, where Einstein wrote diag1,1,1,c2 . One possible explanation for this is that Einstein implicitly used an imaginary time coordinate at this point, which he had explicitly introduced on p. 32R in the context of his treatment of electrodynamics in moving media. If c2 is a constant, this metric represents the Minkowski spacetime of special relativity in quasi-Cartesian coordinates. If c2 is a function of the spatial coordinates, c2c2x1,x2,x3 , it represents the metrical generalization of Einstein’s static theory. At the bottom of the page, under the second horizontal line, Einstein turned to this theory of the static gravitational field. The gravitational field equation of his static theory,(Einstein 1912b, 456, equation (3b)). cΔc-12grad2ckc2σ , is a partial differential equation for the velocity of light, which serves as the single gravitational potential of this theory. In a four-dimensional metric theory, the field equations will be partial differential equations for all ten components of the metric tensor, resulting from the action of some differential operator acting on it. Einstein assumed that, in the special case of a static field and with an appropriate choice of coordinates, the metric tensor would reduce to the form given in equation () with ccx1,x2,x3 .Einstein later made this expectation explicit (Einstein and Grossmann 1913, Sec. 2). In this case, the gravitational field equations would be expected to reduce to equation (). The problem was to reverse this reduction and find the differential operator that enters into field equations for the metric tensor. The first step in his attempt to find this operator was to rewrite the left-hand side of equation () in terms of the components of the metric tensor in equation (), i.e., to rewrite the equation in terms of c2 instead of c . Einstein began with the left-hand side of equation () cΔc-12grad2c . He then wrote down the first- and second-order partial derivatives of c2 with respect to x c2 2c∂c∂x 2∂c∂x2+2c∂2c∂x2 . Since the y - and z -derivatives behave similarly, Einstein could now read off the Laplacian of c2 , Δc22grad2c+2cΔc , as well as the gradient of c2 , gradc22cgradc , in terms of c and its derivatives. At this point, Einstein probably realized that various factors of 2 on the right-hand sides of equations ()–() cancel if one looks at c22 instead of c2 . In any case, he defined a function c22 such that Δgrad2c+cΔc , and gradcgradc . This equation allowed Einstein to express grad2c in terms of : grad22grad2c . Using equations () and (), he finally wrote the expression () in terms of : Δ-34grad2 . Einstein presumably expected that this expression could be recovered from the tensorial field equations. He may have hoped that it would emerge as the 44-component of the left-hand side of these field equation, but he could, of course, not be certain that tensorial equations would yield the familiar static theory in this way.Alternatives are conceivable but less likely. For example, expression () might be the trace of the left-hand side of the unknown gravitational field equations. Finding a Metric Formulation of the Static Field Equation (39R) At the top of p. 39R, Einstein wrote down the vacuum field equations of his 1912 static theory in terms of the variable introduced on the facing page: Δ-34grad20 . Next to this equation, he started to write the word “Umfo[rmen]” (to re-express), then deleted it in favour of “Transformieren” (to transform). Presumably, the point of such transformations was to generalize this field equation for one component of the static field in a special coordinate system to field equations for all components of the field—static or non-static—in more general coordinates. However, instead of dealing with the field equations, Einstein turned to a simpler but related problem. In the Cartesian coordinates used so far, the static character of the spatially flat metric in equation () can be expressed asOriginally, this equation was written as ∂G44∂t0 . ∂G44∂x40 , the vanishing of the time derivative of the 44-component. Einstein now tried to find a covariant formulation of the condition that this metric be static. To this end, he began to transform condition () to primed coordinates. Using equation (), he first expressed the derivative with respect to x4 in terms of primed coordinates: ∂∂x4α14∂∂x′1+α24∂∂x′2+. +. Using equation (), he then did the same for the 44-component of the metric G44∑ρ∑σG′ρσαρ4ασ4 . He now used the inverse transformation for G′ρσ on the right-hand side: G′λμk+∑∑Gρσβρλβσμ=k+G44∑∑β4λβ4μ︸Bλμ . The inclusion of the term k in the first line appears to be an error, perhaps anticipating the need for such a term on the next line, which absorbs all terms in the summation except those containing G44 .In the second line, Einstein presumably forgot to take out the summation signs. If k is a constant, as the notation suggests, then Einstein seems to be considering the special case of linear transformations (in which case only G44 is non-constant). It is evident from equation () that, under arbitrary linear transformations, all components of G′λμ can be non-constant. This expression, however, does not indicate how the G′λμ depend on the primed coordinates, which would require expressing G44 as a function of the primed coordinates. Perhaps looking for such an expression, Einstein began rewriting an expression for G44 but quickly gave up, possibly because such a substitution would just result in the composition of a transformation and its inverse. Finally, he deleted the entire calculation and started afresh in the top right corner of the page. (This part is ruled off by a horizontal and a vertical line.) So Einstein had failed to find a covariant reformulation of the physically-motivated condition () through direct transformation. He now turned his attention to the mathematically more promising relation Div Γ0 , which reduces to equation () in the case of the spatially flat static metric (). That Einstein did indeed find this equation promising is indicated by the word “probable” (“wahrscheinlich”) that he wrote above it. The differential operator “Div” is analogous to the quasi-Cartesian coordinate divergence of a tensor, an operation well known in Minkowski’s four-dimensional spacetime formalism.The notation “Div” for a four-dimensional generalization of the ordinary three-dimensional divergence was introduced by (Sommerfeld 1910b, 650). It is also used in (Laue 1911b, 70). The four-divergence is implicitly used in the derivation of Maxwell’s equations on p. 33L. Note that the ordinary divergence is written with a lower case “d” on p. 5R. Einstein was familiar with the use of this operation in four-dimensional electrodynamics. The symbol “ Γ ” denotes the metric tensor (“Tensor der G ”). Einstein now asked “Is this invariant?” (“Ist dies invariant?”) and wrote out equation () explicitly: ∑μ∂∂xμGλμ0,λ1234 . Using the transformation equations () and (), he transformed this equation to primed coordinates: ∑τατμ∂∑ρ∑σG′ρσαρλασμ∂x′τ0 . Equation () shows that the condition does indeed transform as a vector (i.e., is “invariant”) under linear transformations (i.e., for constant α ’s). This concludes the investigation of the condition that the metric be static. At this point, Einstein presumably returned to the field equations. This is suggested by his derivation on the bottom half of p 39R of the transformation equations for second-order derivatives. To simplify matters, he suppressed two spatial coordinates: “Everything only dependent on x1 and x2 (time).” (“Alles nur von x1 und x2 (Zeit) abhängig”). From the actual calculations at the bottom of p. 39R, it is clear that Einstein focused on linear transformations with symmetric transformation matrices (i.e., a12a21 ).In the 1+1 dimensional spacetime considered by Einstein, all Lorentz transformations are boosts. These are all represented by symmetric matrices. Presumably, this is the physical rationale behind Einstein’s focus on such matrices. The end result of his calculation is ∂2∂x12α112∂2∂x′12+2α11α21∂2∂x′1∂x′2+α222∂2∂x′22 (the last term on the right does not appear in the notebook). This equation shows that, under (symmetric) linear transformations, the set of second-order partial derivatives of a scalar function transforms like the components of the metric tensor, an insight that Einstein put to good use on the next page. Searching for a Generalization of the Laplacian in the Metric Formalism (40L–R) At the bottom of p. 39R, Einstein had found that second-order partial derivatives of a scalar function have the same transformation behavior under linear transformations as the metric tensor. This insight probably prompted the calculations on pp. 40L–R, which are an attempt to find candidates for the differential operator acting on the metric tensor in the field equations on the basis of expressions involving well-known differential operators acting on a scalar field. Given some striking similarities between expressions found on p. 40L and a list of differential invariants in (Wright 1908), Einstein (or Grossmann) probably consulted this book while working on the calculations on this page.Einstein was familiar with this book, as can be inferred from Einstein to Felix Klein, 21 April 1917: “Grossmann (I believe) had the little book by Weight [sic] when we were working together on relativity four years ago” (“Das Büchlei[n] von Weight [sic] hatte Grossmann (glaube ich), als wir vor 4 Jahren zusammen über Relativität arbeiteten.” CPAE 8, Doc. 328). For historical discussion of (Wright 1908), see (Reich 1994, 105–107). Since the field equations have to be of second order and the components of the metric tensor are equivalent—as far as their transformation properties are concernedThe equivalence can only refer to the behavior under coordinate transformations since a direct identification of the metric tensor with the second partial derivatives of a scalar function would result in severe, unacceptable restrictions on the metric.—to second-order derivatives of a scalar, Einstein had to consider expressions containing fourth-order derivatives of the scalar function. In the course of his calculations, he further imposed the condition that all four coordinates enter on the same footing into the differential operator acting on the scalar function. The attempt was abandoned after the first few lines on p. 40R. After a few false starts, Einstein listed a number of expressions with the Laplacian and the gradient operator acting on a scalar function ϕ . First, he wrote down three expressions with “second-order” (“2. Ordnung”) derivatives: Δϕ, ϕΔϕ, grad2ϕ . Then he wrote down a number of expressions with “fourth-order” (“4. Ordnung”) derivatives ΔΔϕ, ΔϕΔϕ, Δgrad2ϕ, ϕΔΔϕ, grad2Δϕ . The list shows some resemblance to a list on pp. 56–57 of (Wright 1908).In Wright’s book the list is formed from combinations of the two Beltrami invariants (Wright 1908, 56–57). At this point in the notebook, the operator Δ does not seem to be related to the Beltrami invariants. Einstein nevertheless may have hoped that Wright’s book would give him some guidance in finding a differential invariant that he could use to construct the left-hand side of the gravitational field equations. The Beltrami invariants explicitly appear on later pages of the notebook (for the first time on p. 7L; for discussion see sec. ). On the next line, the first of these expressions, ΔΔϕ , is expanded, with the help of the definition of the Laplacian in Cartesian coordinates in two dimensions, Δ∂2∂x2+∂2∂y2 . He then expanded the second term of the list, ΔϕΔϕ , asThere is a factor of 2 missing in the third term. Δϕ⋅Δϕ¯+ϕΔΔϕ+∂ϕ∂x∂ϕ∂x+.+.︸grad ϕgradΔϕ . The terms ΔΔϕ and ΔϕΔϕ only contain second-order derivatives of the scalar function ϕ and can therefore be translated into expressions with (differential operators acting on) components of the metric. The underlining of these two terms in the notebook thus confirms our interpretation of the rationale for these calculations. Einstein then wrote “the first two steps” (“Die ersten 2 Schritte”), drew a horizontal line, wrote “2 dimensions” (“2 Dimensionen”), and then crossed out the calculations he had made so far and tried again. Einstein’s next comment makes his strategy explicit: “system of the G equivalent to system ∂2ϕ∂xμ∂xν ” (“System der G äquivalent dem System ∂2ϕ∂xμ∂xν ”). In other words, he now spelled out the equivalence of second-order derivatives of a scalar function and components of the metric tensor as far as their behavior under linear coordinate transformations is concerned. Einstein’s next step was to impose an additional constraint on candidate field equations constructed from the kind of expressions he had been examining above: “The equation should be such that in every term one has the same number of differentiations with respect to every x ” (“Gleichung soll so sein, dass in jedem Glied nach allen x gleich oft diff[erenziert] wird.”). Einstein now had three constraints that any acceptable expression had to satisfy. First, for every factor ϕ there should be two partial-derivative operators ∂∂xμ to enable the translation to components of the metric tensor. Second, there should be two additional partial-derivative operators to yield second-order field equations. Third, for each of the four coordinates the expression should contain an equal number of operators ∂∂xμ . He first considered an expression linear in ϕ : ∂8ϕ∂x12∂x22∂x32∂x42 . Einstein rejected this possibility, writing: “linear impossible, of 8th order in ϕ ” (“Linear unmöglich von 8. Ordnung in ϕ ”). The problem with this expression is that it leads to sixth-order field equations. For instance, using the derivatives with respect to x4 for the translation to components of the metric tensor, one arrives at the expression ∂6g44∂x12∂x22∂x32 . The next possibility he considered was an expression quadratic in ϕ : ∂2ϕ∂x12∂6ϕ∂x22∂x32∂x42 . This term, as Einstein noted, would correspond to an expression with fourth-order derivatives of the metric: “will necessarily be of fourth order” (“wird notwendig 4. Ordnung”). To obtain an expression satisfying all three constraints, one needs an expression cubic in ϕ . Einstein thus wrote: “third degree in ϕ will be of second order, as it has to be” (“dritten Grades in ϕ wird 2. Ordnung, wie es sein muss”) and then wrote down an example of such an expression: ∂2ϕ∂x12∂2ϕ∂x22∂4ϕ∂x32∂x42 . The equivalence under linear coordinate transformations of the metric tensor and the partial derivatives of a scalar function is further explored at the top of p. 40R. There Einstein tried a different way of achieving symmetry between the four spacetime coordinates. He changed what we identified above as the third constraint. Rather than looking at the expression corresponding to the field equations, he now exclusively concentrated on the second-order derivatives of ϕ representing components of the metric in this context. Taking as his starting point the product of two such second-order derivatives (corresponding to g12g34 Here, on top of p. 40R Einstein used the lower case g for the first time, a practice he continued throughout the rest of the notebook.), he ensured that all four coordinates occur on equal footing by adding terms obtained through permutation of the indices. Einstein initially wrote down the cyclic permutations g23g41 and g34g12 . However, g34g12g12g34 , so he crossed out this first attempt and tried again. He now wrote down the terms g13g24 and g14g23 , the only non-redundant terms given the symmetries gikgki and gmngikgikgmn . He then translated these three terms into partial differential operators acting on ϕ : ∂2ϕ∂x1∂x2∂2ϕ∂x3∂x4+∂2ϕ∂x1∂x3∂2ϕ∂x2∂x4+∂2ϕ∂x1∂x4∂2ϕ∂x2∂x3 . This expression is fully symmetric under permutation of the indices. By having a second-order derivative operator symmetric in all four coordinates (such as the d’Alembertian) acting on this expression, one can now construct field equations that meet all three constraints, the two original ones and the modified version of the third one. However, Einstein did not pursue this line of inquiry any further. Transforming the Ellipsoid Equation as a Model for Transforming the Line Element (40R–41L) Einstein’s first exploration of a metric theory of gravitation ends at the horizontal line drawn on p. 40R with the calculations discussed above. The remaining pages of this part, pp. 40R–43L, do not seem to be a direct continuation of these investigations. On the bottom half of p. 40R and the top half of p. 41L, Einstein considered transformations of equations describing three-dimensional ellipsoids. Only these calculations, which bear on Einstein’s exploration of the transformation properties of the metric tensor, will be discussed here. On the bottom half of p. 41L and the top half of p. 41R, Einstein examined some properties of infinitesimal unimodular transformations. These calculations will be discussed in sec.  along with very similar calculations at the bottom of p. 12R. The bottom half of p. 41R, dealing with constrained motion along a two-dimensional surface, will be discussed in the sec. . Finally, most, if not all, of the material on pp. 42L–43L is related to Einstein’s considerations on pp. 23L–R and will be discussed in secs. –. On the bottom half of p. 40R, Einstein transformed the equation for a three-dimensional ellipsoid to its principal-axis form. On the top half of p. 41L, he then tried to determine the class of linear transformations that would leave this description of the ellipsoid invariant. The calculation on p. 40R is analogous to finding coordinate transformations that take the line element in arbitrary coordinates to its standard Minkowski form. This geometrical analogy can also be found in (Einstein and Grossmann 1913, sec. 3): “the real cone ds20 appears brought to its principal axes” (“der reelle Kegel ds20 erscheint auf seine Hauptachsen bezogen”). The calculation on p. 41L is analogous to finding the class of linear transformations leaving the Minkowski line element in its standard diagonal form invariant.These calculations may have been motivated by the following remark in (Wright 1908, 18): “The problem of the equivalence of two quadratic differential forms is reduced to that of the equivalence of two sets of algebraic forms, where one set is obtained from the other by a linear transformation.” The calculation starts from the defining equation for an ellipsoid in Cartesian coordinates x , y , and z : α11x2+2α12xy+. .. ..α33z21 . For this equation to describe an ellipsoid, the left-hand side must be a positive definite quadratic form. Immediately below this expression, Einstein wrote the equation for the same ellipsoid in rotated primed coordinates such that the coordinate axes are aligned with the principal axes of the ellipsoid, x′2A2+y′2B2+z′2C21 , where A , B , and C are the lengths of the three semi-principal axes of the ellipsoid. Next he wrote down the matrix of coefficients of the orthogonal coordinate transformation corresponding to this rotation:Similar transformation schemes appear on p. 39L and on the following p. 41L. More explicitly, the transformation reads x′α1x+β1y+1z , etc. . Under the three columns of this matrix, Einstein wrote 1A , 1B , and  1C , respectively, indicating that he wanted to rescale the coefficients by these factors. He denoted the rescaled coefficients by u1α1A , v1β1A , etc. The matrix of the rescaled coefficients is written below the transformation matrix, u1u2u3v1v2v3w1w2w3 For the remainder of these calculations, Einstein used the geometrical meaning of these quantities. To understand Einstein’s reasoning, it is helpful to examine the two-dimensional case. Consider the diagram on the left. In the primed coordinate system, the equation for the ellipse is: x′2A2+y′2B21 , which can be rewritten as u1⋅x2+u2⋅x21 . In the primed coordinate system, the vectors x, u1 and u2 have components x′,y′ , 1A,0 and 0,1B , respectively. Einstein wrote down the three-dimensional analogue of equation () in the unprimed coordinates:The third term is only indicated by a dot in the notebook. u1x+v1y+w1z2+u2x+v2y+w2z2+u3x+v3y+w3z21 . Expanding the binomials and comparing the resulting coefficients with those of equation (), one findsThe equations for α33 , α31 , and α12 are indicated by dashes in the notebook, and instead of α23 Einstein erroneously wrote α12 . u12+u22+u32α11v12+v22+v32α22w12+w22+w32α33 v1w1+v2w2+v3w3α23w1u1+w2u2+w3u3α31u1v1+u2v2+u3v3α12. The vectors u1,v1,w1 , u2,v2,w2 , and u3,v3,w3 are in the direction of the semi-axes of the ellipsoid (cf. the figure above). It follows that these three vectors are orthogonal and that their norms are the reciprocal of the lengths of the ellipsoid’s semi-axes. The orthogonality of the three vectors is expressed by: u1u2+v1v2+w1w20,u1u3+v1v3+w1w30,u2u3+v2v3+w2w30. Their norms are given by: u12+v12+w121A2,u22+v22+w221B2,u32+v32+w321C2. Equations () and () are the ones written down at the bottom of p. 40R.The last two lines of equations () and () are only indicated by dashes in the notebook. On the next page, Einstein recapitulated, remarking that “the u,v,w determine orientation and size of the ellipsoid” (“Die u,v,w bestimmen Lage und Grösse des Ellipsoids.”). Equation () for the ellipsoid can be written ∑uix+viy+wiz21 , the left-hand side of which can be interpreted as the sum of the squared scalar products of u1,v1,w1 , u2,v2,w2 , and u3,v3,w3 with the radius vector x,y,z . On p. 40R, Einstein had been concerned with the transformation to principal axes for the ellipsoid. On p. 41L, he investigated “arbitrary linear transformations of x , y , z to x′ , y′ , z′ for an invariant ellipsoid function” (“Beliebige lineare Transformationen der x,y,z in x′,y′,z′ bei invarianter Ellipsoidfunktion.”). In other words, he asked which linear transformation leave the principal-axes form of the equation for the ellipsoid invariant. He thus wrote down the equation for the ellipsoid in two different coordinate systems:In writing this expression, Einstein seems to have added the summation signs as an afterthought: additional terms on the right-hand side were deleted in favor of the summation sign, and the summation index “ i ” in that equation is an “ l ” in the notebook. ∑uix+viy+wiz2∑u′ix′+v′iy′+w′iz′2 . Einstein next wrote down transformation matrices for the linear transformations between the three spatial coordinates that parallel those introduced earlier for the four spacetime coordinates: and:Note that the notation here is the reverse of the notation introduced on p. 39L (cf. equation ()). From equation (), it follows that the transformation matrix for the vectors u1,v1,w1 , u2,v2,w2 , and u3,v3,w3 is the transposed of the matrix in equation (), i.e., in Einstein’s notation: Using these transformation equations, one can write: u′12u1β11+v1β21+w1β312,u′22u2β11+v2β21+w2β312,u′32u3β11+v3β21+w3β312. The sum of these three expressions may be more conveniently expressed as: u′12+u′22+u′32∑i13uiβ11+viβ21+wiβ312 In the notebook this equation is written asIn other words, Einstein neglected to change u1,v1,w1 to ui,vi,wi on the right-hand side. u′12+u′22+u′32∑u1β11+v1β21+w1β312 . Similarly, he found: v′12+v′22+v′32∑u1β12+v1β22+w1β322 . The third equation, for w′12+w′22+w′32 , is indicated by a dashed line. At this point, the calculation breaks off. The remaining pages of the part starting from the back of the notebook contain material that was added later and will be covered later. The calculations on the bottom half of p. 41L and on p. 41R will be discussed in section ; the calculations on p. 42L–43L in section . Energy-Momentum Balance Between Matter and Gravitational Field (5R) We now turn to the notes on gravitation that start from the back of the notebook, beginning with p. 5R. The calculations on these pages are not a direct continuation of those on pp. 39L–41R examined in sec. . A clear indication that p. 5R ff. are later is that the calculations on these pages are much more sophisticated mathematically than those on pp. 39L–41R. On p. 5R, Einstein derived the equation for the energy-momentum balance for pressureless dust in the presence of a gravitational field,Essentially the same calculation can be found on p. 20R. For discussion see sec. . P. 5R is also discussed in (Norton 2000, appendices A through C). an argument that later appeared in (Einstein and Grossmann 1913, secs. 2 and 4). It starts with the derivation of the equations of motion of a point particle in a metric field from an action principle, where the action integral is just the proper length of the particle’s worldline. Expressions for the particle’s momentum and the gravitational force acting on the particle are read off from the resulting Euler-Lagrange equations. Einstein generalized these results to expressions for the momentum density and the force density in the case of pressureless dust, or, as it is described in (Einstein and Grossmann 1913, 9), “continuously distributed incoherent masses” (“kontinuierlich verteilter inkohärenter Massen”). He identified the expression for momentum density as part of the stress-energy tensor for pressureless dust. Inserting this stress-energy tensor and a similar expression for the density of the force acting on the pressureless dust into the equations of motion that he started from, Einstein arrived at a plausible candidate for the law of energy-momentum conservation in the presence of a gravitational field, or, more accurately, an equation for the energy-momentum balance between matter and gravitational field.On p. 43Lb, Einstein performed a calculation that is the “inverse” of that on p. 5R. Rather than deriving the law of energy-momentum balance from the equations of motion of a point particle, he derived the equations of motion from the law of energy-momentum balance. This calculation will be discussed in sec. . What made the candidate all the more promising were its transformation properties. In fact, the equation is generally covariant. As was shown explicitly by Grossmann in his part of (Einstein and Grossmann 1913; part II, sec. 4), it expresses the vanishing of the covariant four-divergence of the stress-energy tensor. On p. 5R, Einstein made a similar claim, namely that the result of the expression found at the bottom of the page is always a vector. He performed a calculation for a special case that at least made this claim plausible. The equation thus looked like a promising generalization of the vanishing of the ordinary four-divergence of the stress-energy tensor, which Laue (1911a, b) had made the fundamental equation of relativistic mechanics. Like Laue, Einstein presumably wanted to generalize it from the special case of pressureless dust to arbitrary physical systems.Einstein made this generalization explicit in his paper with Grossmann: “We ascribe to equation (10) a range of validity that goes far beyond the special case of the flow of incoherent masses. The equation represents in general the energy balance between the gravitational field and an arbitrary material process” (“Der Gleichung (10) schreiben wir einen Gültigkeitsbereich zu, der über den speziellen Fall der Strömung inkohärenter Massen weit hinausgeht. Die Gleichung stellt allgemein die Energiebilanz zwischen dem Gravitationsfelde und einem beliebigen materiellen Vorgang dar …;” Einstein and Grossmann 1913, p. 11). Statements almost verbatim the same as this one can be found in the printed text of Einstein’s lecture on the problem of gravitation in Vienna in September 1913 (Einstein 1913, 1253 and 1257). Einstein’s analysis on p. 5R thus provides an excellent example of how physical and mathematical considerations complement each other in the course of Einstein’s development of the general theory of relativity. First, Einstein derived an equation for the energy-momentum balance between matter and gravitational field on the basis of physical arguments centering on the special case of pressureless dust. Then, he confirmed its generalizability to arbitrary physical systems on the basis of mathematical arguments. We shall now examine Einstein’s calculations on p. 5R in detail. He started by writing down the line element, g11dx2+…+g44dt2ds2 . Next to it, he noted that this expression is “always positive for a point” (“immer positiv für Punkt”), i.e., the worldline of a material particle is time-like.From this we can infer that at this point, Einstein’s sign convention is such that the Minkowski metric in its standard diagonal form is diag1,1,1,c2 . This same convention was used in the Nachtrag to (Einstein 1912b) and in (Einstein and Grossmann 1913). On p. [39L], however, the Minkowski metric in its standard diagonal form was given as diag1,1,1,c2 (if we switch from an imaginary to a real time coordinate). As he had first done in a note added in proof to the paper presenting his second static theory (Einstein 1912b, 458), Einstein used this line element to define the Lagrangian—or “Hamiltonian function” (“Hamiltonsche Funktion”) as he called it (Einstein and Grossmann 1913, 7)—for a point particle moving in a given metric field:Given the sign convention on this page (see the preceding footnote), H should be H (cf. Einstein and Grossmann 1913, 7; Einstein 1912b, 458). dsdtH . He then wrote down the corresponding Euler-Lagrange equations, the “equations of motion” (“Bewegungsgleichungen”): d∂H∂x⋅dt-∂H∂x0 . The equation initially had a plus rather than a minus sign on the left-hand side. This sign error is carried through all the way to the end of the calculation. Einstein only corrected it after he discovered that it led to an unacceptable end result (see the discussion following equation ()). The equation, d∂L∂x⋅dt∂Φ∂x , written to the right of equation (), was presumably added at that point and helped Einstein correct his sign error. It is easy to see how equation () could play that role. When the Lagrangian can be written as HL-Φ , where Φ is a potential energy term and L (which stands for “lebendige Kraft”) is the kinetic energy term, equation () is equivalent to equation (). The latter equation is readily recognized as a generalization of Newton’s second law. This then confirmed that the left-hand side of equation () must indeed have a minus sign. Underneath equation (), Einstein wrote down the partial derivative of H with respect to x⋅ : ∂H∂x⋅∂∂x⋅ds2dt212dtds∂∂x⋅gμνdxμdtdxνdtdtdsg1νx⋅ν . ∂H∂x⋅g11x⋅+g12y⋅+.+g14dsdt . This is the x -component of the momentum of a particle of unit mass. He then generalized this result to an expression for the x -component of the momentum density of pressureless dust:Given the sign convention on this page (see footnote 55), G stands for minus the determinant of the metric. ρ0G(g11dxdsdtds+g12dydsdtds+.+.) The derivation of this expression is not given in the notebook, but can easily be reconstructed (see Einstein and Grossmann 1913, secs. 2 and 4). Using equation () for the x -component of momentum Jx and dividing it by the volume V in the same coordinate system, one arrives at an expression for the x -component of momentum density JxV1V∂H∂x⋅1Vg11x⋅+g12y⋅+g13z⋅+g14dsdt . To obtain equation (), the coordinate volume V has to be replaced by the proper volume V0 . The relation between the two is given by (Einstein and Grossmann 1913, 10):This relation can also be found on p. 20R (see equation ()) and at the bottom of p. 27L. VV0dsdt1G , where G is minus the determinant of the metric tensor. Substituting this expression for V on the right-hand side of equation (), one arrives at equation () for what, as Einstein wrote under it, “is the momentum per unit volume” (“ist Bewegungsgrösse pro Volumeinheit”): JxV=1V0Gdtdsdtdsg11x⋅+g12y⋅+g13z⋅+g14 =1V0G(g11dxdtdtdsdtds+g12dydtdtdsdtds+.+g14dtdsdtds) =ρ0G(g11dxdsdtds+g12dydsdtds+.+.), where ρ01V0 is the proper density of a unit mass of dust. The energy density EV of the dust can be found through a similar argument:Cf. footnote 57: EV1V0Gdtds∂H∂x⋅41V0Gdtdsdtdsg4νx⋅νρ0G∑g4νdxνdtdtdsdtdsρ0G∑g4νdxνdsdx4ds . EVρ0G∑g4νdxνdsdx4ds . Equation () for the x -component of momentum density, similar expressions for its y - and z -components, and equation () for the energy density, all divided by G , give four components of the contravariant stress-energy tensor for pressureless dust, which is written down immediately below equation ():Note that the indices are written “downstairs” even though they refer to contravariant components. Einstein’s convention here and elsewhere in the notebook is just the opposite of the one adopted in (Einstein and Grossmann 1913), where all contravariant quantities are indicated by Greek and all covariant ones by Latin characters. Tikbρ0dxidsdxkds . Einstein called this quantity the “tensor of the motion of masses” (“Tensor der Bewegung der Massen”).The superscript ‘b’ stands for “Bewegung” (“motion”). Contracting this tensor with the metric and multiplying by G , one recovers the momentum and energy densities (cf. equations ()–()) JxVG∑g1iTi4b ,      EVG∑g4iTi4b . Einstein thus introduced what he called the “tensor of momentum and energy” (“Tensor der Bewegungsgröße u[nd] Energie.”). Tmn∑GgmνTνnb . In fact, this is not a tensor but a (mixed) tensor density. With the help of equation (), Einstein could begin to rewrite the Euler-Lagrange equations () in terms of a stress-energy tensor, which would allow him to generalize the equation from the special case of pressureless dust to any physical system described by a stress-energy tensor. The second term of equation () represents the gravitational force on a particle. Einstein generalized this term to an expression for the “negative ponderomotive force per volume unit” (“Negative Ponderomotorische Kraft pro Volumeinheit”)The word “Negative” was later added. on pressureless dust:In the expression below, G was corrected from D (which presumably stands for “Determinante”), the subscript m was corrected from some illegible character, and a factor ρ0 was deleted. 12G∑∂gμν∂xmTμνb . As with equation (), the derivation of this expression is not given in the notebook, but can easily be reconstructed (see Einstein and Grossmann 1913, secs. 2 and 4). Defining the x -component of the gravitational force K as ∂H∂x and using the definition of the Lagrangian H , one can write: K1V1V∂H∂x11V∂∂x1dsdt . Using the definition of the line element, one can rewrite this as: K1V1V∂∂x1gμνdxμdxνdt1V12∑μν∂gμν∂x1dxμdxνdsdt1V12∑μν∂gμν∂x1ddtxμddtxνdsdt With the help of equation (), the coordinate volume V can be replaced by the proper volume V0 : K1V1V0G12dtdsdtds∑μν∂∂x1gμνddtxμddtxν12ρ0G∑μν∂∂x1gμνddsxμddsxν . Inserting Tμνb , the stress-energy tensor for pressureless dust, for ρ0dxμdsdxνds , one arrives at the m1 -component of Einstein’s equation (). As he had done before (see equation ()), Einstein assumed that this result, derived for the special case of the stress-energy tensor of pressureless dust, will hold for the stress-energy tensor of any matter. On the next line Einstein wrote down the tensorial generalization of the Euler-Lagrange equations (), using equations () and () and replacing ddt by ∂∂xn :The expression in parentheses is underlined once, and Tμν in the second term is underlined twice. The equation also contains a number of corrections. The minus sign was corrected from a plus sign. This error stems from the sign error in equation (). In the first term, the index n was corrected from m and the factor G was inserted later. In the second term, the factor 12 was deleted once and rewritten. This correction is related to a correction in equation () below. Moreover, G was corrected from D (as in equation ()). Note that Tμν in equation () stands for the contravariant stress-energy tensor, whereas in equation () it stands for the corresponding mixed tensor density.,The same equation appears in slightly different versions and in slightly different notation at various other places in the notebook (see pp. 24L, 26L, 28L, and 43Lb) and in (Einstein and Grossmann 1913, p. 10, equation (10)). ∑νn∂GgmνTνn∂xn-12G∑∂∂xmgμνTμν0 . At this point he dropped the index ‘b’ since he expected the equation to hold for all matter, not just for pressureless dust. Einstein now tried to gain some insight into the transformational behavior of this equation by rewriting its left-hand side as the action of a generalized derivative on an arbitrary tensor. First, he introduced the (contravariant) tensor density corresponding to the (contravariant) stress-energy tensor Tμν in equation (). He wrote: “If we set” (“Setzen wir”)The notation in the notebook (see also pp. 20R, 24L, 24R, 26L, and 43Lb) is different from the notation used in (Einstein and Grossmann 1913), where the contravariant stress-energy tensor is denoted by Θμν and the covariant form by Tμν . GTμνΘμν , then equation () can be rewritten as: ∑νn∂gmνΘνn∂xn-12∑∂∂xmgμνΘμν0 . To write the left-hand side as a differential operator acting on a tensor, Θ should appear with the same indices in both terms. Therefore Einstein relabeled the summation indices in the first term in order to conform with the occurrence of Θμν in the second term, made the assumption that Θμν is symmetric, and rewrote the equation as:The equation originally had a plus rather than a minus sign, an error it inherited from equation (). The factor 12 was deleted once and rewritten. This correction is related to a correction in equation () below. The index ν in gμν is corrected from μ . Both characters Θ are corrected from T ’s. ∑μν∂gmνΘμν∂xμ-12∑∂∂xmgμνΘμν0 . To the right of this equation, he wrote: “In general associated vector” (“Im Allgemeinen zugeordneter Vektor”), and below it “Valid for every symmetric tensor, e.g., Gμν ” (“Gilt für jeden symm[etrischen] Tensor z. B. Gμν ”).The word “symm[etric]” is interlineated. Recall that Einstein had to assume symmetry in going from equation () to equation (). “Associated” (“zugeordnet”) means “covariant” in this context, as is clear from its usage on pp. 6L–R. The notation μν for the contravariant components of the metric (see Einstein and Grossmann 1913, 12, note 4) occurs here for the first time in the notebook. Einstein’s claim is that the quantity on the left-hand side of equation () is a vector as long as Θμν is a symmetric tensor. This claim is correct. The expression is just the covariant divergence of a symmetric tensor, as is shown in (Einstein and Grossmann 1913). The result appears in Grossmann’s part, which suggests that Einstein may have learnt it from him. Einstein checked his claim for the specific example Gμν mentioned above. Insertion of this tensor density into equation () produces:The equation originally had a plus rather than a minus sign, an error it inherited from equation (). Einstein initially omitted the factor of 12 . The factor G was inserted later. ∑μν∂Ggmνμν∂xμ-12∑μνG∂gμν∂xmμν0 , Since ∑gmνμνδmμ , with δmμ the Kronecker delta, the first term is equal to ∂G∂xm , which Einstein wrote underneath the first term of equation (). SinceSee p. 6L for a derivation of this relation. ∑μν∂gμν∂xmμν1G∂G∂xm , the expression in parentheses in the second term is equal to 1G∂G∂xm , which Einstein wrote underneath the second term of equation ().The index m is corrected from μ . Inserting expressions () and () into equation (), one sees that the latter is indeed satisfied. It is at this point that Einstein recognized two errors that had found their way into his calculations. First, equation () inherited the erroneous minus sign from the Euler-Lagrange equations () that formed the starting point of this whole calculation. Second, Einstein omitted a factor 12 when he went from equation () to equation (). Einstein probably only caught these errors when the result of his calculations did not meet this expectation, i.e., when he saw that equation () with a plus rather than a minus and without the factor 12 does not hold. Einstein probably corrected the sign error first, tracing it all the way back to the Euler-Lagrange equations (). Equation () next to these equations was probably added in this context. As to the omitted factor 12 , Einstein originally seems to have been under the impression that expression () is equal to expression () without the factor 12 multiplying the latter in equation (). It is probably for this reason that he deleted this factor in equations () and (), respectively, only to put them back in once he realized that this factor was needed in equation () after all. After making these corrections, Einstein wrote at the bottom of the page: “correct” (“stimmt”). Einstein’s trial calculation supported the claim that his physically motivated expression for energy-momentum balance between matter and gravitational field does indeed lead to a differential operator that acts on a symmetric tensor to produce a vector.On p. 8R, Einstein found that the generally-covariant generalization of the exterior derivative operator acting on the metric also vanishes (see the discussion following equation ()). Exploration of the Beltrami Invariants and the Core Operator (6L–13R, 41L–R) Introduction (6L–13R, 41L–R) Einstein returned to the question of finding field equations for the metric field on p. 6L. He had meanwhile become more sophisticated mathematically. For example, at this point he knew about the Beltrami invariants and carefully distinguished between covariant and contravariant tensors. However, there were still large gaps in his knowledge. He still did not know about the Riemann tensor or covariant differentiation, which severely handicapped his search for satisfactory field equations. Most of the calculations on pp. 6L–13L are investigations of the covariance properties of various expressions that might either be part of the field equations or play a role in their construction. These calculations did not lead to any promising candidates for the left-hand side of the field equations, but they led to several clusters of important results, ideas, and techniques that Einstein was able to put to good use once he learned about the Riemann tensor (see p. 14L). First of all, one can begin to discern the double strategy discussed in sec. . On p. 6L Einstein started with two generally-covariant operators acting on a scalar function, known as the Beltrami invariants. The second Beltrami invariant is a generally-covariant generalization of the Laplacian of a scalar function and as such provided a natural point of departure in Einstein’s search for gravitational field equations. The basic challenge was to get from an operator acting on a scalar function to an operator acting on the metric tensor. Einstein did not immediately see how to achieve this goal. On p. 7L, he therefore temporarily abandoned his mathematically-oriented approach for a physically-oriented one. He wrote down a version of what we call the “core operator,” an expression constructed out of the metric tensor and its coordinate derivatives in such a way that for weak fields it reduces to the d’Alembertian acting on the metric. The problem with this core operator is that its transformation properties are unclear. Einstein addressed this problem by trying to relate the physically well-understood core operator to the mathematically well-understood Beltrami invariants. Two key components of Einstein’s mathematical strategy first emerge in the course of the calculations documented on these pages. The first is the idea is to start from expressions with a well-defined covariance group (either generally covariant or covariant under unimodular coordinate transformations) and then to extract candidates for the left-hand side of the field equations by imposing additional conditions, such as the condition that the correspondence principle be satisfied (i.e., that the field equations reduce to the Poisson equation of Newtonian theory for weak static fields). The other key component is the use of what Einstein would later call “non-autonomous transformations” (“unselbständige Transformationen”).See Einstein to H. A. Lorentz, 14 August 1913 (CPAE 5, Doc. 467). See sec.  for further discussion. To investigate the covariance properties of various expressions constructed out of the metric tensor and its derivatives, he wrote out the transformation law for such an expression under general coordinate transformations and identified those terms that would have to vanish if the expression were to transform as a tensor. The vanishing of these terms gives conditions on the transformation matrices that depend explicitly on the components of the metric (see pp. 7L-R, 9R, 10L). Because of this dependence, such transformations are called “non-autonomous.” Initially, Einstein investigated the behavior of his candidate field equations under such non-autonomous coordinate transformations. Eventually he realized that the complexity of the relevant calculations could be reduced considerably by combining the notion of non-autonomous transformations with the basic idea of the mathematical strategy, namely to extract candidate field equations from expressions with well-known covariance properties by imposing additional conditions. The covariance properties of field equations constructed in this fashion are determined by the covariance properties of these additional conditions, which typically will be much simpler than the field equations themselves. Einstein could thus focus on determining the class of non-autonomous transformations under which these simpler additional conditions transform tensorially.See p. 9R and the discussion in footnote 134 for the first example of this type of argument. An important example of such an additional condition is what we shall call the “Hertz restriction,” in which the so-called “Hertz expression,” ∑∂μν∂xν , is set equal to zero.Named after Paul Hertz (see footnote 22). The Hertz restriction reappears in the course of Einstein’s exploration of the Riemann tensor on p. 22R (see sec. ). On pp. 10L–11L, Einstein investigated whether the class of non-autonomous transformations under which the Hertz expression transforms as a tensor includes the transformations from quasi-Cartesian coordinates to rotating and linearly accelerating frames in the special case of Minkowski spacetime. These transformations were crucially important to Einstein’s attempts to extend the relativity principle from uniform to non-uniform motion and to establish the equivalence of rotation and acceleration in Minkowski spacetime to corresponding gravitational fields. Later in the notebook, Einstein applied the same strategy to candidate field equations constructed out of the Riemann tensor. As before, the covariance properties of the additional condition(s) determine the covariance properties of the candidate field equations. Einstein regarded such conditions as essential parts of the theory, restricting the class of admissible coordinate systems. We therefore refer to these conditions as “coordinate restrictions.” They should be distinguished from “coordinate conditions.” Mathematically, one and the same equation can express a coordinate restriction or a coordinate condition, but the two have a very different status. Coordinate conditions may always be imposed on generally-covariant equations to choose a suitable class of coordinate systems for some particular purpose. Consequently, different coordinate conditions may be used for different purposes, just as different gauge conditions can be used for different purposes. In contrast, coordinate restrictions are an integral part of the theory, imposing a limitation on the allowed class of coordinate systems in which the theory is expected to hold. With coordinate restrictions, one does not have the freedom to pick different conditions for different purposes. On p. 9L, building on the energy-momentum balance equation derived on p. 5R, Einstein arrived at two important insights related to energy-momentum conservation. First, drawing on his experience with the 1912 static theory, he realized that for the field equations to be compatible with energy-momentum conservation their left-hand side should be the sum of a core operator and a quantity representing the stress-energy of the gravitational field. Secondly, he found a way to construct a candidate for this stress-energy tensor out of the first Beltrami invariant. Einstein’s main concern in this part of the notebook, however, was not how to satisfy the conservation principle but how to satisfy simultaneously his other three heuristic requirements, the correspondence principle, the relativity principle, and the equivalence principle.See sec. for discussion of these requirements. By the time he made the entries at the bottom of p. 12R, Einstein had established a number of results related to the latter three principles. He had (i) introduced the core operator (p. 7L); (ii) found its relation to the second Beltrami invariant with the determinant of the metric playing the role of the scalar function in the latter’s definition (pp. 8R–9R); (iii) investigated unimodular transformations since the determinant of the metric transforms as a scalar under this restricted class of transformations only; (iv) recognized the importance of the Hertz restriction in getting from the second Beltrami invariant first to the core operator and then to weak field equations with the d’Alembertian acting on the metric (p. 10L); (v) derived conditions for the classes of non-autonomous transformations under which the weak field equations and the Hertz expression transform as a tensor and a vector, respectively (pp. 10L–11R); and (vi) developed a strategy for checking whether such non-autonomous transformations include the important special cases of the transformation to rotating and accelerating frames in Minkowski spacetime (pp. 11L, 12L–R, 41L–R). However, serious difficulties on all these counts remained. On pp. 13L–R, Einstein therefore bracketed the question of the covariance properties of his candidate field equations and turned to the compatibility with energy-momentum conservation instead, provisionally requiring only covariance under autonomous unimodular linear transformations. On p. 13R he returned to the physical strategy trying to find the left-hand side of the field equations by starting from the weak field equations and imposing energy-momentum conservation. An ingenious simpler version of this strategy was used on pp. 26L–R to derive the Entwurf field equations. On p. 13R Einstein did not yet see how to generate field equations in this way. Given this impasse and the difficulties he had run into in his investigation of the covariance properties on pp. 6L–12R, one can well imagine Einstein turning to his mathematician friend Marcel Grossmann for fresh ideas.As is related in (Kollros 1956, 278), Einstein allegedly turned to Grossmann at one point and said: “Grossmann, you have to got to help me, otherwise I am going mad” (“Grossmann, Du mußt mir helfen, sonst werd’ ich verrückt!”). On the next page, p. 14L, the Riemann curvature tensor makes its first appearance in the notebook with Grossmann’s name written next to it. Einstein only returned to the physical strategy emerging on p. 13R after a series of failed attempts to extract field equations from the Riemann tensor (see pp. 14L–25R, 42L–43L covered in sec. ). Experimenting with the Beltrami Invariants (6L–7L) Beltrami’s two differential invariants (more precisely “differential parameters”) are generally-covariant scalars constructed out of the metric, its first- and second-order derivatives, and some arbitrary, at least twice-differentiable scalar function ϕ .See (Bianchi 1910, secs. 22–24) or (Wright 1908, sec. 53). The first Beltrami invariant can be defined as Δ1ϕμν∂ϕ∂xμ∂ϕ∂xν , the second asEquations () and () are equivalent to (Bianchi 1910, sec. 23, eq. (8), and sec. 24, eq. (19)), respectively. The second Beltrami invariant is alluded to in the 1914 review article on the Entwurf theory. In sec. 8 of this paper, Einstein points out that the covariant divergence of the contravariant vector gμν∂ϕ∂xν is equal to “the well-known generalization of the Laplacian Δϕ , Φ∑μν1G∂∂xμggμν∂ϕ∂xν ” (“die bekannte Verallgemeinerung des Laplaceschen Δϕ …;” Einstein 1914b, 1051–1052; see also Einstein 1916, 797). This generalization is nothing but the second Beltrami invariant. Following (Weyl 1918), Einstein also used the second Beltrami invariant, again without identifying it by name, in lectures in 1919 (see CPAE 7, Doc. 19, [p. 22], and Doc. 20, [p. 1]). Δ2ϕ1G∂∂xμμνG∂ϕ∂xν . At the top of p. 6L, Einstein wrote down the contraction of the contravariant metric with the gradient of some scalar function ϕ ∑μν∂ϕ∂xν . Next to this expression he wrote: “vector” (“Vektor”).He initially wrote and then deleted “associated vector” (“zugeordneter Vekt”). “Associated” stands for “covariant” in this context. Expression () is a contravariant vector. Cf. the use of the term “zugeordnet” on p. 5R (see the discussion following equation () in sec. ) and on p. 6R (see the remarks following equation ()). Einstein now applied a differential operator to this vector. In this way, he obtained the second Beltrami invariant 1G∂μνG∂ϕ∂xν∂xμ . Next to this expression, he wrote: “scalar” (“Skalar”). Einstein then investigated a condition, under which expression () would reduce to the ordinary Laplacian acting on a scalar. He called this condition a “plausible hypothesis” (“Naheliegende Hypothese”): ∑∂Gμν∂xν0. This condition determines so-called “isothermal coordinates,”See, e.g., (Bianchi 1910, sec. 36-37) or (Wright 1908, sec. 57). or, as they are now called, harmonic coordinates. The corresponding harmonic coordinate restriction came to play an important role later in the notebook in Einstein’s analysis of the Ricci tensor (see pp. 19L–20L and the discussion following equation () in sec. ). In the last four lines of p. 6L, Einstein rewrote equation () to eliminate the derivative of the determinant G , leaving only derivatives of components of the metric: G∂μν∂xμ+μν12G∂G∂xμ0 . Dividing by G , he arrived at: ∑∂μν∂xμ+12μν1G∂G∂xμ0 . He then rewrote the second term using that the components of the contravariant metric are defined as μνΓμνG where Γμν is the minor of the μν -component of the covariant metric. Einstein thus rewrote the second term of equation () as: 1G∂G∂xμ∑ρσ∂gρσ∂xμΓρσG∑ρσ∂gρσ∂xμρσ . Here the calculation breaks off. A possible explanation is that Einstein wanted to check whether the spatially flat static metric, which reappears on the next page, satisfies the “plausible hypothesis” (). The first term of equation () clearly vanishes for this metric, but the second term does not. On p. 6R Einstein made a fresh start, denoting the gradient of the scalar function introduced at the top of p. 6L by: αν∂ϕ∂xν . This is a covariant vector as is indicated by the label “associated vector” (“zugeordneter Vektor”) written next to it. Recall that on the facing page 6L, Einstein started to write and then deleted this same label next to expression (), the contravariant form of this vector. Writing the contravariant form as aννμαμ , Einstein wrote down the second Beltrami invariant (cf. equation ()) as: ∑1G∂G aν∂xν . Einstein now replaced all covariant elements in this expression by their contravariant counterparts and vice versa. To this end, he defined ξμΣgμνxν (with inverse xνΣμνξμ ).Note that in these definitions, Einstein used finite coordinates xν instead of infinitesimals dxν , as he had done on p. 39L (cf. equations () and ()). It seems doubtful whether, at this time, Einstein realized that only coordinate differentials are proper vectors, and that these formal definitions have no invariant significance for finite coordinates. He thus arrived at: ∑1Γ ∂Γ αν∂ξν , where Γ is the determinant of the contravariant metric and ∂∂ξν∑νσ∂∂xσ . If Einstein thought that in going from the second Beltrami invariant () to the expression () he had constructed another scalar invariant, he was wrong. But he proceeded to rewrite expression () in terms of ∂∂xν and G in order to compare it with the second Beltrami invariant: ∑Gνσ∂∂xσ1G∂ϕ∂xν . Next to this expression, he noted that it is a “scalar” (“Skalar”). It is not. On the next line, Einstein expanded expression () to: ∑νσνσ∂2ϕ∂xν∂xσ+νσ∂ϕ∂xν12G∂G∂xσ , losing a minus sign in the differentiation of 1G . The plus sign should thus be a minus sign. To compare this expression with the second Beltrami invariant, he expanded expression () on p. 6L for the latter to ∑νσνσ∂2ϕ∂xν∂xσ+νσ∂ϕ∂xν12G∂G∂xσ+∂∂xvϕ∂σν∂xσ . In the notebook there is actually a line connecting expression () on p. 6L to expression () on p. 6R. Comparing the expressions () and (), Einstein noted: “Should there only be one such scalar, it has to be the case that ∑σ∂σν∂xσ0 ” (“Soll es nur einen derartigen Skalar geben so muss ∑σ∂σν∂xσ0 ”). Recall, however, that expression () is not a scalar and that there is a sign error in the subsequent expression () for this alleged scalar. Einstein did not discover these errors until the top of p. 7L. On p. 6R, he was under the impression that the second Beltrami invariant and expression (), which he took to be a scalar, were identical once condition (), which we shall call the “Hertz restriction,”The reason for naming it after Paul Hertz is explained in footnote 22. was imposed. Adding this condition, he now proceeded as if he had two expressions—the second Beltrami invariant () and quantity constructed out of it in expression ()—for one and the same scalar invariant. To turn this scalar invariant into a candidate for the left-hand side of the field equations, Einstein substituted G for the scalar ϕ in expression (). The resulting field equations will be invariant under unimodular transformations since G transforms as a scalar under such transformations. In a separate box to the right of equations ()–(), Einstein wrote down how G and Γ transform under a coordinate transformation with determinant P :This follows from the multiplication theorem for determinants. G′P2G,Γ′1P2Γ. If P1 , i.e., for unimodular transformations, G and Γ are indeed scalars. At the bottom of p. 6L, Einstein had just gone through the derivation of another result for G : 12G∂G∂xνρσ∂gρσ∂xν . Inserting G for ϕ in expression () and using equation (), Einstein arrived at the following candidate field equations: ∑μνρσμν∂ρσ∂gρσ∂xν∂xμ0 “or” (“oder”), equivalently,Note that 0∂∂xνgρσρσρσ∂gρσ∂xν+gρσ∂ρσ∂xν . ∑μνρσμν∂gρσ∂ρσ∂xν∂xμ0 . Note that Einstein omitted a factor G2 in the expressions on the left-hand sides of both these equations. These expressions, however, retain the transformation behavior under unimodular transformations of the supposedly-invariant expression () from which Einstein constructed them. Since G is a scalar under unimodular transformation, a scalar divided by G remains a scalar. Einstein now turned to “special cases μν and ρσ ” (“Spezialfälle μν ρσ ”) of diagonal metric tensors. He quickly focused on one such case, namely the spatially flat static metric first introduced on p. 39L. He accordingly corrected “special cases” (“Spezialfälle”) to “special case” (“Spezialfall”) and wrote down the diagonal components of the static metric:In the notebook, the “ 1 ” seems to be corrected from “ 0 .” The contravariant components have the wrong sign. Einstein originally wrote them down correctly, but then changed the signs. g11g22g331g44c2112233+1441c2, as well as its determinant Gc2 , which he could read off from the matrix diag1,1,1,c2 written directly above this expression. Einstein now inserted the static metric into equations () and () for the index combinations ρσ4 and μν1,2,3 , the only combinations giving a non-vanishing contribution. He found ∑ν∂+1c2∂c2∂xν∂xν0 , and ∑∂c2∂1c2∂xν∂xν0 , “respectively” (“b[e]z[iehungs]w[eise]”).Inserting the components () of the static metric (with or without correcting the sign error noted in the preceding note) into the left-hand sides of equations () and (), one finds the left-hand sides of equations () and () with the opposite sign. Probably, Einstein wrote down equation () using 1122331 and 441c2 , then adjusted the result once he changed the sign of μν . Both equations are equivalent toFor equations () and (), one finds ∂∂xν1c2∂c2∂xν∂∂xν2c∂c∂xν2∂2logc∂xν20 and ∂c2∂1c2∂xν∂xν∂2c∂c∂xν∂xν2∂2logc∂xν20 , respectively. ∂2logc∂xν20 . This equation has the form of the four-dimensional generalization of the Poisson equation and therefore looks promising as a component of candidate field equations. Next to equation () wrote: “In this way not distinguishable” (“So nicht unterscheidbar.”). Presumably, this refers to Einstein’s query in the middle of the page concerning the relation between expression () for the second Beltrami invariant and expression () for what Einstein took to be an alternative scalar invariant. Since the spatially flat static metric satisfies condition (), which reduces the former expression to the latter, it is clear this special case cannot be used to distinguish the two. At the top of p. 7L, Einstein returned to the more general considerations above the horizontal line on p. 6R, where he had written down expression () for what he thought was an alternative scalar invariant. In fact, it is not a scalar and contains a sign error. Einstein had compared this expression to expression () for the second Beltrami invariant and had noticed that the former reduces to the latter if the Hertz condition () is imposed. Einstein began his considerations on p. 7L by writing down an expression that can be obtained from the second Beltrami invariant () on p. 6L by imposing the Hertz condition ():The labeling of the indices suggests that this expression was obtained directly from expression () on p. 6L. μνG∂G∂ϕ∂xν∂xμ and wrote next to it that this is a “scalar” (“Skalar”). In the next two lines, Einstein tried to construct a vector out of the scalar quantities he had formed on the preceding two pages. On the assumption that Einstein meanwhile caught the sign error in expression (), one can readily understand these two lines: “If one forms [the second Beltrami invariant] Δ2ϕ in two ways, it follows that ∑ν1G∂μνG∂xν [is] a vector” (“Bildet man Δ2ϕ auf zwei Arten, so folgt ∑ν1G∂μνG∂xν ein Vektor”). The conclusion was subsequently deleted. If the error in expression () is corrected, then the difference between expressions () and () becomes: ∂ϕ∂xννσ1G∂G∂xσ+∂σν∂xσ∂ϕ∂xν1G∂νσG∂xσ . Except for the labeling of its indices, the term in parentheses on the right-hand side is exactly equal to Einstein’s expression () above. This explains why Einstein initially expected expression () to be a vector. Since expression () contracted with the covariant vector ∂ϕ∂xi (for arbitrary ϕ ) is the difference between (what Einstein took to be) two scalars, the contraction must also be a scalar and expression () itself must be a contravariant vector. However, Einstein presumably recognized that expression () is in fact not a vector. The expression is equal to 1G∂μν∂xν+1G∂G∂xν . The second term is a vector. For the entire expression to be a vector, the first term would have to be a vector as well. This, however, is not the case, as Einstein presumably knew. One can thus understand why he eventually deleted the claim that expression () is a vector. This immediately told Einstein that the starting point of this entire line of reasoning, the assumption that expression () is a scalar, had to be mistaken. The reconstruction given above leaves one question unanswered: how did Einstein discover the sign error in expression ()? There is a plausible answer to this question. Suppose Einstein went through the same argument that we just described before he corrected this sign error. He would then have arrived at the conclusion that ∂ϕ∂xν∂νσ∂xσ is a scalar and that ∂νσ∂xσ therefore has to be a vector. Einstein already knew this to be false (see p. 39R). This then might well have alerted him to the sign error in expression (). He may then have repeated the construction of a vector out of the difference between his two scalars () and (), using the corrected version of the former.It remains unclear exactly what Einstein took the relation between expressions () and () to be. His comment on p. 6R (“Should there only be one such scalar, it has to be the case that …”) suggests that he thought of them as two different quantities that coincide only if an additional condition is imposed. However, his comment on p. 7L (“If one forms Δ2ϕ in two ways …”) suggests that he thought of them as different expressions of the same quantity. In that case, however, expression () should vanish, whereas Einstein only says that it is a vector. Investigating the Core Operator (7L–8R) On pp. 6L–6R Einstein had tried to construct field equations for the metric tensor using as his starting point a familiar object from differential geometry, the generally-covariant second Beltrami invariant. The calculations documented on these pages had not produced any promising results. On pp. 7L–8R, he tried a different approach inspired by physical rather than mathematical considerations. He generalized the Laplace operator acting on the scalar potential of Newtonian gravitational theory to an operator acting on the tensorial potential μν . We shall call this object the “core operator.” It is a combination of two simpler operators, explicitly called “divergence” (“Divergenz” [p 7R]) and “exterior derivative” (“Erweiterung” [p. 8L]) in the notebook.Cf. (Einstein and Grossmann 1913, Part I, sec. 5). Einstein first examined the covariance properties of the core operator as a whole (pp. 7L-R). He then switched to generalizing the two constituent operators. The aim of both investigations was to see whether the core operator would provide him with a basis for extending the relativity principle from uniform to non-uniform motion. One can discern two strategies with which Einstein tried to achieve his aim. The first strategy was to find a special type of non-linear coordinate transformations under which the core operator transforms as a tensor. The second strategy was to generalize the core operator to a differential operator that transforms as a tensor under ordinary non-linear transformations. We introduce some special notation to facilitate a concise discussion of these two strategies. Consider an object, O∂, g —constructed out of the metric, gμν , and the derivative operator, ∂∂xα —that transforms as a tensor under arbitrary linear transformations, Tlinear : x→x′ . The transformation law for O under the inverse transformation, Tlinear1 , can schematically be written as: O∂, gTlinear1O∂′, g′ . If one now expresses ∂′ and g′ on the right-hand side in terms of ∂ and g with the help of Tlinear , one, of course, just reproduces O in unprimed coordinates: Tlinear1OTlinear∂, TlineargO∂, g . Einstein’s strategy for dealing with arbitrary non-linear transformations, T : x→x′ , involves two steps analogous to the two steps above. In general, O will not transform as a tensor under T . The first step is to introduce the object, O˜∂, g , that one obtains when applying the inverse transformation T1 to O∂′, g′ as if O does transform as a tensor under T , more specifically, as if O transforms under T the same way it transforms under Tlinear : O˜∂, gT˜1O∂′, g′ . The tilde on T˜1 merely indicates this special use of the transformation rules. The second step is to express ∂′ and g′ in terms of ∂ and g with the help of T . Since in general O does not transform as a tensor under T , this operation will in general not just reproduce O . In addition to O , it will produce a (sum of) term(s), C , constructed out of ∂ , g , and the transformation matrices p and for T and T1 :For the definition of these transformation matrices, see equations ()–() below. T˜1OT∂, TgO∂, g+C∂, g, p, This two-step procedure is common to both strategies distinguished above. The two strategies differ in the way they make use of equation (). In the first strategy, Einstein uses equations such as equation () to read off the condition on the transformation matrices p and for T that needs to be satisfied for O to transform as a tensor under T . This will be the case if O˜∂, gO∂, g , i.e., in view of equations ()–(), if C∂, g, p, 0 . This condition gives a set of differential equations involving the transformation matrices, the components of the metric, and derivatives of both. Inserting a specific metric into equation () and solving the resulting equation for the transformation matrices, pg and g , one arrives at a special type of coordinate transformations. In the case of ordinary coordinate transformations, the transformation matrices are functions only of the coordinates. In the case of these special coordinate transformations, however, they depend both on the coordinates and on the metric. Following a suggestion by Paul Ehrenfest, Einstein later introduced the term “non-autonomous transformations” (“unselbständige Transformationen”) for such transformations.See Einstein to H. A. Lorentz, 14 August 1913 (CPAE 5, Doc. 467): “One can consider two fundamentally different possibilities. 1) Transformations which are independent of the existing gμν -field, which Ehrenfest designated as ‘autonomous transformations;’ according to my knowledge group theory has only dealt with this kind of transformations. 2) Transformations whose [matrices] would have to be determined by differential equations for the gμν -field considered as given, which hence have to be adapted to the existing gμν -field. Such transformations have—as far as I know—not yet been systematically studied. (‘non-autonomous transformations’)” (“Zwei Möglichkeiten prinzipiell verschiedener Art kommen da in Betracht. 1) Transformationen, welche von dem vorhandenen gμν -Feld unabhängig sind, welche Ehrenfest als ‘selbständige Transformationen’ bezeichnete; nur mit solchen hat sich meines Wissens bisher die Gruppentheorie beschäftigt. 2) Transformationen, deren […] erst durch Differentialgleichungen zu dem als gegebenen zu betrachtenden gμν -Feld zu bestimmen wären, die also dem vorhandenen gμν -Feld angepasst werden müssen. Solche Transformationen sind—soviel ich weiss—noch nicht systematisch untersucht worden. (‚unselbständige Transformationen‘)”). For further discussion of non-autonomous transformations—or, as Einstein later called them “justified” (“berechtigte”) transformations between “adapted” (“angepaßte”) coordinates” (Einstein and Grossmann 1914, 221; Einstein 1914b, 1070)—see “Untying the Knot …” sec. 3.3 (in this volume). For a modern discussion of such transformations, see (Bergmann and Komar 1972). The transformation rule for such non-autonomous transformations can be written as O∂′, g′TgO∂, g , where the matrices p and for the transformation Tg must satisfy a condition for non-autonomous transformations of the form of equation (). In the case of the second strategy, Einstein looked upon the right-hand side of equation () as a generally-covariant expression that in the special case of a diagonal Minkowski metric reduces to the object O he started from. One can then set g′μνdiag1, 1, 1, 1 in the transformation laws gμνpαμpβνg′αβ and μναμβν′αβ and express the transformation matrices p and in terms of gμν and its derivatives. Substituting the resulting expressions pg and g into C in equation (), one arrives at a new expression D that depends only on the metric and its derivatives: D∂, gC∂, g, pg, g . One can now define the generalization Ogen of the original object O : Ogen∂, gO∂, g+D∂, g . Ogen reduces to O in the special case of the Minkowski metric in pseudo-Cartesian coordinates. The construction of Ogen only guarantees that Ogen transforms as a tensor under arbitrary transformations in Minkowski spacetime. Einstein, however, expected and made an attempt to prove (at the top of p. 8L), that Ogen would transform as a tensor under arbitrary transformations for any metric field. Einstein applied the first of the two strategies described above to find non-autonomous transformation under which the core operator as a whole transforms as a tensor. He then applied the second strategy to generalize the two constituent operators of the core operator and thereby the core operator itself to expressions that transform as tensors under arbitrary transformations. He did not see these calculations through to the end. He came to realize that the generalized operators produced by this strategy degenerate when applied to the metric tensor. Einstein thereupon abandoned this second strategy altogether. The first strategy and the concept of non-autonomous transformations, however, continue to play an important role in the notebook. Covariance of the Core Operator under Non-autonomous Transformations (7L–R) Underneath the horizontal line on p. 7L, Einstein wrote down the core operator: ∑∂μν∂iκ∂xν∂xμ . For weak static fields represented by a metric tensor of the form diag1,1,1,c2x,y,z the 44-component of this operator reduces to minus the Laplacian acting on c2 , the square of the gravitational potential of Einstein’s 1912 theory. On pp. 7L–8R, Einstein studied the transformation properties of the core operator using transformation matrices pμν and μν , which are defined as follows. Under a transformation from coordinates xμ to x′μ , a contravariant vector aμ transforms as a′μ∑pμνaν with pμν∂x′μ∂xν , while a covariant vector transforms asNotice that the definitions of pμν and μν differ from the definitions of these quantities in (Einstein and Grossmann 1913, 24), where they are defined as pμν∂xμ∂x′ν and μν∂x′ν∂xμ . In other words, the roles of p and are interchanged as are the indices μ and ν . This is related to the fact that in the Zurich Notebook contravariant quantities (with some exceptions such as the contravariant components μν of the metric) are generally denoted by Latin letters and covariant quantities by Greek ones, whereas in (Einstein and Grossmann 1913) this is just the other way around. α′μ∑μναν with μν∂xν∂x′μ . The inverse transformation of a contravariant vector is given by: aμ∑vμa′ν with vμ∂xμ∂x′ν ; the inverse transformation of a covariant vector by: αμ∑pvμα′ν with pνμ∂x′ν∂xμ . It follows that ∑pμαναδμν and that ∑pαμανδμν , where we availed ourselves of the Kronecker delta, which Einstein does not use in the notebook. Einstein considered the transformation of the core operator from x′μ to xμ . Substituting ′μνpμαpνβαβ and ∂∂x′μμα∂∂xα intoExpression () is a concrete example of O∂′, g′ introduced in equation (). ∑μν∂′μν∂′iκ∂x′ν∂x′μ , one arrives at the equation written directly under expression () for the core operator in the notebook:There is also a summation over m and l . ∑αβμνρσμα∂pμρpνσρσνβ∂pilpκmlm∂xβ∂xα . The contraction of pνσ and νβ gives δβσ . Einstein also set μα∂∂xαpμρ�∂∂xρ… , tacitly making the erroneous assumption that μα∂pμρ∂xα0 , possibly on the basis of the following incorrect application of the chain rule:This error was committed repeatedly by Einstein. See, e.g., pp. 7R, 8L. It was eventually discovered on p. 10L (see sec. below). μα∂pμρ∂xα∂xα∂x′μ ∂2x′μ∂xα∂xρ∂2x′μ∂x′μ∂xρ0 . With these simplifications, Einstein arrived at the following expression for the core operator in x′μ -coordinates in terms of quantities in xμ -coordinates:The notebook has ∂∂ρ and ∂∂σ instead of ∂∂xρ and ∂∂xσ . ∑μν∂′μν∂′iκ∂x′ν∂x′μ∑∂ρσ∂pilpκmlm∂xσ∂xρ . To further investigate the covariance properties of the core operator, Einstein adopted the two-step procedure described in the introduction of sec. . The first step is to write down the law according to which the core operator would transform if it transformed as a tensor under the transformation from x′μ to xμ . Adopting Einstein’s notation αβ∑∂μν∂αβ∂xν∂xμ for the core operator, one can write this tensorial transformation law asEquations () and () form a concrete example of a combination of equations () and (): O˜∂, gT˜1O∂′, g′T˜1OT∂, Tg . αβ∑iακβ′iκ . The second step of the procedure is to rewrite ′iκ , the core operator in the primed coordinate system, in terms of quantities in the unprimed coordinate system. Inserting (the erroneous) expression () for ′iκ found earlier, one arrives at the equation given in the notebook at this point: αβ∑iακβ∂ρσ∂pilpκmlm∂xσ∂xρ . If αβ transforms as a tensor under the transformation from xμ to x′μ and vice versa, the right-hand side of this equation should be equal to the core operator ∑∂ρσ∂αβ∂xσ∂xρ . This is trivially true for linear transformations.For linear transformations, equation () reduces to: αβ∑iακβpilpκm∂∂xρρσ∂lm∂xσ∑δαlδβm∂∂xρρσ∂lm∂xσ∑∂ρσ∂αβ∂xσ∂xρ . Einstein wanted to find more general transformations for which the right-hand side of equation () reduces to expression (). For such transformations the sum of all terms on the right-hand side of equation () that involve derivatives of the components of the transformation matrix pμν should vanish. Einstein set out to collect such terms. First, he considered the differential operator ∂∂xσ in front of the innermost set of parentheses on the right-hand side of equation (). He rewrote this part of the equation as: pilpκm∂lm∂xσ+lm∂pilpκm∂xσ . Then he turned to the differential operator ∂∂xρ in front of the outermost set of parentheses on the right-hand side of equation (). First, using equation (), he wrote down the term that comes from having ∂∂xρ act on ρσ : ∂ρσ∂xρpilpκm∂lm∂xσ+lm∂pilpκm∂xσ . Finally, he wrote down the four terms that result from applying ∂∂xρ to expression () and contracting it with ρσ : ρσpilpκm∂2lm∂xρ∂xσ+ρσ∂lm∂xσ∂pilpκm∂xρ+ρσ∂lm∂xρ∂pilpκm∂xσ +ρσlm∂2∂xρ∂xσpilpκm The core operator αβ transforms as a tensor under transformations with transformation matrices pμν if the components of the transformation matrices satisfy the condition that the sum of all terms in expressions () and () that involve derivatives of pμν vanish for a given metric field with contravariant components μν . Collecting such terms, one arrives at the condition:This is a concrete example of condition () in the introduction to sec. . ∂ρσ∂xρlm∂pilpκm∂xσ+ρσ∂lm∂xσ∂pilpκm∂xρρσ∂lm∂xρ∂pilpκm∂xσ+ρσlm∂2∂xρ∂xσpilpκm0 This condition on the transformation matrices pμν is the condition for what Einstein would later call “non-autonomous” transformations under which the core operator transforms as a tensor (see the introduction to sec. ).This condition would still need to be corrected for the error made in going from equation () to equation (). Looking at this condition, one readily sees that it will be satisfied if ∂pilpκm∂xσ0 for all index combinations. Except for a meaningless summation sign, which Einstein seems to have left in by mistake, and a relabeling of the indices the left-hand side of this equation is just the expression at the top of p. 7R: ∑∂pαlpβm∂xσ . This expression is introduced with the comment: “where p is differentiated at least once. iα κβ ” (“wobei p mindestens einmal abgeleitet wird. iα κβ ”).To understand the change in the labeling of the indices—from i and k to α and β —note that the core operator (see equation ()) is obtained by contracting the sum of expressions () and () with iακβ . In first-order approximation, iα and κβ can be replaced by the Kronecker deltas δiα and δκβ . The restriction to terms in which “ p is differentiated at least once” identifies those terms in expressions () and () on p. 7L that must vanish if the core operator is to transform as a tensor under the coordinate transformation described by the matrix pμν . Unfortunately, condition () restricts the allowed transformations to linear transformations, whereas Einstein was looking for non-linear transformation under which the core operator transforms as a tensor. Einstein now drew a horizontal line and made a fresh start. He wrote down the contraction of the core operator with the covariant metric: ∑iκμνgiκ∂μν∂iκ∂xν∂xμ writing “scalar” (“Skalar”) next to it. The expression will transform as a scalar for transformations under which the core operator transforms as a tensor. Conversely, Einstein may have expected that transformations under which expression () transforms as a scalar are transformations under which the core operator transforms as a tensor. He may have felt that the former were easier to find than the latter. On the next line, however, Einstein returned once more to the condition derived on p. 7L for the transformation matrices pμν and μν (see equations ()–()). We can only make sense of the expression Einstein wrote down if we assume that he now focused on infinitesimal transformations. Consider the right-hand of equation (): ∑iακβ∂ρσ∂pilpκmlm∂xσ∂xρ . Following Einstein’s notation on p. 10L (see equation ()) for an infinitesimal transformation, pilδil+pilx , and neglecting terms smaller than those of first order in pilx , one can write the product in the innermost parentheses in expression () as δil+pilxδκm+pκmxlmiκ+pilxlκ+pκmxim Inserting this last expression into expression () and collecting all terms involving derivatives of the transformation matrices, one finds, to first order, iακβ∂ρσ∂pilxlκ+pκmxim∂xσ∂xρ Since only terms involving derivatives of pilx matter, iα and κβ can be replaced by Kronecker deltas in first-order approximation. This then gives ∂ρσ∂pαlxlβ+pβlxαl∂xσ∂xρ . If the core operator transforms as a tensor under the transformation described by pμν , the sum of all terms in expressions () that involve derivatives of the components of pμνx must vanish (cf. the discussion following equation ()). This condition can be satisfied by requiring that the sum of all terms in ∂ρσ∂pαlxlβ∂xσ∂xρ “in which p is differentiated at least once” (“wobei p mindestens einmal abgeleitet wird”) vanish. Einstein wrote down expression ()—albeit with pαl rather than with pαlx —with this remark next to it, which supports our reconstruction of the purpose behind it. The condition for infinitesimal transformations resulting from expression () is much simpler than condition () for finite transformations. Einstein’s calculations nonetheless break off at this point. Generalizing the Constituent Parts of the Core Operator: Divergence and Exterior Derivative Operators (7R–8R) Rather than continuing the search for non-autonomous non-linear coordinate transformations under which the core operator transforms as a tensor, Einstein, on pp. 7R–8R, tried to find a generalization of the core operator that would transform as a tensor under ordinary non-linear transformations. In other words, he switched from the first to the second of the two strategies that we distinguished in the introduction of sec. . He applied this strategy to the two constituent components of the core operator, the divergence and the exterior derivative. On p. 7R, under the heading “Divergence of a tensor” (“Divergenz des Tensors”), he tried to generalize the divergence operator. On pp. 8L-R, under the heading “Exterior derivative of a tensor” (“Erweiterung des Tensors”), he tried to generalize the exterior derivative operator. To generalize the divergence operator, Einstein started from the ordinary divergence of a second-rank tensor in primed pseudo-Cartesian coordinates on Minkowski spacetime and then wrote down the expression in unprimed arbitrary coordinates that one would get if the divergence in primed coordinates transformed as a tensor under this transformation. In the primed coordinates the Minkowski metric has the standard diagonal form. Since the components of this metric are constants, there will be a simple relation between the metric in unprimed coordinates and the matrices for the transformation between primed and unprimed coordinates. Using this relation, one can eliminate the transformation matrices from the expression for the divergence transformed from primed to unprimed coordinates as if it were a tensor. This expression will then be entirely in terms of components of the metric in unprimed coordinates and their derivatives. Einstein expected that this procedure would yield the generalized divergence operator that he had found on p. 5R (essentially the covariant divergence), but he was unable to prove this conjecture. These considerations begin beneath the second horizontal line on p. 7R. Under the heading “Divergence of the Tensor” (“Divergenz des Tensors”), Einstein wrote down the ordinary divergence of a second-rank contravariant tensor in a primed coordinate systemThe quantity a′μ is a concrete example O∂′, g′ in equation (). a′μ∑∂T′μν∂x′ν μ-Vektor . In the line above this equation, Einstein characterized the primed coordinate system with the remark: “original system ( ′ ) shall have constant g , .” (“Ursprüngliches System ( ′ ) habe konstante g , .”). Presumably, what he had in mind was a pseudo-Cartesian coordinate system on Minkowski spacetime. Einstein now applied the same two-step procedure that we encountered on p. 7L. First, he wrote down how a′μ would transform if it were to transform as a vector under transformations from the special primed to arbitrary unprimed coordinates. He then took expression () for the vector in primed coordinates and used the standard transformation rules to rewrite the building blocks of this vector in terms of their counterparts in the unprimed coordinate system. He thus arrived at the following expression for the vector aμ “in the unprimed system” (“Im un­gestrichenen System”):Equation () is a concrete example of a combination of equations () and (): O˜∂, gT˜1O∂′, g′T˜1OT∂, Tg . aσ∑μσa′μ∑μσντ∂∂xτpμipνκTiκ . He simplified the right-hand side of equation (), using the same erroneous relation, …ντ∂∂xτ…pνκ……ντpνκ∂∂xτ……∂∂xκ… , that he had used earlier (cf. equations ()–()). In this way he obtainedCorrecting Einstein’s mistake in going from equation () to equation (), one finds: aσ∑μσντ∂∂xτpμipνκTiκ∂∂xτTστ+Tiτμσ∂pμi∂xτ+Tσκντ∂pνκ∂xτ . This is a concrete example of equation (). The first term on the right-hand side corresponds to O∂, g ; the last two terms to C∂, g, p, . aσ∑μσ∂∂xκpμiTiκ∑∂Tσκ∂xκ+∑μiκTiκμσ∂pμi∂xκ . Einstein now indicated how he wanted to generalize the ordinary divergence operator: “This sum is to be expressed by the g resp.  . In doing so one has to use the fact that the primed g and are constant.” (“Diese Summe ist durch die g bzw. auszudrükken. Dabei ist zu benutzen, dass die gestrichenen g bzw. konstant sind.”). In other words, using the simple form of the metric in the special primed coordinates, he wanted to express the components of the transformation matrices and their derivatives in the last term of equation () in terms of the components of the metric and their derivatives in the arbitrary unprimed coordinates. The final step producing the sought-after generalization of the divergence operator is to replace the components of the Minkowski metric in the unprimed coordinates in the resulting expression by components of an arbitrary metric. Underneath the sentence explaining the aim of his calculation, indicating the connection to the last term of equation () by a vertical line, Einstein wrote: ∑Tiκ∂gσκ∂xi-12∂giκ∂xσ . He thus expected that aσ could be written as: aσ∑∂Tσκ∂xκ+∑Tiκ∂gσκ∂xi-12∂giκ∂xσ . One sees immediately that this cannot be correct since the right-hand side is a sum of a contravariant and a covariant term. Einstein’s expectation derives from his experience with the covariant divergence of the stress-energy tensor on p. 5R. If in equation () found on p. 5R one sets G1 , its left-hand side reduces to: ∑∂gmνTνn∂xn-12∑∂∂xmgμνTμν . Relabeling indices and using that the stress-energy tensor Tμν is symmetric,At the beginning of the calculation on p. 7R (see equation ()), Tμν was an arbitrary tensor. At this point in the calculation, however, it becomes essential that Tμν is symmetric. one can rewrite expression () as: ∑∂gσκTκi∂xi-12∑∂∂xσgiκTiκ∑gσκ∂∂xiTiκ+∑Tiκ∂gσκ∂xi-12∂giκ∂xσ . Comparing this last expression with Einstein’s expectation for the form of a generalized divergence operator in equation (), one easily recognizes the basic problem. He expected the generalized divergence of a contravariant tensor to be a contravariant vector, whereas the operator he had found on p. 5R turns a contravariant tensor into a covariant vector (see expression ()). Einstein did not recognize the problem at first and tried to show that equation () for aσ can indeed be rewritten as equation (). This can be inferred from the last two lines on p. 7R. He tried to show that ∂gσκ∂xi-12∂giκ∂xσ in equation () is equal toSince μσ∂xσ∂x′μ , σ is a contravariant index. Expression () can thus never be equal to expression () in which σ is a covariant index. μσ∂pμi∂xκ in equation (). To this end he eliminated the metric in unprimed coordinates in expression () in favor of the metric in primed coordinates, whose components are constants: ∑∂pμσpνκg′μν∂xi-12∂∂xσpμipνκg′μν . Finally, he wrote down one contribution coming from the second term in expression (), the one involving a derivative of pμi , which also occurs in expression (): 12∂pμi∂xσpνκg′μν . At this point the calculation breaks off. The method by which Einstein tried to generalize the ordinary divergence operator on p. 7R does not guarantee that the result will be a tensor. The main problem comes from the final step in which an expression derived for one special metric, the Minkowski metric, is assumed to transform as a tensor for an arbitrary metric (at least under unimodular transformations because of the restriction to G1 ). The deleted calculation at the top of p. 8L may have been an attempt to prove that the method employed on p. 7R actually does produce an object that transforms as a tensor under arbitrary coordinate transformations. If that is indeed the purpose of this calculation, the strategy chosen by Einstein is clear. He tried to prove that a transformation from arbitrary unprimed coordinates to arbitrary double-primed coordinates can be decomposed into two transformations of the type considered on p. 7R, namely a transformation from the unprimed coordinates to special primed coordinates followed by a transformation from these special primed coordinates to the double-primed coordinates. Since any metric can locally be transformed to a Minkowski metric in standard diagonal form, the form of the metric in the special primed coordinates, this would guarantee that the method of p. 7R does indeed produce a tensor. These considerations may lie behind Einstein’s question at the top of p. 8L, “Do symmetrical transformations form a group?” (“Haben symmetrische Transformationen Gruppeneigenschaft?”), and behind the subsequent investigation of a transformation from unprimed to primed to double-primed coordinates. What remains unclear, however, is why Einstein focused on symmetric transformations. To determine whether symmetric transformations given by x′ν∑σpνσxσ , with pνσpσν form a group, Einstein considered the components p″λσ∑p′λνpνσ of the matrix for the composite transformation x″λ∑νp′λνx′ν∑νσp′λνpνσxσ and switched the indices of these components: p″σλ∑νp′σνpνλ . He did not pursue this calculation any further. Either he concluded (correctly) that symmetric transformations do not form a group or he did not see how to settle the question either way. He now turned to rotational transformations, perhaps as an example of a class of transformations that certainly do form a group, or to check whether such transformations are symmetric. He wrote down transformation laws for a rotation over an angle α as well as for its inverse: x′xα+yαy′-xα+yαxx′α-y′αyx′α+y′α These expressions show that rotation does not belong to the class of symmetric transformations. Perhaps this is why he deleted the calculation at the top of p. 7R. Another possibility is that he realized that the restriction to symmetric transformations was not necessary to show that the method of p. 7R produces a tensor. Under the heading “Exterior derivative of the Tensor” (“Erweiterung des Tensors”), Einstein now turned to the second differential operator relevant to generalizing the core operator. This takes up the remainder of p. 8L and the first two lines on p. 8R. As in the case of generalizing the divergence operator, Einstein’s starting point is the ordinary exterior derivative of a contravariant second-rank tensor in special relativity. He wrote: “In ordinary space [i.e., Minkowski spacetime in pseudo-Cartesian coordinates]The quantity Δ′μνρ is a concrete example of O∂′, g′ in equation (). Δ′μνρ∂T′μν∂x′ρ , is a tensor of three manifolds [i.e., of third rank]” (“Im gew. [gewöhnlichen] Raum ist … Tensor von 3 Mannigfaltigkeiten”). He then “introduced transformations with constant coefficients” (“Subst[itutionen] von konst. Koeffizienten eingeführt”) and confirmed that “for such transformations ∂Tαβ∂xσ is also a tensor” (“Für solche Transformationen ist ∂Tαβ∂xσ auch Tensor”). As indicated by the arrows in equation (), this is because the components pμν of the transformation matrix can put in front of the derivative operator ∂∂xσ . Einstein now tried to generalize the tensor Δ′μνρ . He started by asking the question: “What is this tensor called when arbitrary substitutions are admitted?” (“Wie heisst dieser Tensor, wenn bel[iebige] Subst[itutionen] zugelassen werden?”). Clearly he was not familiar with the notion of a covariant derivative at this point. Einstein used the same two-step procedure that he had used on p. 7R (see equation ()). First, he wrote down how Δ′μνρ would transform if it transformed as a tensor under transformations from special primed to arbitrary unprimed coordinates. He then took expression () for Δ′μνρ and used the standard transformation rules to rewrite its building blocks in terms of their counterparts in unprimed coordinates. He thus arrived at the following expression for the tensor Δμνρ in unprimed coordinatesEquation () is a concrete example of a combination of equations () and (): O˜∂, gT˜1O∂′, g′T˜1OT∂, Tg . Δμνρ∑μνρmμnνprρΔ′mnr∑mμnνprρrα∂pmδpnεTδε∂xα Einstein initially wrote ρα but eventually corrected it to rα . The right-hand side of equation () gives a sum of three terms:Expression () is a concrete example of the right-hand side of equation (): O∂, g+C∂, g, p, . ∂Tμν∂xρ+∑mμnνprρrα∂pnε∂xαpmδTδε+∑mμnνprρrαpnε∂pmδ∂xαTδε . Using that mμpmδδμδ , Einstein simplified the second term, still with ρα instead of rα and omitting Tδε : nν∂pnε∂xαραprρ . This expression does not allow for further simplification. This may be what drew Einstein’s attention to the error in equation (), which he then corrected. The correct expressions ()-() can be simplified further. This yields: ∂Tμν∂xρ+∑nν∂pnε∂xρTμε+∑mμ∂pmδ∂xρTδν . To generalize the exterior derivative of Tμν , one proceeds in the same way as in generalizing the divergence of Tμν on the basis of expression (). Using that the metric in primed coordinates is the Minkowski metric in standard diagonal form, one first expresses the components of the transformation matrices and their derivatives in the last two terms of equation () in terms of the components of the Minkowski metric and their derivatives in the arbitrary unprimed coordinates. In the resulting expression, one then substitutes the components of an arbitrary metric for the components of the Minkowski metric in unprimed coordinates. At the top of p. 8R, Einstein tried to find the relation between the transformation matrices and the Minkowski metric in the unprimed coordinates. This relation is given by the first equation on p. 8R: ∂gρσ∂xτ∑βαgρβαβ∂pασ∂xτ+∑αβgασβα∂pβρ∂xτ . Although the derivation of this equation is not recorded in the notebook, it is easily reconstructed. It is basically the same calculation as in equations ()–(), only for gμν instead of Tμν .The starting point of the derivation is the observation that in the primed coordinates in which the Minkowski metric takes on its standard diagonal form, the exterior derivative of the metric vanishes: ∂g′αβ∂x′0 . The contraction of the left-hand side of this equation with the transformation matrices pαρpβσpτ obviously still vanishes: ∑pαρpβσpτ∂g′αβ∂x′0 . Expressing the primed quantities in terms of their unprimed counterparts, one finds that ∑pαρpβσpτλ∂αμβνgμν∂xλ0 , which can be rewritten as ∂gρσ∂xτ+pβσ∂βν∂xτgρν+pαρ∂αμ∂xτgμσ0 . Since pβσβνδσν and, consequently, 0∂pβσβν∂xμpβσ∂βν∂xμ+βν∂pβσ∂xμ , equation () is equivalent to ∂gρσ∂xτ-gρνβν∂pβσ∂xτ-gμσαμ∂pαρ∂xτ0 . Bringing the second and the third term to the right-hand side and relabeling indices ( νβ→βα in the second term, μα→αβ in the third term), one finds an expression for the exterior derivative of the covariant metric, ∂gρσ∂xτgρβαβ∂pασ∂xτ+gασβα∂pβρ∂xτ , which is just equation () given at the top of p. 8R. Underneath this equation, Einstein wrote down a similar equation for the contravariant metric:It looks as if Einstein first started to write down equation () next to equation () rather than underneath it. ∂μν∂xσ∑αβμαpβα∂βν∂xσ+∑αβανpβα∂βμ∂xσ . The derivation of this equation is fully analogous to the derivation of equation (). The only difference is that the starting point is now ∂′αβ∂x′0 rather than ∂g′αβ∂x′0 .Equation () can also be obtained by substituting the contravariant metric μν for the contravariant tensor Tμν in equation (), using equation () and the fact that ∂′αβ∂x′0 and by relabeling indices. Einstein did not proceed any further. With hindsight, however, knowing that the generalization that Einstein was looking for is just the covariant derivative, one can easily complete his chain of reasoning. With the help of equation () and two equations like it with different permutations of the indices ρ , σ , and τ , one can show that the relation between the transformation matrices and the Minkowski metric in the unprimed coordinates that Einstein was looking for is given by αβ∂pασ∂xτβστ12βα∂gασ∂xτ+∂gατ∂xσ-∂gστ∂xρ , where the curly brackets represent the Christoffel symbols of the second kind. This can easily be verified by inserting equation () back into equation ().Inserting equation () into the right-hand side of equation (), one recovers the left-hand side: ∑βαgρβαβ∂pασ∂xτ+∑αβgασβα∂pβρ∂xτ = ∑gρββστ+∑gασαρτ = ∂gρσ∂xτ . Inserting equation () into expression (), one finds the generally-covariant analogue of the exterior derivative of the contravariant tensor Tμν :The final expressions in equations () and () are concrete examples of the quantities Ogen∂, gO∂, g+D∂, g defined in equation (). ∂Tμν∂xρ+∑nν∂pnε∂xρTμε+∑mμ∂pmδ∂xρTδν=∂Tμν∂xρ+∑νερTμε+∑μδρTδν Inserting equation () into the right-hand side of the equation in footnote 107, one finds the covariant divergence of Tμν : ∂∂xτTστ+μσ∂pμi∂xτTiτ+ντ∂pνκ∂xτTσκ∂∂xτTστ+σiτTiτ+τκτTσκ . These last two equations show how close Einstein came to finding the correct generally-covariant generalizations of the two constituents of the core operator, the divergence and the exterior derivative, using the second one of the two strategies that we distinguished in the introduction of sec. . Why did Einstein not pursue this calculation beyond the first two lines on p. 8R? It seems unlikely that the complexity of having to solve equation () for αβ∂pασ∂xτ would have deterred him. On pp. 14R–18R, we shall see Einstein pursue far more cumbersome calculations with great tenacity. A more plausible answer is that Einstein realized at this point, if not earlier, that the generalization of the exterior derivative he was in the process of constructing cannot be used to build a generalized core operator that could serve as the left-hand side of the gravitational field equations. The problem is that the exterior derivative of the metric vanishes. On p. 5R, Einstein had already found that the covariant divergence of the metric vanishes, but that does not mean that the generalization of the core operator, which is essentially the divergence of the exterior derivative of the metric, vanishes. If the exterior derivative of the metric vanishes, however, the core operator vanishes as well. Equation (), the second equation on p. 8R, is, in fact, the statement that the covariant exterior derivative of the contravariant metric vanishes. This can be seen as follows. From equations () and () it follows that pβσ∂βν∂xμβν∂pβσ∂xμνσμ . Substituting this expression into equation (), one finds, ∂μν∂xσ-∑μανασ-∑ανμασ Comparison with equation () shows that equation () expresses the vanishing of the covariant exterior derivative of the metric.When Grossmann introduced the Christoffel symbols in his part of the Entwurf paper, he added a footnote saying: “On the basis of these formulae one can easily prove that the exterior derivative of the fundamental tensor vanishes identically” (“Auf Grund dieser Formeln beweist man leicht, dass die Erweiterung des Fundamentaltensors identisch verschwindet,” Einstein and Grossmann 1913, part 2, sec. 2). Einstein did not have to find the relation between the transformation matrices and the metric and rewrite equation () in the form of equation () to see that the generalization of the exterior derivative acting on the metric would vanish. It is, in fact, a direct consequence of the method that Einstein used to construct this generalization. Whatever the exact form of the sought-after operator acting on the metric in the arbitrary unprimed coordinates, its form in the special primed coordinates used to construct it is ∂′μν∂x′ρ (cf. equation ()). The primed coordinates were chosen in such a way that the metric—be it the Minkowski metric or an arbitrary metric—takes on the form of the standard diagonal Minkowski metric in these coordinates. So the generalized exterior derivative of the metric vanishes in the special primed coordinates. Now this quantity was constructed to transform as a tensor under arbitrary coordinate transformations. So it will vanish in all coordinate systems.In the Entwurf paper, Einstein mentioned this problem as one of the obstacles to formulating generally-covariant field equations: “These operations [i.e., the divergence and the exterior derivative operators] degenerate if they are applied to the fundamental tensor gμν . From this it seems to follow that the equations sought will be covariant only with respect to a particular group of transformations, which for the time being, however, is unknown to us” (“Aber es degenerieren diese Operationen, wenn sie an dem Fundamentaltensor gμν ausgeführt werden. Es scheint daraus hervorzugehen, daß die gesuchten Gleichungen nur bezüglich einer gewissen Gruppe von Transformationen kovariant sein werden, welche Gruppe uns aber vorläufig unbekannt ist.” Einstein and Grossmann 1913, part 1, sec. 5). At this point, Einstein had no choice but to abandon his second strategy for finding field equations on the basis of the core operator: generalizing this operator to an expression that transforms as a tensor under arbitrary (autonomous) coordinate transformations does not work. He returned to the first strategy of finding non-autonomous non-linear transformations under which the core operator itself—if necessary with correction terms—transforms as a tensor. Trying to Extract Field Equations and a Gravitational Stress-Energy Tensor from the Beltrami Invariants (8R–9R) On pp. 7L-8R Einstein had examined the transformation properties of candidate field equations based on the core operator, the natural analogue of the Poisson equation in a theory in which the gravitational potential is represented by a tensor rather than a scalar. Einstein’s approach on these pages had thus been along the lines of what we call the physical strategy. On pp. 8R–9R, Einstein returned to the mathematical strategy, more specifically to the exploration of the Beltrami invariants introduced on p. 6L. He tried to extract the core operator from the second Beltrami invariant, using (some power of) the determinant of the metric as the arbitrary scalar function in the definition of this invariant. The connection between the core operator and the Beltrami invariant might throw light on the covariance properties of the former. Einstein’s investigation of the covariance properties of the core operator on pp. 7L–R had remained inconclusive. On p. 8R, Einstein returned to the basic expression for the first and the second Beltrami invariants. The first Beltrami invariant can be used to find a candidate for the quantity representing gravitational energy-momentum, the second to find a candidate for the left-hand side of the field equations. On p. 9L Einstein managed to write the second Beltrami invariant as a sum of two contributions, the first of which is the contraction of the metric with the core operator. On the bottom half of p. 9L, Einstein tried to relate the second contribution to gravitational energy-momentum. This is the first time in the notebook that Einstein, drawing on his experience with the 1912 static theory,In (Einstein 1912b, sec. 4), it was pointed out that the field equation originally proposed for the theory for static gravitational fields, Δckcσ (where the variable speed of light c doubles as the gravitational potential, k is a constant, and σ is the mass density), is in conflict with the action-equals-reaction principle and thereby with energy-momentum conservation. Einstein remedied the problem         by adding the gravitational energy density to the source term: Δckcσ+12kgrad2cc . introduces the notion that gravitational energy-momentum should enter the field equations on equal footing with the energy-momentum of matter. Expecting gravitational energy-momentum to be represented by a generally-covariant tensor, Einstein turned to the first Beltrami invariant to find a candidate for the stress-energy tensor of the gravitational field. The contraction of the metric with this supposed gravitational stress-energy tensor, however, turns out to be slightly different from the second contribution to the expression for the second Beltrami invariant, and Einstein abandoned the idea of interpreting this contribution in terms of gravitational energy-momentum. On p. 9R Einstein tried to find the infinitesimal non-autonomous transformations under which this contribution to the second Beltrami invariant transforms as a scalar. The first contribution, the contraction of the metric and the core operator, would then be the difference between two scalars (for this restricted class of transformations) and therefore be such a scalar itself. This in turn would suggest that the core operator transform as a tensor under these transformations. Since the rationale behind the return to the Beltrami invariants on p. 8R was presumably to avoid non-autonomous transformations, which Einstein had found difficult to handle (see pp. 7L–8R), it is not surprising that the Beltrami invariants no longer explicitly appear in the notebook after these calculations on pp. 8R–9R. Einstein, however, continued to use the restriction to unimodularity in his calculations on the following pages. This suggests that he still hoped to find some connection between the field equations and the Beltrami invariants. Underneath the first horizontal line on p. 8R, Einstein substituted (some power α of) the determinant G of the metric for the arbitrary function ϕ in the definition of the first and the second Beltrami invariants (see equations () and (), respectively): ϕ1μν∂G∂xμ∂G∂xν , ϕ21G∂∂xμμνG∂Gα∂xν . Since G is a scalar only under unimodular transformations, the Beltrami invariants ϕ1 and ϕ2 above are no longer generally-covariant scalars but invariants under this restricted class of transformations only. To establish the connection with the core operator, ∑∂μν∂iκ∂xν∂xμ (see equation ()), Einstein rewrote ∂G∂xν as ∂G∂xν∑iκ∂giκ∂xνGiκ∑G∂giκ∂xνiκG∑giκ∂iκ∂xν , where Giκ is the minor of giκ . In the second step, Einstein used that iκGiκG , and in the third step that ∂∂xν∑giκiκ0 . Next to this expression he wrote “of zeroth power” (“nullter Potenz”).This is the only place in the notebook where the word “Potenz” occurs. Einstein only wrote “Potenz” after writing and deleting first “of zeroth kind” (“nullter Art”) and then “of zeroth order” (“nullter Ordnung”). At the top of p. 9L he switched back to “zeroth order” (“nullter Ordnung”). In (Bianchi 1910, Ch. II, sec. 22), the “order” (“Ordnung”) of a “differential parameter” (“Differentialparamet-er”)—i.e., an expression constructed out of the metric and its derivatives and a number of arbitrary functions and their derivatives—is defined as the highest-order derivative of the arbitrary functions occurring in it. Since there are no arbitrary functions in ϕ1 and ϕ2 in equations () and (), these quantities are of zeroth order in this sense. Inserting the first expression for ∂G∂xν in equation () into equation (), Einstein arrived at: ϕ1∑iκlmμνμν∂∂xμgiκ ∂∂xνglmGiκGlm , “or” (“oder”), as he wrote next to it, using the second expression for ∂G∂xν in equation (), ∑μν∂∂xμgiκ ∂∂xνglmiκlm . A factor G2 is omitted in this last expression. This is inconsequential: since both ϕ1 and G are scalars under unimodular transformations, ϕ1G2 is too. After drawing a horizontal line, Einstein turned to the second Beltrami invariant, ϕ2∑∂Gμν∂Gα∂xν∂xμ . Once again, he omitted an inconsequential factor G (cf. equation ()).The expression written next to equation (), ∂∂xμGμν∂ψ∂xν , is likewise a scalar under unimodular transformations. With the help of equation (), Einstein rewrote this as α∑∂∂xμGμνGα-1 ∂∂xνgiκGiκ≈∑∂∂xμGα+12μνiκ∂∂xνgiκ . As indicated by the proportionality sign, the overall factor α is dropped from the expression. Einstein now set α12 and transferred the differentiation operator ∂∂xν from the covariant to the contravariant metric using that ∂∂xν∑giκiκ0. The resulting expression contains the core operator: ∑∂μνiκ∂giκ∂xν∂xμ∑∂μνgiκ∂iκ∂xν∂xμ . At the bottom of the page, Einstein briefly turned to the first Beltrami invariant and wrote down a “Different expression for the above scalar ϕ1 ” (“Anderer Ausdruck für obigen Skalar ϕ1 ”): ∑gμμ′ μν∂G∂xν μ′ν′∂G∂xν' (which is indeed equivalent to equation () since gμμ′ μνμ′ν′νν′ ). The phrase “different expression for ϕ1 ” (“Anderer Ausdruck für ϕ1 ”) is repeated two lines farther down. Einstein returned to this expression on p. 9L to extract a candidate for the stress-energy tensor of the gravitational field. Einstein drew a horizontal line and copied the final expression for ϕ2 from equation (), ∑∂∂xμGα+12μνiκ∂∂xνgiκ . Underneath this expression he began to evaluate one of the derivatives in expression (), ∂G∂xν∑giκ∂Giκ∂xν , but then deleted the erroneous right-hand side. Einstein continued his investigation of the Beltrami invariants on p. 9L. He began by rewriting the first Beltrami invariant () modulo a factor G2 . Starting from expression () on p. 8R, he transferred the derivative operators from the covariant to the contravariant metric, using that ∂∂xμ∑giκiκ0 . He thus arrived at:The quantity ϕ1 in equation () needs to be multiplied by G2 to obtain the first Beltrami invariant ϕ1 as defined in equation (). See footnote for a discussion of the term “zeroth order” (“nullter Ordnung”) written next to equation (). ϕ1∑giκglmμν∂∂xμiκ∂∂xνlm . He similarly rewrote expression () for the second Beltrami invariant, setting α12 as he had done on p. 8R: . Einstein expanded this expression to . The first term is the core operator () contracted with the covariant metric. The second term required further attention. Einstein tried to use the relation 0∑∂∂xμgiκiκ+giκ∂∂xμiκ to deal with it. Differentiating this relation, he did indeed recover, up to a contraction with μν , the second term on the right-hand side of equation () but at the price of introducing three other terms: 0∑∂2giκ∂xμ∂xνiκ+∂∂xμgiκ∂∂xνiκ+.+giκ∂2iκ∂xμ∂xν Rather than rewriting the second Beltrami invariant with the help of equation (), Einstein tried to interpret the second term in equation () for ϕ2 with the help of the first Beltrami invariant modulo a factor G2 , i.e., the quantity ϕ1 at the top of p. 9L (see equation ()). In the passage in the middle of p. 9L, set off by two horizontal lines, Einstein tried to interpret the difference between ϕ2 and the core operator contracted with giκ as minus a candidate gravitational stress-energy tensor contracted with giκ . As was explained in the introduction to this subsection, the second Beltrami invariant would then in all likelihoodIn all likelihood, because it obviously cannot be ruled out that the expression is not a tensor even though its contraction with the metric tensor is a scalar. Its contraction with an arbitrary second-rank tensor might not be. yield a candidate for the left-hand side of the field equations which is (i) a tensor under unimodular transformations and (ii) equal to the core operator minus the gravitational stress-energy tensor. Setting this candidate equal to the stress-energy tensor for matter, one sees that such field equations satisfy the requirement that all energy-momentum enters the field equations on the same footing. Einstein naturally assumed at this point that gravitational energy-momentum like the energy-momentum of matter would be represented by a tensor.(Einstein and Grossmann 1914, 218, footnote 1) is the first place where Einstein explicitly stated in print that the assumption that gravitational energy-momentum can be represented by a tensor is erroneous. In this footnote he identified this assumption as the flaw in an argument in (Einstein 1914a) that appeared to restrict the covariance of the Entwurf field equations to linear transformations. For discussion of this argument and its flaws, see (Norton 1984, sec. 5), “What Did Einstein Know …” sec. 2 and “Untying the Knot …” sec. 3.3 (both in this volume). By analogy to both the stress-energy tensor for the electromagnetic field and the one for the gravitational field of his 1912 static theory,(Einstein 1912b, 456–457). he furthermore expected the gravitational stress-energy tensor to be quadratic in first-order derivatives of the metric. Any such object, transforming as a tensor at least under unimodular transformations, would have to be constructed out of the first Beltrami invariant, multiplied possibly by other scalars under unimodular transformations such as the determinant G of the metric. The quantity ϕ1 in equation () is such an object. Moreover, ϕ1 looks very similar to the term in ϕ2 to be written as the contraction of giκ with the gravitational stress-energy tensor.Using ∂giκ∂xμgilgκm∂lm∂xμ , one can rewrite the second contribution to ϕ2 in equation () as ∑μν∂giκ∂xμ∂∂xνiκ∑μνgilgκm∂lm∂xμ∂∂xνiκ , which closely resembles the expression for ϕ1 in equation (). Einstein thus tried to extract a gravitational stress-energy tensor from ϕ1 . Einstein began this attempt with expression () on p. 8R. Dividing this equation by G2 , one arrives at the first equation in the passage set off between two horizontal lines on the bottom half of p. 9L: ϕ11G2∑gμμ′ μν∂G∂xν μ′ν′∂G∂xν' . The “presumable gravitation tensor” (“vermutlicher Gravitationstensor”) that can be read off from this expression is: 1G2∑νν′μν∂∂xνGμ′ν′∂∂xν′G∑νν′ii′κκ′μν∂∂xνgiκiκμ′ν′∂∂xν′gi′κ′i′κ′ . (In the second step, Einstein used equation ().) For the remainder of the argument on p. 9L, however, Einstein used the expressionOn p. 8R, Einstein had written the second Beltrami invariant in terms of this function ψ (see footnote 121). ∑μνμ′ν′∂∂xνψ∂∂xν′ψ , defining ψ through lgGψ . He drew a line from expression () to the expression on the left-hand side of equation () and encircled these two expressions. On the next line, he explicitly made the claim that this choice of a gravitational stress-energy tensor is unique: “[This] is the only tensor in which we differentiate only once” (“Ist der einzige Tensor, in dem nur einmal diff[erenziert] wird”). To see whether his gravitational stress-energy tensor () would be acceptable from a physical point of view as well, Einstein, writing “divergence calculated” (“Divergenz gebildet”), substituted it for the stress-energy tensor Tμν of matter in equation (), ∑νn∂GgmνTνn∂xn+12G∑∂∂xmgμνTμν0 , for the energy-momentum balance between matter and gravitational field. On p. 5R, Einstein derived this equation for the stress-energy tensor of pressureless dust. He then postulated the same equation for the stress-energy tensor of any matter. Gravitational energy-momentum, however, plays a special role and one cannot simply substitute the gravitational stress-energy tensor for Tμν in equation ().Einstein subsequently recognized that gravitational energy-momentum cannot be handled in the same way as the energy-momentum of matter. On p. 13R, he took a first step in this direction (see sec. , especially the discussion following equation ()). On p. 19R, he had essentially arrived at the treatment of gravitational energy-momentum that he would use and defend in the ensuing years (see the discussion following equation () in sec. ; see also pp. 20L, 21L, 24R–26R). On p. 9L, Einstein did not recognize the special status of gravitational energy-momentum and demanded that expression (), his candidate for a gravitational stress-energy tensor, like any other stress-energy tensor Tμν , satisfy the energy-momentum balance equation (): ∑∂Ggmμμνμ′ν′∂∂xνψ∂∂xν′ψ∂xμ′12∑G∂gμμ′∂xmμνμ′ν′∂∂xνψ∂∂xν′ψ . Einstein began to simplify both sides of this equation. Using (in slightly modernized notation involving the Kronecker delta) ∑gmμμνδmν , he wrote the left-hand side as: ∑∂Gμ′ν′∂∂xmψ∂∂xν′ψ∂xμ′ He used that ∂gμμ′∂xmμνμ′ν′gμμ′ ∂∂xmμνμ′ν′ , and that ∑gμμ′μ′ν′δμν to rewrite the right-hand side of equation () as 12∑G∂νν′∂xm∂∂xνψ∂∂xν′ψ . At this point, Einstein abandoned the ill-conceived condition (). He also abandoned his attempt to interpret the second term in equation () for the Beltrami invariant ϕ2 as the contraction of giκ with the gravitational stress-energy tensor () extracted from ϕ1 . Perhaps Einstein had come to realize these two expressions are not quite the same.The former can be written as ∑glmμν∂∂xμil∂κm∂xν , the latter as ∑glmμν∂∂xμiκ∂lm∂xν (cf. footnote 126). However, he retained the notion that the left-hand side of the field equations be the sum of two terms, each transforming as a tensor, one term being the core operator, the other term representing gravitational energy-momentum.Exploiting this general feature, Einstein developed a strategy for finding field equations compatible with the conservation principle. An embryonic version of this strategy can be found on p. 13R. On pp. 26L–R, he used the mature version of this strategy to find the Entwurf field equations. At the bottom of p. 9L and the top of p. 9R, Einstein returned to the investigation of the covariance properties of the core operator through its relation with the Beltrami invariant ϕ2 in equation ().At the bottom of p. 9L, Einstein wrote ϕ2∑ , drew and then deleted a line connecting this incomplete expression to equation () in the middle of the page. A completed version of the expression occurs at the top of p. 9R. The final pair of equations on p. 9L, a′μpμνaν and α′μμναν , give the transformation laws for contravariant and covariant vectors, respectively (see equations () and ()). The second term in expression () for ϕ2 , which Einstein had tried to interpret as a correction term to the core operator representing gravitational energy-momentum, now had to be dealt with in a new way.From a modern perspective the question arises why Einstein continued to concentrate on the covariance properties of the core operator itself rather than on the covariance properties of the sum of the core operator and the gravitational stress-energy (pseudo-)tensor. The answer to this question is that Einstein tacitly assumed (see footnote ) that both terms of this sum would separately transform as a tensor. At the top of p. 9R, Einstein copied equation () for the second Beltrami invariant ϕ2 (see the line connecting the two expressions for ϕ2 on pp. 9L–R) leaving the exponent α undetermined rather than setting α12 : ϕ2∑∂Gα+12giκμν∂∂xνiκ∂xμ . In the top right corner of p. 9R, using equation () for ∂G∂xν , Einstein calculated the derivative of Gα+12 ∂Gα+12∂xμα+12Gα+12∑∂gρσ∂xμρσ . With the help of this relation, ϕ2 can be rewritten as:The factor α+12Gα+12 in front of the summation sign was added later. In the second term ∂∂x should be ∂∂xμ . . As is indicated by the grouping of the factors in the expression under the first summation sign, this summation can be rewritten as (cf. equation ()) ∑giκμν∂∂xνiκρσ∂∂xμgρσ1G2∑μν∂G∂xν∂G∂xν . As Einstein noted, this means that the first term on the right-hand side of equation () is a scalar: “Obvious because ∂G∂xν vector of the second kind [i.e., a covariant vector]” (“Selbstverst[ändlich] weil … Vektor zweiter Art”). The second term on the right-hand side of equation () can be written as Gα+12 times the sum of two terms (see equation ()): the core operator () contracted with gik and the term μν∂giκ∂xμ∂iκ∂xν , which Einstein had tried in vain on p. 9L to write as the contraction of the metric with a gravitational stress-energy tensor. On the remainder of p. 9R Einstein investigated under which (infinitesimal) unimodular non-autonomous transformations expression () would be invariant. Under such a restricted class of transformations the core operator contracted with giκ and multiplied by Gα+12 would also be a scalar since, by virtue of equations (), () and (), it is equal to Gα+12giκiκϕ2-α+12Gα-32∑μν∂G∂xν∂G∂xν-Gα+12∑μν∂giκ∂xμ∂iκ∂xν , where we used Einstein’s notation iκ of p. 7L for the core operator (see equation ()). The first two terms on the right-hand side transform as scalars under arbitrary unimodular transformations. The left-hand side will thus transform as a scalar under the restricted class of unimodular transformations under which the third term on the right-hand side transforms as a scalar. Presumably, the core operator would then transform as a tensor under these transformations. Einstein thus set out to determine exactly how the class of unimodular transformations would have to be further restricted. As he wrote on the second line below equation (): “Substitutions must be restricted more” (“Subst[itutionen] müssen mehr eingeschränkt werden”).The calculation on p. 9R thus provides the first example of a strategy that Einstein routinely availed himself of in investigating the covariance properties of candidate field equations extracted from the Riemann tensor (cf. the discussion in sec. ). Rather than trying to find the class of non-autonomous transformations under which the candidate field equations themselves transform as a tensor, he tried to find the class of non-autonomous transformations under which the coordinate restriction with the help of which these field equations could be constructed out of an object of broad covariance transformed as a tensor. In this case that coordinate restriction is to transformations leaving expression () invariant. To find the condition for non-autonomous transformations leaving expression () invariant, Einstein used the usual two-step procedure.See the introduction to sec.  for a discussion of how in general one finds the condition for non-autonomous transformations (i.e., transformations for which the transformation matrices depend on the metric and its derivatives) under which a given expression transforming as a tensor under (ordinary) linear transformations retains such transformation behavior under non-linear transformations. He started with the expression in primed coordinates ∑′μν∂∂x′μg′iκ∂∂x′ν′iκ . Using the transformation laws for the various ingredients of this expression (see equations () and () repeated at the bottom of p. 9L), he wrote expression () in terms of quantities in unprimed coordinates: ∑′μν∂∂x′μg′iκ∂∂x′ν′iκ∑pμαpνβαβμlνm∂iδκεgδε∂xl∂piδ′pκε′δ′ε′∂xm , The right-hand side can be simplified by using that the matrices p and are each other’s inverse (see equations () and ()): ∑lm∂iδκεgδε∂xl∂piδ′pκε′δ′ε′∂xm . At this point, Einstein simplified the derivation by restricting himself to infinitesimal transformations. With the help of the Kronecker delta, the transformation matrices can then be written as iδδiδ+iδx,piδ′δiδ+piδ′x. For such infinitesimal transformations, expression () reduces to: ∑lm∂gδε∂xl∂∂xmδε+terms of order px and x . Einstein denoted these infinitesimal correction terms as “transformation infinitely small” (“Transformation unendlich klein”). Expression () shows that expression () transforms as a scalar, if the sum of all terms of order px and x vanish.This is a concrete example of the condition C∂, g, p, 0 in sec.  (see equation ()). This then is the condition on the transformation matrices for infinitesimal non-autonomous transformations leaving expression () invariant. This condition consists of four terms, which are obtained by differentiating one of the four transformation matrices in expression () and replacing the remaining three by Kronecker deltas:Lacking the Kronecker delta, Einstein indicated this procedure as follows. For the term in which the first of the four coefficients iδ,κε,piδ′,and pκε′ in expression () is differentiated he wrote underneath expression (): “1 κε iδ′ κε′ .” ∑lm∂iδx∂xl∂∂xmiκgδκ , ∑lm∂κεx∂xl∂∂xmiκgiε , ∑lm∂piδx∂xm∂∂xlgiκδκ , ∑lm∂pκεx∂xm∂∂xlgiκiε . Einstein wrote the last two expressions next to the first two, separating the two pairs by a vertical line. By rewriting expression () asEquation () can be obtained from () with the help of (in modernized notation) iκ,mgδκiκgδκ,m and iδxpδix . The latter relation follows from pδαiα δδα+pδαxδiα+iαx=δδi+pδix+iδxδδi . ∑lm∂pδi∂xl∂∂xmgδκiκ (where the superscript “ x ” was dropped), Einstein showed that it was equal to expression () written next to it.Expressions () and () turn into one another if the summation indices l and m and the summation indices δ and i are switched. Similarly, expression () is equal to expression () written next to it. The condition for infinitesimal non-autonomous transformations under which expression () transforms as a scalar can thus be written as the vanishing of the sum of expressions () and (): ∑lm∂piδx∂xm∂∂xlgiκδκ+∑lm∂pκεx∂xm∂∂xlgiκiε0 . An additional condition on the transformation coefficients was the requirement that the transformations be unimodular. Einstein thus turned to the determinant 1+p11p12p13p14p211+p22p23p24p31p321+p33p34p41p42p431+p44 of the transformation matrix for the infinitesimal transformation in equation ().Following Einstein, we have dropped the subscript “ x .” To first order in pμν , the determinant () is equal to 1+p11+p22+p33+p44 . So the condition that an infinitesimal transformation be unimodular is simply that the trace of pμν (and of μν ) vanish: ∑pααx0,∑ααx0. As with the conditions for non-autonomous transformations on pp. 7R and 8R, Einstein made no attempt to find transformation matrices depending on the metric that satisfy the conditions () and (). So the calculation of p. 9R did not lead to the identification of any specific non-autonomous transformations under which the core operator would transform as a tensor. Exploring the Covariance of the Core Operator under Hertz Transformations (10L–12R, 41L–R) On the preceding pages, Einstein had investigated the covariance properties of the core operator (), ∑∂∂xμμν∂iκ∂xν , and of expression (), ∑μν∂giκ∂xμ∂iκ∂xν , by deriving the conditions for non-autonomous transformations under which these objects transform as tensors.See the introduction of sec. for discussion of the concept of non-autonomous transformation. On p. 7L, Einstein examined expression (), deriving conditions () and () for non-autonomous transformations—finite and infinitesimal, respectively—under which the expression would transform as a tensor. On p. 9R, he examined expression (), deriving conditions () and () for infinitesimal unimodular non-autonomous transformations under which this expression would transform as a scalar. None of these investigations had been carried through to the end. Errors were made and left uncorrected. Einstein had not even begun the task of finding non-autonomous transformations that would actually be solutions of such conditions. On pp. 10L–11L, however, he made a sustained effort to find infinitesimal unimodular non-autonomous transformations under which a much simpler object, the Hertz expression, ∑∂μν∂xν , transforms as a vector. The task at hand was still to determine the covariance properties of the core operator. By focusing on the Hertz expression first, Einstein could split this task into two separate and more manageable tasks. The core operator () can be written as the sum of two terms, the first of which contains the Hertz expression: ∑∂μν∂xμ∂iκ∂xν+∑μν∂2iκ∂xμ∂xν . On pp. 10L–10R, Einstein focused on the first term, deriving the condition for unimodular non-autonomous transformations under which the Hertz expression transforms as a vector. We shall call such transformations “Hertz transformations.” Einstein only dealt with infinitesimal Hertz transformations, which simplified his calculations considerably. The restriction to unimodular transformations in the calculations on these and the following pages indicates that he still wanted to connect the core operator to the second Beltrami invariant ϕ2 (see equations ()–()). On p. 10L, in a first attempt to derive the condition for Hertz transformations, Einstein made a mistake he only discovered on p. 10R after he had already moved on to the second term in expression (). By that time, he had convinced himself that rotation in Minkowski spacetime is a Hertz transformation. Not surprisingly, therefore, when Einstein had derived the correct condition for Hertz transformations, he carefully checked once more whether the class of Hertz transformation include the important special cases of rotation and uniform acceleration in Minkowski spacetime before returning to the investigation of the second term in expression (). On p. 11L, Einstein established that the matrices pμν and μν for transformations to rotating frames in Minkowski spacetime with infinitesimally small angular velocity do indeed satisfy the conditions for infinitesimal Hertz transformations. He noticed that the transformation matrices for finite rotations satisfy these conditions for infinitesimal transformations as well. From this he seems to have drawn the erroneous conclusion that finite transformations to rotating coordinates in Minkowski spacetime are also Hertz transformations (see sec. ). Einstein immediately recognized, however, that transformations to uniformly accelerating coordinate systems in Minkowski spacetime are not Hertz transformations, not even for infinitesimally small accelerations. He initially thought he could circumvent this problem by modifying the transformation (see sec. ). So the class of Hertz transformations initially did seem to include rotation and uniform acceleration in Minkowski spacetime. On p. 11R, Einstein turned to the second term in expression () for the core operator (see sec. ). He imposed the “Hertz restriction,” i.e., the condition that the Hertz expression vanish.See sec. for the definition of a coordinate restriction. Under this restriction, expression () reduces to its second term. Einstein now checked whether this term transforms as a tensor under Hertz transformations. He discovered that it does not. At the bottom of p. 11R, he wrote: “Leads to difficulties” (“Führt auf Schwierigkeiten”). The class of Hertz transformations thus needs to be restricted further. If the metric is set equal to the Minkowski metric in its standard diagonal form, the condition expressing this further restriction reduces to the requirement that the matrices pμν and μν of the infinitesimal transformations be anti-symmetric. It turns out that the requirement of anti-symmetry is all that is needed to satisfy the conditions defining the class of infinitesimal Hertz transformations as well. In other words, the core operator transforms as a tensor under all infinitesimal anti-symmetric transformations from an inertial pseudo-Cartesian coordinate system in Minkowski spacetime. At the top of p. 12L, Einstein therefore made an “attempt” (“Versuch”) to find anti-symmetric transformations corresponding to infinitesimal rotation in Minkowski spacetime. He went back to p. 11L and noted how the transformation matrix for infinitesimal rotation needs to be changed to make it anti-symmetric. He realized that this is not feasible. The transformation matrix for infinitesimal uniform acceleration is not anti-symmetric either—be it in its original form, or in the modified form introduced on p. 11L. It seems that at this point Einstein deleted the modified form and accepted that the Hertz restriction rules out the important special case of uniform acceleration in Minkowski spacetime. The upshot then was that the strategy Einstein adopted on p. 10L to study the covariance of the core operator by splitting it into the two terms (see expression ()) did not produce any physically interesting transformations, not even infinitesimal non-autonomous ones, under which the core operator would transform as a tensor. Einstein’s first reaction was to change the definition of the core operator by inserting an extra factor of G . Such extra factors, however, do not affect the argument on pp. 9R–12L. Rather than going through this argument again, Einstein, as he did on p. 11L, turned his attention to the important test cases of rotation and acceleration in Minkowski spacetime. On pp. 12L–R, he carefully examined rotation (see sec. ). On pp. 12R, 41L–R, he systematically studied (autonomous) infinitesimal unimodular transformations with a view to recovering the transformation to uniformly accelerating frames in Minkowski spacetime. His attempt, however, to find a unimodular transformation corresponding to acceleration failed (sec. ). Since the case of uniform acceleration was drawn from his 1912 static theory, Einstein now reexamined an important insight connecting the 1912 theory based on one potential (the variable speed of light) and the metric theory based on ten potentials (the components of the metric tensor). This is the insight that the equation of motion of a test particle in a gravitational field can be obtained from a variational principle with the line element serving as the Lagrangian. A consideration of constrained motion on a curved surface probably reassured him that this insight was sound (sec. ). Deriving the Conditions for Infinitesimal Hertz Transformations (10L–R) At the top of p. 10L, Einstein wrote down the transformation law for the Hertz expression: . The structure of this quantity is considerably simpler than that of expression () on p. 9R. The condition for non-autonomous transformations under which the Hertz expression transforms as a vector will likewise be considerably simpler than condition () on p. 9R, for non-autonomous transformations under which expression () transforms as a scalar. Condition () is the one referred to in the header of p. 10L: “For comparison with this condition” (“Zum Vergleich mit dieser Bedingung”). Einstein applied his usual two-step procedure to find the condition characterizing what we called “Hertz transformations,” i.e., unimodular non-autonomous transformations under which the Hertz expression transforms as a vector.On p. 10L Einstein only gives the condition for infinitesimal Hertz transformations. See footnote 146 below for a self-contained derivation of the condition for finite transformations. See the introduction to sec.  for general discussion of the two-step procedure. Einstein rewrote the right-hand side of equation () as ∑pμα∂ασ∂xσ+∑ασ∂pμα∂xσ , where he used that the matrices νσ and pνβ —connected by the line in equation ()—are the inverse of one another and assumed that ∑νσ∂pνβ∂xσ vanishes. This assumption holds for infinitesimal unimodular transformations (see note 148 below) but not in general, as Einstein soon came to realize.Einstein had made use of the relation ∑νσ∂pνβ∂xσ0 in situations in which this was not warranted before (see equation () and footnote 98). Einstein was now ready for the first step of his two-step procedure. For non-autonomous transformations under which the Hertz expression transforms as a vector, the transformation of the Hertz expression from primed to unprimed coordinates is given by: ∂κλ∂xλμκ∂′μν∂x′ν . Einstein omitted this step and immediately wrote down the second step, expressing the right-hand side in terms of unprimed quantities. Using the abbreviation ακ for the Hertz expression,On p. 6R, Einstein had used the notation ακ for a covariant vector (see equation ()). and using equations ()–(), he wrote: ακ∑μκpμα∂ασ∂xσ+∑μκ∂∂xσpμαασ . The first term on the right-hand side is equal to ακ . Einstein thus concluded that the condition for non-autonomous transformations under which the Hertz expression transforms as a tensor is: ∑μκ∂∂xσpμαασ0 . For infinitesimal unimodular transformations this is correct (see note 148 below), but in general, there will be additional terms.The condition for finite Hertz transformations will be important for understanding Einstein’s argument on p. 11L (see sec. ). The derivation of this condition is fully analogous to the derivation of condition () for infinitesimal Hertz transformations. For finite transformations, equation () needs to be replaced by: ακμκα′μμκ∂′μν∂x′νμκνσ∂pμαpνβαβ∂xσ This gives a sum of three terms ακμκνσpμαpνβ∂αβ∂xσ+μκνσ∂pμα∂xσpνβαβ+μκνσpμα∂pνβ∂xσαβ , which can be rewritten as ακακ+μκ∂pμα∂xσασ-∂νσ∂xσpνβκβ , where in the third term on the right-hand side the relation νσ∂pνβ∂xσ∂νσ∂xσpνβ was used, which follows from ∂∂xσνσpνβ . Hence the condition for finite Hertz transformations is: μκ∂pμα∂xσασ-∂νσ∂xσpνβκβ0 . To simplify condition (), Einstein restricted himself to infinitesimal transformations, pμνδμν+pμνxμνδμν+μνx with pμνxνμx (see p. 9R and footnote 138). As he wrote underneath the second term on the right-hand side of equation (): “for infinitesimal transformations” (“für infinitesimale Transformationen”) ∑∂pκαx∂xσασ0 . This equation, he continued, “Is a system of four conditions for the px if this should always vanish. Furthermore, the determinant should always be equal to 1. ∑pααx0 .” (“Ist für die px ein System von 4 Bedingungen, wenn dies stets verschwinden soll. Ferner soll Determinante stets gleich 1 sein. ∑pααx0 .”). As Einstein notes here, for infinitesimal unimodular transformations detpdet1 . It follows that the trace of the matrices pμνx and μνx vanishes:Cf. equation (). The condition is illustrated by a simple example written underneath it: 1+ε1 1+ε21+ε1+ε21 ,It turns out that for infinitesimal unimodular transformations ∑νσ∂pνβ∂xσ0 and that equation () is correct. This, in turn, means that conditions ()–() correctly specify the class of infinitesimal Hertz transformations. The aborted calculation in the top-right corner of p. 10L suggests that Einstein realized this. He began by writing down the term he had omitted in going from equation () to equation (): ∑νσpμα∂pνβ∂xσαβ . Underneath this expression, he wrote: ∑∂νσ∂xσ . One can indeed transfer the derivative operator ∂∂xσ in expression above from pνβ to νσ . This follows from: 0∂∂xσνσpνβνσ∂pνβ∂xσ+∂νσ∂xσpνβ . Condition () for infinitesimal unimodular transformations implies that ∂νσ∂xσ vanishes: ∂νσ∂xσ∂νσx∂xσ∂pσνx∂xσ∂pσσx∂xν0 . where we used that νσxpσνx (see footnote 138) and that ∂pσνx∂xσ∂∂xσ∂x′σ∂xν∂∂xν∂x′σ∂xσ∂pσσx∂xν . This concludes the proof that the expression in the top-right corner of p. 10L vanishes for infinitesimal unimodular transformations and that the transition from equation () to equation () is justified in this case. ∑pααx∑ααx0 . Together, equations () and () thus determine the class of infinitesimal Hertz transformations (i.e., infinitesimal unimodular non-autonomous transformations under which the Hertz expression transforms as a vector). Raising the question, “Is it possible to have both?” (“Ist beides möglich?”), Einstein now set out to solve equations () and () for the transformation matrix pμνx in the special case that μν in equation () is a constant diagonal metric (be it Euclidean or Minkowskian). He first considered the two-dimensional and then the three-dimensional case. Einstein introduced the coordinate transformations x′Xx,y,y′Yx,y. The corresponding differentials can be written as: dx′∂X∂xdx+∂X∂ydy,dy′∂Y∂xdx+∂Y∂ydy. . From this one can read off the matrix pμν∂X∂x∂X∂y∂Y∂x∂Y∂y . Condition () then reduces toEinstein originally wrote X′ and Y′ and then deleted the primes. ∂∂xX+∂∂yY2 . To the right of equation (), Einstein specified that the metric be diagonalHe first wrote down the Euclidean metric diag1,1 and later changed it to the Minkowski metric diag1,1 . 111 221120 . Inserting these values into condition (), one finds: ∂p1αx∂xσασ∂p11x∂x111+∂p12x∂x222∂p11x∂x1+∂p12x∂x20,∂p2αx∂xσασ∂p21x∂x111+∂p22x∂x222∂p21x∂x1+∂p22x∂x20. Inserting equation () for pμν into equation (), one finds: ∂2X∂x2+∂2X∂y20,∂2Y∂x2+∂2Y∂y20. Einstein omitted the terms with p12x and p21x in equation () at this point and used the erroneous set of equations ∂2X∂x20, ∂2Y∂y20 . Einstein eventually discovered his error. The calculation on pp. 10L–R based on equations () is deleted and a fresh start is made at the bottom of p. 10R, based on the correct set of equations () (see equations ()–() below). But first we shall discuss the deleted calculations on pp. 10L–R Integrating equations (), Einstein arrived at ∂X∂xψy ∂Y∂yχx with arbitrary functions ψ and χ . The unimodularity condition () requires that: ψy+χx2 . Einstein concluded that ψ and χ would “both [be] constant” (“beide konstant.”). So the only infinitesimal Hertz transformations in the two-dimensional case are linear transformations. In an attempt to find non-linear transformations, Einstein turned to the next simplest case, a Euclidean space of “three dimensions” (“drei Dimensionen”) with a constant diagonal metric. Starting point of the calculation is the coordinate transformation (cf. equation ()): x′Xx,y,z,y′Yx,y,z,z′Zx,y,z. The matrix pμν is thus given by (cf. equation ()): pμν∂X∂x∂X∂y∂X∂z∂Y∂x∂Y∂y∂Y∂z∂Z∂x∂Z∂y∂Z∂z . The analogue of equation (), expressing unimodularity, is ∂X∂x+∂Y∂y+∂Z∂z3 , while the analogues of conditions ()—again with omission of non-diagonal terms of pμνx — are ∂2X∂x20, ∂2Y∂y20, ∂2Z∂z20 . Integrating these equations, Einstein wrote (cf. equation ()): ∂X∂xψ1y,z , ∂Y∂yψ2x,z , ∂Z∂zψ3x,y . Inserting these expressions into equation () and introducing ψixψi-1 , he obtained: ψ1xy,z+ψ2xx,z+ψ3xx,y0 . Taking the derivative of equation () with respect to z and dropping the superscript x , Einstein found, at the bottom of p. 10L: ∂ψ1∂z+∂ψ2∂z0 . At the top of p. 10R, Einstein integrated this last equation, writing the result as ψ1+ψ2χ3x,y . Taking the derivative of equation () with respect to x and y , one similarly finds ψ2+ψ3χ1y,z , and ψ1+ψ3χ2x,z , respectively. Einstein explicitly wrote down equation (), but not equation (). Since both ψ1y,z and ψ2z,x depend on z , while their sum, according to (), does not, Einstein could write them in the form ψ1y,zψ1y+ζ , ψ2z,xψ2x-ζ , where ζ is some function of z . He used equation () to write ψ3 as: ψ3x,yψ1y-ψ2x . Now insert equations ()–() into the equations ()–(), keeping in mind that all ψis in equations ()–() are actually ψixs related to the ψis in equations ()–() through ψi1+ψix . ∂X∂x1+ψ1xy+ζ , ∂Y∂y1+ψ2xx-ζ , ∂Z∂z1-ψ1xy-ψ2xx . Integrating these equations, one findsInstead of equations ()–(), Einstein wrote down the equations: Xxψ1y+ζx-η+ζ+ω1y,z,Yyψ2x-ζy-ζ+ξ+ω2z,x,Zzψ1y-ψ2zz-ξ+η+ω3x,y, where ξψ2x and ηψ1y . These equations contain a number of mistakes. In the third line, ψ1y-ψ2z should be ψ1y+ψ2x . In all three lines, ψix and ψi are conflated and the integration constants ωi are missing after the first equality sign. Einstein also defined and then deleted the quantities δxψ1 and δyψ2 (cf. equation () at the bottom of p. 12R where δx is essentially defined as dX .). Xx1+ψ1xy+ζ+ω1y,z , Yy1+ψ2xx-ζ+ω2x,z , Zz1-ψ1xy-ψ2xx+ω3x,y . Einstein now considered a special case. Writing “Specified” (“Spezialisiert”), he set ψ1xy , ψ2xx , and the quantities ωi to zero and ζαz , with α≪1 . Equations ()–() then reduce to:As a consequence of Einstein’s conflation of ψix and ψi (see the preceding note), the notebook has Xαxz , Yαyz and Z0 instead of equations (). Xx+αxz,Yy-αyz,Zz. Next to this transformation, Einstein wrote: “Is torsion and, in the case zt , uniform rotation. Torsion very special case.” (“Ist Torsion & im Falle zt gleichformige Drehung. Torsion ganz spe­zieller Fall.”). The transformation () is indeed a very special case of the much more general transformation ()–(); it does not, however, correspond to torsion, nor, with zt , to rotation. The transformation setting an inertial pseudo-Cartesian coordinate system in Minkowski spacetime rotating with angular velocity ω is given by: x′xωt+yωt,y′xωt+yωt,t′t. For infinitesimal ω , this transformation reduces to: x′x+ωyt,y′y-ωxt,t′t, which is clearly not the same as transformation () with zt . Einstein did not realize this until later. He did, however, realize at this point that he had made an error in evaluating condition () for the special case of a constant diagonal metric.See equation () for the correct form of the condition in the two-dimensional case and equation () for the incorrect form in the three dimensional case used by Einstein in his derivation of the general transformation ()–(). Under the header, “Conditions of integrability” (“Integrabilitätsbedingungen”), he partly corrected this error:The crucial residual error in the equation below is that it has ∂pyxx∂y instead of ∂pxyx∂y . Inserting μνdiag1,1,1 into condition (), ∑∂pκαx∂xσασ0 , one finds ∂pxxx∂x+∂pxyx∂y+∂pxzx∂z =∂2X∂x2+∂2X∂y2+∂2X∂z2 for the κ1 component instead of equation () in the notebook. ∂pxxx∂x+∂pyxx∂y+.∂∂x∂X∂x+∂∂y∂Y∂x+∂∂z∂Z∂x0 . This condition, however, is still satisfied by the solutions of the erroneous conditions () that Einstein had used up to this point.Inserting equations ()–() into the three terms on the right-hand side of equation (), one finds zero, ∂ψ2xx∂x , and ∂ψ2xx∂x , respectively. A line drawn from the header above equation () to these solutions (see equations ()–() and footnote 151), suggests that Einstein actually checked this. This would explain why he initially simply proceeded with the next part of his investigation of the transformation properties of the core operator. On the next line he wrote the contraction of the core operator () with some arbitrary covariant tensor Tiκ :So far Einstein had used Latin letters for contravariant objects (see, e.g., equation () for aμ and Tμν on p. 7R) ∑Tiκ∂∂xμμν∂iκ∂xνSkalar . The core operator will transform as a tensor under all transformations under which its contraction with Tiκ transforms as a scalar. When the Hertz restriction (i.e., ∑∂μν∂xμ0 ) is imposed, the contraction () reduces to: ∑Tiκμν∂2iκ∂xμ∂xν . The Hertz restriction is invariant under Hertz transformations. Hence, if Einstein could show that expression () transforms as a scalar under infinitesimal Hertz transformations, he would have shown that the core operator transforms as a tensor under such transformations. This is precisely the problem that Einstein takes up at the top of p. 11R, where he raises the question whether expression () is a scalar. On p. 10R, however, he did not proceed beyond equation (). Expressions () and () were deleted and Einstein returned to condition () for infinitesimal Hertz transformations. It was probably at this point that in the upper-right corner of p. 10R, he wrote down the κ1 component of condition () for the special case of a diagonal Euclidean metric: ∂p11∂x1+∂p12∂x2+∂p13∂x30 . Inserting equation () for pμν into equation (), one finds: ∂2X∂x2+∂2X∂y2+∂2X∂z20 . For the κ2 and κ3 components, one similarly finds: ∂2Y∂x2+∂2Y∂y2+∂2Y∂z20 , ∂2Z∂x2+∂2Z∂y2+∂2Z∂z20 . Einstein, however, still continued to use the transformations ()–() he found as a solution of conditions (), an erroneous version of conditions ()–() and similar conditions for Y and Z . Inserting equation () for X into equation (), he found: ΔX0∂2ω1∂y2+∂2ω1∂z2 . The solution of this harmonic differential equation is: ω1α+βy+z+δyz+εy2-z2 . He added one more line,Cf. equation () at the bottom of p. 12R where δx is essentially defined as dX . δxkonst.+α1zδy+α2yδz+ , before he realized that the replacement of conditions () by conditions ()–() invalidated much of the subsequent calculation on pp. 10L–R. He deleted this calculation—equations ()–(), with the exception of equation () at the top of p. 10R—and made a fresh start. Einstein went back to the two-dimensional case examined on p. 10L. The unimodularity condition () for this case can still be written as (see equation ()): ∂∂xX+∂∂yY2 , He then turned to condition () for non-autonomous transformations with μνδμν (see equations ()–()). For the first component of condition (), he now correctly wrote: ∂p11∂x1+∂p12∂x20 . Substituting p11∂X∂x p12∂X∂y (cf. equation ()) into in equation (), he found: ∂2X∂x2+∂2X∂y20 . For the second component of condition (), he similarly wrote ∂2Y∂x2+∂2Y∂y20 . Equations () and () are easily solved: Xα1xy+α2x2-y2+x , Yβ1xy+β2x2-y2+y . Einstein started to calculate p11∂X∂x for these coordinate transformations, but proceeded no further than p11α1y+2 . He then substituted equations ()–() into the unimodularity condition () and found α1y+2α2x+β1x-2β2y0 . The constants of integration α1 , α2 , β1 , and β2 thus have to satisfy β12α2 , α12β2 . Equations ()–() with conditions ()–() give the general form for infinitesimal Hertz transformations for the special case of the two-dimensional Euclidean metric μνδμν . Checking Whether Rotation in Minkowski spacetime Is a Hertz Transformation (11L) On p. 11L Einstein checked whether transformations to uniformly rotating and uniformly accelerating frames in Minkowski spacetime are included in the class of (infinitesimal) Hertz transformations that he had studied on pp. 10L–R. At the top of the page, under the header “Rotation” (“Drehung”), he dealt with the former; under a horizontal line in the middle of the page, under the header “Acceleration” (“Beschleunigung”), he dealt with the latter. Einstein concluded that both infinitesimal and finite rotations in Minkowski spacetime are Hertz transformations. This is true for infinitesimal but not for finite rotations. Einstein also found, however, that uniform accelerations in Minkowski spacetime, whether infinitesimal or finite, are not Hertz transformations (see sec. ). The condition for some unimodular non-autonomous transformation to be a Hertz transformation (i.e., a unimodular transformation under which the Hertz expression, ∂μν∂xν , transforms as a vector) is: μκ∂pμα∂xσασ-∂νσ∂xσpνβκβ0 (see footnote 146). For infinitesimal transformations the second term vanishes and this condition reduces to: ∂pκα∂xσασ0 (see equation ()). For infinitesimal transformation the condition for unimodularity reduces to the requirement that the trace of pμνxpμν-δμν and μνxμν-δμν vanish: pααxααx0 . (see equation ()). At the top of p. 11L, next to a drawing indicating rotation, Einstein wrote down the transformation from an inertial pseudo-Cartesian coordinate system in Minkowski spacetime to a coordinate system uniformly rotating with angular velocity ω :Anticipating the next step in the calculation, Einstein wrote dt′dt instead of t′t . x′xωt+yωty′xωt+yωtt′t. The differentials of the coordinates transform as: dx′ωt dx+ωt dy+xωt+yωtωdtdy′ωt dx+ωt dy+xωtyωtωdtdt′ 0 dx+ 0 dy + dt. For an infinitesimal rotation, equation () reduces to: dx′= dx+ωtdy+yωdtdy′=ωtdx+ dy�xωdtdt′= dt, from which one can read off the components of the transformation matrix pαβ , the “table of p ” (“Tabelle der p ”):The 13-component originally had an additional term xωt . This term should be xω2t and can therefore be neglected. The 23-component likewise has a deleted term yωt , which should be yω2t . The expressions yω and xω underneath the 31- and 32-components were added later (see equation () in sec. ). pαβ1ωtyωωt1xω001 . For this matrix,  the trace pααx vanishes, and so does ααx .It follows from pμαναδμν that μνxpνμx (see footnote 138). Hence, condition () is satisfied. Inserting the diagonal Minkowski metric for αβ in equation (), one readily verifies that pαβ in equation () satisfies conditions () as well: ∂pκ1∂x+∂pκ2∂y-∂pκ3∂t0 for κ1, 2, 3 . It follows that infinitesimal rotations are indeed infinitesimal Hertz transformations. Next to his “table of p ” Einstein accordingly wrote: “Correct” (“Stimmt”). In what looks like a later addition to the page, Einstein checked whether finite rotations are Hertz transformations too. In the lower-left part of p. 11L, Einstein wrote down the “table of the p ” (“Tafel der p ”) for a finite rotation, which can be read off from equation ():Einstein omitted a factor ω in p13 and p23 . pμνωt ωt -xωt+yωtωωt ωt -xωt-yωtω0 0 1 . Inverting this matrix, one finds the corresponding “table of the ” (“Tafel der ”):The notebook has 31y and 32x . The inversion is done with the formula ik1i+kΔikp , where Δik is the co-factor of pik and pdetpik1 . The expression for 31 is found as follows: 31=ωt -xωt+yωtωωt -xωt-yωtω =ωωt-xωt-yωt-ωt-xωt+yωt =ωy A completely analogous calculation gives 32ωx . Inserting equations () and () into pμανα , one readily verifies that this gives δμν . μνωtωt0ωtωt0yωxω1 . Einstein noted that the second term in condition () for finite Hertz transformations vanishes for this transformation matrix. He noted that ∑∂μν∂xν0 is “always fulfilled” (“immer erfüllt”). Since the transformation to rotating coordinates is also unimodular— detpik1 for the matrix in equation ()—Einstein presumably concluded that finite rotations, like infinitesimal ones, are Hertz transformations. This conclusion, however, is not warranted. The problem is that the first term of equation () does not vanish for finite rotations. Substituting the diagonal Minkowski metric for ασ in the first term of equation (), one finds μκ∂pμ1∂x+∂pμ2∂y-∂pμ3∂t . Since ∂pμ1∂x∂pμ2∂y∂p33∂t0 , this expression reduces to -1κ∂p13∂t-2κ∂p23∂t . This expression does not vanish for the coefficients pμν in equation () for a finite rotation. For κ1 , for instance, it is equal to xω2 .Using equation (), one finds that ∂p13∂tω2-xωt-yωt , ∂p23∂tω2xωt-yωt Inserting these expressions into the κ1 component of expression () and using equation (), one finds -11∂p13∂t-21∂p23∂tω2ωt-xωt-yωt-ωtxωt-yωt , which is equal to ω2x . It follows that condition () is not satisfied in the case of finite rotations in Minkowski spacetime, which means that these transformations cannot be Hertz transformations.A more direct way to arrive at this conclusion is to note that the Hertz expression, ∂μν∂xν , vanishes for the standard diagonal Minkowski metric but not for the Minkowski metric in rotating coordinates. The contravariant form of the latter is given by: μν-1+ω2y2ω2xy0ωyω2xy-1+ω2x20ωx0010ωyωx01 For this metric, the Hertz expression is: ∂μν∂xνω2x,ω2y,0,0≠0 . Checking Whether Acceleration in Minkowski Spacetime is a Hertz Transformation (11L) On the lower half of p. 11L, under the heading “Acceleration” (“Beschleunigung”), Einstein checked whether a transformation to a uniformly accelerated frame in Minkowski spacetime is a Hertz transformation. He started from the transformation equations that he had found for this case in the course of the work on his theory for static gravitational fields:See (Einstein 1912b, 456). ξx+c2dcdxt2τct, where c is the variable speed of light that served as the gravitational potential in Einstein’s static theory. Einstein assumed c to be of the formInitially, Einstein wrote cc0eax but then deleted the factor c0 . ceax . In the notebook, Einstein used x,t instead of ξ,τ and x′,t′ instead of x,t . The transformation () then becomes: xx′+cx′2dcx′dx′t′2tct′. With the ansatz ()—in terms of x′ rather than x —the transformation () turns into: xx′+a2e2ax′t′2teax′t′. Inverting this transformation while neglecting terms quadratic in a and smaller, Einstein found x′x-a2t2t′t1-ax. As he had done for rotation, Einstein considered the infinitesimal transformation dx′dx-atdtdt′-atdx+1-axdt. From the transformation matrix pij1atat1-ax , one immediately sees that its elements do not satisfy the two conditions () and () for infinitesimal Hertz transformations (see also equations () and ()). Inserting μνδμν in equation (), we find that the κ1 component is: ∂p11x∂x1+∂p12x∂x2a≠0 , Condition () is not satisfied either: ∑pααxax≠0 . Both problems could be fixed by setting: p111+ax . It looks as if Einstein considered this modification. He changed the first line of equation () to dx′1+2xdx-atdt , and added the remark: “is also correct for a suitable shift of scale” (“stimmt auch bei geeigneter Massstabverschiebung”). He subsequently deleted this remark. For the time being, however, Einstein seems to have been satisfied that he could ensure in this fashion that transformations to uniformly accelerating frames in Minkowski spacetime would be included in the class of Hertz transformations. The modification () of the transformation matrix (), however, is not allowed. The form of the transformation (), of which transformation () is a special case, was derived from the equivalence of the propagation of light in the two systems ( ξ,τ ) and ( x,t )See (Einstein 1912a, sec. 1). dξ2-dτ2dx2-c2dt2 . With the adjustment () this fundamental equation would no longer be valid. Trying to Find Hertz Transformations under which the Core Operator Transforms as a Tensor (11R) Having satisfied himself for the time being that the class of Hertz transformations includes transformations to rotating and uniformly accelerating frames in Minkowski spacetime, Einstein once again turned his attention to the core operator. Picking up on an idea that had made its first appearance on p. 10R (see equations ()–()), Einstein examined under which transformations the contraction of the core operator and some arbitrary covariant second-rank tensor would transform as a scalar. On p. 10R, Einstein had written this contraction as (see equation ()): ∑Tiκ∂∂xμμν∂iκ∂xνSkalar . At the top of p. 11R Einstein wrote down a very similar expression (down to the labeling of the indices), ∑Tiκxμν∂∂xμ∂∂xνiκ , and asked: “Is this a scalar?” (“Ist dies ein Skalar?”). If Tiκ in equation () is replaced by Tiκx ,It is not clear why Einstein added an ‘x’ added to Tiκ at this point. This superscript was Einstein standard notation for first-order deviations from constant values. The argument on p. 11R works with both Tiκ and Tiκx . and, more importantly, if the Hertz restriction, ∑∂μν∂xν0 , is imposed, the expressions on 10R and 11R are equivalent. So if expression () transforms as a scalar under Hertz transformations, so will expression (). Einstein set out to find what further coordinate restrictions over and above the Hertz restriction would be needed for expression () to transform as a scalar. To this end he once again used his two step procedure.See the introduction to sec.  for a general discussion of this procedure and equations ()–() for a concrete example involving a scalar. He wrote expression () in primed coordinates, and expressed its various components in unprimed coordinates: S∑iα1κα2Tα1α2xpμβ1pνβ2β1β2μσ∂∂xσντ∂∂xτpiε1pκε2ε1ε2 . Einstein connected the factors pμβ1 and μσ by a solid V-shaped line to indicate that they combine to form (in modern notation) δβ1σ , Underneath this line, he wrote β1σ . He likewise connected pνβ2 and ∂∂xσ by a dashed V-shaped line and directly underneath wrote In the first step he used that 0∂∂xσpνβ2ντpνβ2∂ντ∂xσ+ντ∂pνβ2∂xσ . He replaced β1 in β1β2 by σ , using that pμβ1μσδβ1σ . Expression () has exactly the form of the last term in equation () at the top of p. 10L. At that point, Einstein had drawn the conclusion that the vanishing of this expression (see equation ()) was the condition for non-autonomous transformation under which the Hertz expression transforms as a tensor. Equation () does indeed express the Hertz restriction for infinitesimal transformations, but not for finite ones (see the discussion following equation () and footnote 148). In the second step in equation (), Einstein nonetheless used equation (). On the basis of equation (), Einstein could move ντ outside the scope of the differential operator ∂∂xσ in equation (). As he wrote: “hence ντ can be taken outside” (“also ντ heraus setzbar”). The factor ντ combines with pνβ2 to give δτβ2 . In a separate box Einstein summarized the simplifications pμβ1μσδβ1σ and ντpνβ2δτβ2 in equation (): β1σβ2τ. With these simplifications, equation () becomes: S′∑iα1κα2Tα1α2xστ∂∂xσ∂∂xτpiε1pκε2ε1ε2 . Underneath this equation Einstein wrote: “Let us restrict ourselves to an infinitesimal substitution” (“Beschränken wir uns auf infinitesimale Substitution”). Whether Einstein realized it or not, this immediately takes care of the problem that to arrive at equation () he had used the Hertz restriction in a form that only holds for infinitesimal transformations. Einstein’s task now was to identify all those terms in equation () that would have to vanish for the right-hand side to reduce to expression () that he had started from. These are all terms in which the elements pμν of the transformation matrix are differentiated at least once.Einstein had made this observation twice on p. 7R in comments on equations () and (). The condition determining under which subclass of infinitesimal Hertz transformations expression () transforms as a scalar is obtained by setting the sum of all these terms equal to zero. If in a product of several matrix elements pμν of an infinitesimal transformation one is differentiated, all others can be replaced by Kronecker deltas. Equation () can thus be rewritten as: S′=δiα1δκα2Tα1α2xστ∂∂xσ∂∂xτpiε1δκε2ε1ε2+δiε1pκε2ε1ε2 =Tiκxστ∂∂xσ∂∂xτpiε1ε1κ+pκε2iε2 As Einstein put it: “If one of the p ’s is differentiated at all, for instance piε1 , then one has to set κε2 , iα1 , κα2 ” (“Dann muss, falls überhaupt eines der p diff wird, z. B. piε1 κε2 , iα1 , κα2 gesetzt werden”). The second step in equation () is indeed to set α1i , α2κ and to set ε2κ if piε1 is differentiated and ε1i if pκε2 is. Einstein could thus rewrite equation () as:The second term in parentheses in equation () is written underneath the first. Both S and στ in equation () are interlineated. Note that στ is placed within the scope of the differential operators ∂∂xσ and ∂∂xτ , whereas in equations () and () it was not. Because of the Hertz restriction, however, ∂στ∂xσ∂στ∂xτ0 , so this makes no difference. S′∑Tiκx∂∂xσ∂∂xτστpiε1ε1κ+pκε2ε2i+S , adding: “where p is to be differentiated at least once” (“wobei p mindestens einmal zu differenzieren ist”). The last term, S , is obtained if the two differential operators both act on ε1κ or ε2i instead of at least one of them acting on piε1 or pκε2 . For S to be a scalar, the sum of all terms containing derivatives of pμν in the first term on the right-hand side of equation () should vanish. Einstein first considered the expressionThree lines farther down, Einstein added: “Already sum over ε1 ” (“Schon Summe über ε1 ”), drawing a line from expression (333) to this comment. The comment refers, perhaps, to the similarity between the sum over ε1 in ∂piε1∂xτε1κ in expression (333) and the sum over α in equation (), one of the conditions for infinitesimal Hertz transformations. Tiκ∂∂xσστ∂piε1∂xτε1κ . Since ∂piε1∂xτ∂piτ∂xε1 (see footnote 148) this can be rewritten as: Tiκ∂∂xσστ∂piτ∂xε1ε1κ . Because of the Hertz restriction, ∂μν∂xν0 , στ can be taken outside the scope of ∂∂xσ and ε1κ can be taken inside the scope of ∂∂xε1 . Einstein thus arrived at: ∑Tiκστ∂∂xσ∂∂xε1piτε1κ . He repeated this result on the next line, Tiκ∂∂xσστ∂2∂xε1∂σ(piτε1κ , but did not pursue the calculation any further. He went back to equation () and, contrary to what he did in expression (), treated the two derivative operators ∂∂xσ and ∂∂xτ on an equal footing. For the term with piε1ε1κ in equation () he wrote: ∑Tiκστ∂piε1∂xσ∂ε1κ∂xτ +∂piε1∂xτ∂ε1κ∂xσ+∂2piε1∂xσ∂xτε1κ . He indicated that the term with pκε2ε2i gives a similar contribution by noting: “+ the same with i & κ exchanged” (“+ dasselbe mit vert[auschten] i & κ ”). Here the calculation seems to break off abruptly with Einstein concluding: “Leads to difficulties” (“Führt auf Schwierigkeiten.”). However, the considerations at the top of p. 12L (and some additions to p. 11L resulting from them) can be seen as a natural continuation of the search on p. 11R for Hertz transformations under which the core operator transforms as a tensor. Einstein may therefore only have added this final remark on p. 11R after running into difficulties on pp. 12L and 11L. Checking Whether Rotation in Minkowski Spacetime Is a Hertz Transformation Under Which the Core Operator Transforms as a Tensor (12L, 11L) At the bottom of p. 11R, Einstein had derived a condition determining under which subclass of infinitesimal Hertz transformations expression ()—the contraction of the core operator and an arbitrary second-rank covariant tensor—transforms as a scalar. He had found that, given the metric field, the matrices pμν for such transformations must satisfy the condition that the sum of expression () and a similar expression obtained by switching the indices i and κ vanish. For the special case of a flat diagonal metric, μνδμν , this condition reduces to: στ∂2piκ∂xσ∂xτ+∂2pκi∂xσ∂xτ0 , which is satisfied if pμνxpμν-δμν is anti-symmetric, i.e., pμνxpμνx .In that case, conditions () and () for infinitesimal Hertz transformations found on p. 10L are automatically satisfied as well, the latter because pκσx is traceless, the former because (cf. footnote 148): ∂pκσx∂xσ∂pσκx∂xσ∂pσσx∂xκ , which once again vanishes because pκσx is traceless. At the top of p. 12L, under the heading: “Attempt. Infinitesimal transformation is anti-symmetric. Rotation modified” (“Versuch. Infinitesimale Transformation ist schief symmetrisch. Drehung modifiziert”), Einstein turns to the investigation of anti-symmetric infinitesimal transformations. This quickly aborted attempt can thus be seen as a natural continuation of the considerations on p. 11R. Einstein wrote down the transformation law for the differentials dxμ under an infinitesimal coordinate transformation: dx′νdx+pνκxdxk . and noted the condition of anti-symmetry pνκxpκνx . As on p. 10L, he expressed the coefficients of the transformation in terms of the functions Xμ describing the coordinate transformation (cf. equations ()–()), pνκ∂Xν∂xκ , with the help of which he rewrote condition () as ∂Xν∂xκ∂Xκ∂xν . The comment “Rotation modified” (“Drehung modifiziert”) at the top of p. 12L indicates that Einstein was interested in the special case of rotation in Minkowski spacetime at this point. If the matrix pμνx for this transformation were anti-symmetric—which, of course, it is not—an infinitesimal rotation in Minkowski spacetime would be an example of a non-autonomous transformation under which expression () transforms as a scalar. It would then, presumably, also be a transformation under which the core operator transforms as a tensor. Einstein thus explored whether the matrix pμνx for rotation can meaningfully be made anti-symmetric. If this was indeed the point of modifying the matrix pμνx for rotation, Einstein had already achieved his goal without such modification. What he appears to have overlooked is that infinitesimal rotations in Minkowski spacetime already are infinitesimal Hertz transformations under which expression () transforms as a scalar. That they are Hertz transformations was shown in p. 11L (see sec. ). Moreover, the matrix pμνx for such transformations satisfies condition (). After all, pμνx is linear in xμ and condition () involves only second-order derivatives of pμνx . It is true that condition () is not satisfied, since pμνx≠pνμx , but that condition, although sufficient, is not necessary to meet condition (). Hence, Einstein did not need to modify pμνx for rotation at all. Einstein seems to have missed this and returned to p. 11L to see whether the matrix () for pμνx for infinitesimal rotation in Minkowski spacetime could be made anti-symmetric. As he indicated underneath the matrix on p. 11L, he replaced p31xp32x0 by ωy and ωx , respectively: 1ωt+yωωt1xω001 ωy +ωx . With this anti-symmetrized matrix the differential of the time coordinate transforms as: dt′-yωdx+xωdy+dt . Einstein wrote this equation in a separate box in the left margin of p. 11L. For this equation to be a coordinate transformation, dt′ has to be an exact differential, i.e., it must be possible to write it as: dt′∂ϕ∂xdx+∂ϕ∂ydy+∂ϕ∂tdt . Comparison of equations () and () gives ∂ϕ∂xyω and ∂ϕ∂yxω . This implies that ∂2ϕ∂x∂y≠∂2ϕ∂y∂x . Hence dt′ in equation () is not an exact differential. Einstein seems to have gone through this same argument himself, although the only trace of this in the notebook are the terms ∂ϕ∂x ∂ϕ∂y ∂ϕ∂z , written underneath equation () in the same separate box. In any event, he concluded that a transformation characterized by equation (), which would yield an antisymmetric matrix pμνx , is “impossible” (“unmöglich”). This is the last word in the separate box on p. 11L, and it signals the end of this whole line of reasoning, which started on p. 10L and ended with the first horizontal line on p. 12L. Einstein seems to have reached the conclusion that the core operator does not transform as a tensor under infinitesimal rotations in Minkowski spacetime. Initially, Einstein, it seems, considered changing the form of the core operator. Following the first horizontal line on p. 12L, Einstein changed the core operator () on p. 7L to: 1G∑∂Gμν∂iκ∂xν∂xμ . The extra factors of G make this expression resemble the second Beltrami invariant more closely (cf. equation () on p. 6L). Einstein did not even begin the search for non-autonomous transformations under which this modified core operator transforms as a tensor. Instead he drew another horizontal line and turned to a closer examination of the important special case of rotation in Minkowski spacetime that had spelled trouble for the original form of the core operator. Deriving the Exact Form of the Rotation Metric (12L–R) In the middle of p. 12L, under the heading, “The Rotational Field in First Approximation” (“Drehungsfeld in erster Annäherung”), Einstein wrote down the line element g11dx2+…+g44dt2ds2 , and defined H , the “Lagrangian function” (“Lagrange’sche Funktion”) for a point particle in a metric field, in terms of a potential term Φ and a kinetic term L : Φ-LHdsdt . The accompanying diagram (see below) and the subsequent calculations make it clear that Einstein considered the motion of the particle in a coordinate system xμx,y,z,t rotating counterclockwise at constant angular velocity ω around the z -axis, coinciding with the z′ -axis of an inertial coordinate system x′μx′,y′,z′,t′ . The line element in the inertial coordinate system is given by: ds2ημνdx′μdx′ν1-v′2dt′2 where ημνdiag1,1,1,1 (coordinates are chosen such that c1 ) and v′ix⋅′idx′idt′ . The relation between velocity in the inertial frame and velocity in the rotating frame is given by: v′v+ω→xx⋅-ωyy⋅+ωxz⋅x⋅-ωrωty⋅+ωrωtz⋅ , where ω→0,0,ω and r2x2+y2 . With the help of equation () and the transformation equation dt′dt , the Lagrangian () can be written as Hds2dt21-v′2≈1-v′22 . Comparison with HΦ-L leads to the identification 2Lv′2 . Einstein presumably arrived at this equation simply on the basis of the interpretation of L as the kinetic energy. Using equation (), he found: 2Lx⋅-ωrωt2+y⋅+ωrωt2+z⋅2 , which he expanded to: 2Lx⋅2+y⋅2+z⋅2-2ωrωtx⋅ + 2ωrωty⋅+ω2r2 . Substituting xrωt and yrωt (see equation ()) and introducing the potential energy ΦA ,As follows directly from equation () and as Einstein subsequently realized (see p. 12R), A cannot be chosen freely but has to be equal to 1. Einstein wrote the Lagrangian H as Φ-LA-ω2r22-x⋅2+y⋅2+z⋅22+ωyx⋅-ωxy⋅ . Given equation (), Einstein could find ds2dt2 by squaring equation (). Writing “ ds2dt [ dt should be dt2 ] calculated up to and including ω2 & x⋅2 ” (“ ds2dt berechnet bis und mit ω2 u. x⋅2 ”), he arrived at: ds2dt2=A2+ω2y2x⋅2+ω2x2y⋅2-Aω2r2 -A-ω2r22x⋅2+y⋅2+z⋅2+2Aωyx⋅-2Aωxy⋅ A simpler way of finding an expression for ds2dt2 is to use equations () and (): ds2dt21-v′21-ω2r2-x⋅2-y⋅2-z⋅2+2ωyx⋅-2ωxy⋅ . If we neglect terms containing both ω2 and x⋅i2 in eq. () and insert A1 , we recover equation (). At the top of p. 12R, Einstein used equation () in combination with ds2gμνdxμdxν , the expression for the line element in rotating coordinates, to identify the components of gμν , the Minkowski metric in rotating coordinates: This matrix contains several errors. First, the ω2 -terms in g11 and g22 come from terms in equation () containing both ω2 and x⋅i2 , which are negligible. This mistake was partly corrected.The term ω2r22 in the expressions for g11 , g22 , and g33 may have been deleted in the course of Einstein’s evaluation of the determinant of this metric. Second, the factors of 2 in g14 , g24 , g41 , and g42 should be 1. The term 2Aωydxdt in ( dt2 times) equation (), for instance, should be set equal to g14dx1dx4+g41dx4dx1 , leading to the identification g14g41Aωy .Einstein made the same mistake on p. 42R (see footnote 308) and in the Einstein-Besso manuscript (CPAE 4, Doc. 14, pp. [41–42]). Largely due to this error, Einstein convinced himself at that point that the rotation metric is a solution of the field equations of his Entwurf theory (see Janssen 1999, 145–146, and “What Did Einstein Know …” sec. 3 (in this volume). Immediately below the matrix (), Einstein noted that A1 (cf. note 174) Next, Einstein computed the determinant G of (), again retaining only terms up to order ω2 or x⋅i2 . Since so many of the elements of the matrix () are zero, there are only a few contributions to its determinant: Gg11g22g33g44-g11g33g422+g412 . Inserting the values given in the matrix () into this expression, setting A1 , and neglecting terms smaller than of order ω2 or x⋅i2 , we find a result very similar to the following expression in the notebook at this point:Expression () contains a number of errors. First, the last three minus signs in the second expression in ordinary brackets were all corrected from plus signs. The first two should indeed be minus signs but the third should be a plus sign. Secondly, the terms 4ω2x2 and 4ω2y2 , coming from the second term in equation (), should both be inside the curly brackets with a minus sign. G[-1+ω2r22+ω2y21-ω2r22-ω2x2-ω2r22-ω2r2+4ω2x2 +4ω2y2. Einstein rewrote the right-hand side of this equation as: = -1+ω2r22+ω2y2+ω2r22+ω2x2+ω2r22+ω2r2-4ω2x2+4ω2y2 and then as: = -1+2.5ω2r2-ω2r2-3ω2x2+5ω2y2 . These last two equations inherit the errors made in equation () (see note 177). Einstein may have realized that these equations contained some errors. He subsequently deleted all three equations ()–(). However, he retained the main result of his calculation on the lower half of p. 12L and the upper half of p. 12R, expression () for the Minkowski metric in rotating coordinates, which still contains several errors. Trying to Find Infinitesimal Unimodular Transformations Corresponding to Uniform Acceleration (12R, 41L–R) In the middle of p. 12R, under the heading, “Substitutions with Determinant 1. Infinitesimal in 2 Variables” (“Substitutionen mit Determinante 1. Infinitesimal in 2 Variablen”), Einstein wrote down the transformation law for coordinate differentials, dx′dx+p11xdx+p12xdy,dy′dy+p21xdx+p22xdy, under the transformation x′x+Xx,y,y′y+Yx,y, with (cf., e.g., p. 10L and equations ()–()): pijx∂X∂x∂X∂y∂Y∂x∂Y∂y . Next to the transformation law (), he wrote the unimodularity condition, i.e., the condition that the determinant of the transformation matrix equals 1 (cf. equations ()–()): p11x+p22x0 . With the help of equation (), this condition turns into ∂X∂x+∂Y∂y0 , which is automatically satisfied if there is a generating function ψx,y determining X and Y via X∂∂yψ, Y∂ψ∂x . With the help of equation (), the coefficients pijx can also be expressed in terms of ψ :In these equations, pij should be pijx . p11∂2ψ∂x∂y, p12∂2ψ∂y2,p21∂2ψ∂x2, p22∂2ψ∂x∂y. Using these expressions, Einstein rewrote the first line of the transformation () as dx′1+∂2ψ∂x∂ydx+∂2ψ∂y2dy . He did not bother to write down the corresponding equation for dy′ . Equation () in turn can be rewritten as dx′dx+∂∂x∂ψ∂ydx+∂∂y∂ψ∂ydydx+d∂ψ∂ydx+dX . This last expression already follows directly, of course, from equation (). Writing δx and δy for X and Y and using equation (), Einstein wrote δx∂∂yψ, δy∂∂xψ. He could thus write transformation () as x′x+∂ψ∂y,y′y-∂ψ∂x, which appear at the bottom of p. 12R enclosed in a box. In another part of the notebook, on pp. 41L–R, immediately following Einstein’s earliest considerations on gravitation (see sec. ), Einstein did the same calculation as on the bottom half p. 12R but this time pursued it a little further. At the bottom of p. 41L, he examined some specific choices for the generating function ψ . At the top of p. 41R, he chose one of the coordinates to be the time coordinate and compared the transformation for a particular choice of ψ with the transformation to a uniformly accelerated frame of reference, a transformation familiar from his papers on the static gravitational field. It seems plausible that the calculation on pp. 41L–R is just a continuation of the one on p. 12R. The calculation breaks off after Einstein failed to recover the transformation to a uniformly accelerating frame of reference in this manner. Below the horizontal line in the middle of p. 41L, under the heading, “Simplest Substitutions, whose Determinant = 1” (“Einfachste Substitution, deren Determinante = 1”), Einstein, as on p. 12R, began by writing down the transformation equations () for coordinate differentials, albeit in a more compact form than on p. 12R and leaving open the dimension of the space(-time) under consideration: dx′νdxν+∑pνσxdxσ . As on p. 12R (see equation ()), Einstein used the relations pμνx∂Xμ∂xν . Underneath equation (), Einstein wrote: “ Xν are homogeneous and of second degree in the coordinates. Only two coordinates are being transformed” (“ Xν sind homogen u. zweiten Grades in den Koordinaten. Es werden nur zwei Koordinaten transformiert”). These comments suggest that Einstein was interested at this point in the special case of uniform acceleration, which he explicitly considered at the top of p. 41R. In that case only two coordinates, x and t , transform non-trivially. Moreover, the function X in x′x+X (cf. equation ()) has to be proportional to t2 and cannot have a constant term to get the desired form x′x+fxt2 . In other words, X has to be “of second degree” and “homogeneous.” Einstein began by writing down the condition of unimodularity (cf. equations ()–()) ∂X∂x+∂Y∂y0 , which is satisfied automatically if there is a generating function ψx,y such that (cf. equation ()) X∂∂yψ, Y∂∂xψ . As before (see equation ()), the relation between the matrix pijx and the function ψ is given by p11x∂2ψ∂x∂y, p12x∂2ψ∂y2,p21x∂2ψ∂x2, p22x∂2ψ∂x∂y. Up to this point the argument on p. 41L is identical to the argument on p. 12R. Einstein now considered two specific choices for the generating function, namely ψr3 and ψr2x . For these two cases he evaluated the four elements of the matrix in equation (). Using that rx2+y2 , so that ∂r∂xxr and ∂r∂yyr , one recovers the results given by Einstein at this point, except for an overall factor of 3 in the case of ψr3 . Einstein effectively did the calculation with ψr33 . This function gives: ∂ψ∂xrx∂ψ∂yry ∂2ψ∂x2r+x2r∂2ψ∂x∂yxyr∂2ψ∂y2r+y2r The function ψr2x gives: ∂ψ∂xr2+2x2∂ψ∂y2xy ∂2ψ∂x26x∂2ψ∂x∂y2y∂2ψ∂y22x To the right of equations ()–(), Einstein wrote down two matrices 1 y � 1 ,       yxϕx0 y , which appear to be related to the considerations at the top of p. 41R. Apart from the 21-component, the second matrix is proportional to the matrix pijx for ψr2x , which is found upon substitution of equation () into equation (): pijx2y x3xy . As we shall see below, Einstein probably changed the 21-component of the second of the two matrices () to zero in the course of his calculations on p. 41R. At the top of p. 41R (the first four lines in the top left corner and the first two in the top right corner), Einstein examined the transformation generated by ψy2x , a modification of the function ψr2x considered at the bottom of p. 41L. Replacing y by t , he then compared this transformation to the transformation to a uniformly accelerated frame that he had considered in the context of his 1912 theory for static gravitational fields. Inserting equation () into equation () for two dimensions and setting ψr2x , one finds, using equation (): dx′=dx+p11xdx+p12xdy =dx+∂2ψ∂x∂ydx+∂2ψ∂y2dy =dx+2ydx+2xdy,dy′=dy+p21xdx+p22xdy =dy-∂2ψ∂x2dx-∂2ψ∂x∂ydy =dy-6xdx-2ydy. This result corresponds to the matrix pijx in equation () above. Einstein wrote at the top of p. 41R: dx′dx+αydx+xdy,dy′dy-αydy. This corresponds to the matrix pijxαy x0y, which, except for the factor α , is the second of the two matrices () on p. 41L. It seems that Einstein adjusted his choice of the function ψ to get the matrix pijx in equation () instead of the one in equation (). It is easily seen that to achieve this ψr2x should be replaced by ψy2x . If that is done, one finds (cf. equation ()–()): ∂ψ∂xy2∂ψ∂y2yx ∂2ψ∂x20∂2ψ∂x∂y2y∂2ψ∂y22x Using these results in equation () and writing α instead of 2, one finds Einstein’s equation (). Einstein now replaced y by t The second line of equation () originally had y instead of t . and integrated equations (), using that tdx+xdtdxt : x′x+αxt,t′t-αt22. He compared this transformation to the transformation to a uniformly accelerating frame x′x+12c∂c∂xt2,t′ct, which he had obtained in the context of his theory for static fields of 1912 (Einstein 1912b, 456) and which he had already used on p. 11L (see equation ()). The expressions for t′ in equations () and () are quite different. So even after changing the generating function from ψr2x (with r2x2+t2 ) to ψt2x , Einstein was unable to recover the transformation to a uniformly accelerating frame of reference by integrating the infinitesimal unimodular transformation () generated by ψ . On the remainder of p. 41R, Einstein went through a calculation showing that motion constrained to a curved surface in three-dimensional space is along a geodesic (see sec. ). The purpose of this calculation may simply have to been to reassure himself after the disappointing results of p. 12R and pp. 41L-R that at least the conceptual basis of his theory was sound. Geodesic Motion along a Surface (41R) On p. 41R, starting with the expression md2xdt2 , Einstein considered the motion of a particle in three-dimensional space constrained to move on a two-dimensional surface, but otherwise free of external forces. He proved that the trajectory of such a particle is a geodesic of the surface by showing that the line element of the trajectory is an extremal on the surface. Einstein had earlier recognized that the equation of motion of a particle in a static gravitational field follows from a variational principle for a four-dimensional line element.See the “Nachtrag” to (Einstein 1912b). At this point, he presumably realized that a similar result holds for the equation of motion in a general gravitational field.See Einstein to Ludwig Hopf, 16 August 1912: “The work on gravitation is going splendidly. Unless everything is just an illusion, I have now found the most general equations” (“Mit der Gravitation geht es glänzend. Wenn nicht alles trügt, habe ich nun die allgemeinsten Gleichungen gefunden.” CPAE 5, Doc. 416). In his earlier lecture notes on mechanics, Einstein had also treated constrained motion along a surface, but the concept of a geodesic line is not to be found in any of his published writings up to this point.See CPAE 3, Doc. 1, [34–38], [75–76]. It is not entirely clear why Einstein considered the problem of constrained motion in this context. Einstein had long been familiar with the link between the physical concept of constrained motion and the geometric concept of a geodesic line from a course on infinitesimal geometry that he had taken as a student at the ETH with Carl Friedrich Geiser.Einstein had registered for this course in winter semester 1897/1898 (see CPAE 1, Appendix E). Marcel Grossmann’s notes on these lectures contain a page with very similar calculations (Bibliothek ETH, Zurich, Hs. 421:15). Einstein started from the x -component of Newton’s equation of motion for a particle of mass m , constrained to move on the surface fx,y,z0 . md2xdt2λ∂∂xf . The analogous equations for the y – and z –components are indicated by dashed lines. As is clear from the accompanying figure, reproduced below, the right-hand side of this equation is the normal force that constrains the particle to move along the surface f0 . This normal force is proportional to the gradient of f , which defines the normal direction, and which must be multiplied by a Lagrange multiplier λ determined by the magnitude of the force. Einstein next absorbed m into a new Lagrange multiplier λ′ , and then changed independent variables, substituting the arc length s for the time t .There are no forces parallel to the surface, hence the speed of the point particle is constant. He thus arrived at d2xds2λ″∂∂xf , where λ″dt2ds2λ′ . The rest of the proof is intended to show that this is the equation of a geodesic line on the curved surface f0 .Cf. Grossmann’s notes on Geyser’s lectures as well as very similar passages in Einstein’s lecture notes on mechanics (CPAE 3, Doc. 4, [pp. 75ff.]). Einstein wrote down the equation for the surface, f0 , drew a horizontal line, and sketched the figure below. He then wrote down the coordinates: The coordinates ( xs , ys , zs ) refer to points on the actual path. To prove that this path is a geodesic, he considered a nearby curve, produced by small variations ( ξ , η , ζ ). Thus a point with coordinates ( x+ξ , y+η , z+ζ ) is a point on this nearby curve. Next, Einstein considered a nearby point on the actual path, with coordinates ( x+dx , y+dy , z+dz ), and a corresponding point on the path obtained through variation. He only wrote down the x-coordinate of the latter point: . Since Einstein was considering constrained motions, the path obtained through variation must also lie in the surface f0 , i.e., ( x+ξ , y+η , z+ζ ) must also be the coordinates of a point on the surface f0 . One can bring out the meaning of Einstein’s figure more clearly by adding the path obtained through variation and labeling the various points. Einstein now calculated the square of the line element ds′ for the path obtained through variation, discarding terms of order dξ2 : ds′2=dx+dξ2+.+. =ds2+2dxdξ+.+. =ds21+2dxdsdξds+.+.⋅. He then took the square root of this equation: ds′ds1+dxdsdξds+.+.⋅ . Substituting x⋅ for dxds and ξ⋅ for dξds , he wrote the variation in the line element as: ds′-dsx⋅ξ⋅+.+.ds . For a geodesic δ∫ds vanishes, which means that: ∫x⋅ξ⋅+.+.ds0 . Through integration by parts Einstein transformed this equation into: ∫x⋅⋅ξ+.+.ds0 , which will hold if the acceleration, with components ( x⋅⋅ , y⋅⋅ , z⋅⋅ ), is perpendicular to the variation, with components ( ξ , η , ζ ). Since these variations are arbitrary (except that they have to lay within the surface), they should be perpendicular to the normal to the surface: ∂∂xfξ+∂∂yfη+∂∂zfζ0 . Hence, as Einstein wrote, “if” (“wenn”) condition () holds, then the actual path will be a geodesic: “from which the assertion” (“woraus die Behauptung”). Emergence of the Entwurf Strategy (13L–R) On pp. 6L–12R (and pp. 41L–R), Einstein had tried in vain to find field equations invariant under a broad enough class of transformations—autonomous or non-autonomousSee the introduction to sec.  for discussion of the distinction and footnote for the origin of the terminology.—to meet the requirements of the relativity principle and the equivalence principle (see sec. ). He had pursued a combination of what we have called the mathematical and the physical strategy (see sec. ). Mathematically, the generally-covariant second Beltrami invariant (), with (some power of) the determinant G of the metric playing the role of the arbitrary scalar function ϕ in its definition, looked like an especially promising point of departure. Field equations constructed out of the Beltrami invariant in this way are invariant under arbitrary (autonomous) unimodular transformations. On p. 12R and p. 41R, however, just prior to the entries on pp. 13L–R, Einstein had reached the conclusion that the important special case of an (autonomous) transformation to a uniformly accelerating frame of reference is not a unimodular transformation, not even infinitesimally. This must have been an important setback. From a physics point of view, the core operator (), which for weak fields reduces to the d’Alembertian acting on the metric, looked most promising. The drawback was that the core operator does not transform as a tensor under any autonomous non-linear transformations. It might, however, transform as a tensor under a class of non-autonomous non-linear transformation that would include the important special cases of rotation and uniform acceleration in Minkowski spacetime (see pp. 11L–12L). And even if this turned out not to be true for the core operator taken by itself, it might still be true for the sum of the core operator and some correction terms. Such correction terms were needed anyway to guarantee energy-momentum conservation (see p. 9L and sec. , especially the passage following equation ()). With the help of such terms, it might furthermore be possible to connect field equations based on the core operator to the Beltrami invariants, which would throw light on their covariance properties. The Hertz restriction—the restriction to Hertz transformations, i.e., non-autonomous transformations under which the Hertz expression () transforms as a vector—played an important role in connecting the core operator to the Beltrami invariants. On p. 11L, Einstein had found that, at least infinitesimally, rotation in Minkowski spacetime was a Hertz transformation. He had also found, however, that uniform acceleration in Minkowski spacetime is not. Still, the physically motivated core operator clearly held more promise overall than the mathematically more elegant Beltrami invariants. It is not surprising therefore that Einstein, on pp. 13L–R, bracketed the problem of the covariance of the field equations for the time being. He now began looking for field equations based on the core operator initially demanding only that such equations be invariant under arbitrary autonomous unimodular linear transformations. Presumably, he would check later whether these equations were also invariant under non-autonomous non-linear transformations such as uniform rotation and acceleration in Minkowski spacetime as was required by the equivalence principle. That Einstein restricted himself to unimodular transformations suggests that he eventually still wanted to connect the field equations to the Beltrami invariants. On p. 13L, Einstein began an inventory of expressions involving the metric and its derivatives that transform as vectors and tensor under linear transformations and out of which he could therefore construct the correction terms to the core operator on the left-hand side of the field equations. On p. 13R, he substituted the core operator for the stress-energy tensor of matter in expression () for the energy-momentum balance between matter and field derived on p. 5R. In this way, it seems, Einstein hoped to identify the correction terms to the core operator that would guarantee the compatibility of the field equations with energy-momentum conservation. A variant of this strategy would subsequently lead to the Entwurf field equations. On p. 13R, however, Einstein quickly gave up on this line of reasoning. Marcel Grossmann then handed him a new mathematical quantity, which was far more promising than the Beltrami invariants. At the top of the very next page, p. 14L, the Riemann tensor makes its first appearance in the notebook. Pp. 14L–24L along with pp. 42L–43L are given over to attempts to extract field equations from this quantity along the lines of the mathematical strategy (see sec. ). Only after these attempts had failed did Einstein return to the strategy we see emerging on pp. 13L–R (see p. 24R and pp. 26L–R and secs.  and ). Bracketing the Generalization to Non-linear Transformations: Provisional Restriction to Linear Unimodular Transformations (13L) On p. 13L, under the heading, “Differential Covariants for Linear Substitutions, if one sets G1 ” (“Differentialkovarianten für lineare Substitutionen, falls G1 gesetzt wird”), Einstein started an inventory of quantities constructed out of the metric tensor and its derivatives that transform as vectors or tensors under (unimodular) linear transformations. All quantities that made it onto the list on p. 13L involve one and only one first-order derivative of the metric. Hence, they all fall under the heading “First Order” (“Erster Ordnung”) on the third line of p. 13L. Originally, this heading was numbered “1)” but the number was subsequently deleted and no quantities involving second-order derivatives of the metric are listed. This may be because Einstein was interested in constructing a stress-energy tensor for the gravitational field, which is a quantity quadratic in first-order derivatives of the metric (see sec. ). In addition to the “order” (“Ordnung”), Einstein also considered the “degree” (“Grad”) and the “multiplicity” (“Mannigfaltigkeit”) of the expressions he constructed. The degree of an expression is the number of factors of gμν and μν it contains. Its multiplicity is simply its rank.Einstein used this same terminology on p. 8L (see equation ()). Einstein denoted every free index of the vectors and tensors he constructed either by a dot (for a contravariant index) or a dash (for a covariant index). A contravariant vector is accordingly called a “point vector” (“Punktvektor”), a covariant vector a “plane vector” (“Ebenenvektor”).This terminology may have been inspired by Grassmann’s “Ausdehnungslehre” (Grassmann 1862). For evidence of Einstein’s reading of Grassmann in this period, see Einstein to Michele Besso, 13 May 1911 (CPAE 5, Doc. 267), and Einstein to Conrad Habicht, 7 July 1913 (CPAE 5, Doc. 450). Later in the notebook Einstein used this same terminology for tensors as well (see, e.g., pp. 17L–R). Similarly, a tensor with two covariant and one contravariant index, for instance, is called a “ ⋅ ⋅ tensor.” Einstein distinguished two ways of forming such vectors and tensors, “internal” (“Innere”) and “exterior” (“Aussere”) differentiation. In the case of “inner” differentiation, the index of the derivative operator is contracted with one of the indices of the components of the metric, so that a four-divergence is formed. In the case of “outer” differentiation, the index of the derivative operator is different from the indices of the components of the metric, so that a four-gradient is formed. With this explanation of Einstein’s terminology the list on p. 13L becomes largely self-explanatory. The first item on the list is the Hertz expression, a point vector of first order and first degree obtained through “internal” differentiation: ∑∂μν∂xν Punktvektor. Note that μν cannot be replaced by gμν in this expression since that would involve contraction over two covariant indices. The next items on the list are therefore ∑∂μν∂xi  ⋅ ⋅ - Tensor, andExpression () is, in fact, in Einstein’s terminology, a � � �Tensor . Einstein inadvertently may have thought for a moment that the character of the three indices in expression () would be just the opposite of those in expression (). ∑∂gμν∂xi  - - ⋅ Tensor, both obtained through “external” differentiation. On the next line, Einstein turned to expressions of “first order” and “second degree” (“zweiten Grades”) and began by writing down the four different possible vectors of this kind that can be obtained by contracting expressions ()–() with gμν and μν in various ways:Einstein omitted the summation sign in expression (). ∑gμσ∂μν∂xν Ebenenvektor, ∑gμν∂μν∂xi Ebenenvektor, ∑μν∂gμν∂xi Ebenenvektor, ∑μi∂gμν∂xi Ebenenvektor, Expressions () and () are connected by a curly bracket. Not only are they equal to one another (because of the relation ∂∂xigμνμν0 ), they both vanish because of the restriction to unimodular coordinates (for which G1 ) and the relation gμν∂μν∂xi1G∂G∂xi (cf. equation ()). Finally, under the heading, “In addition the tensors of third multiplicity” (“Dazu die Tensoren dritter Mannigfaltigkeit”), Einstein wrote down all third-rank tensors of first order and second degree that can be constructed out of expressions ()–(). The Hertz expression () can be turned into a tensor of this kind through multiplication with either the covariant or the contravariant metric, ∑giκiκ∂μν∂xν , giving, in Einstein’s terminology, a “ � tensor” and a “ � � � tensor,” respectively. Expressions () and () can be turned into third-rank tensors of first order and second degree by contracting them with gμν and μν . This leads to the last four expressions on p. 13L: ∑gκμ∂μν∂xi  �- [Tensor], ∑κi∂μν∂xi � � � [Tensor], ∑κμ∂gμν∂xi �- [Tensor], ∑κi∂gμν∂xi �- [Tensor]. Trying to Find Correction Terms to the Core Operator to Guarantee Compatibility of the Field Equations With Energy-Momentum Conservation (13R) The starting point of the considerations on p. 13R is equation () for the energy-momentum balance between matter and gravitational field derived on p. 5R. This equation is equivalent to the vanishing of the covariant derivative of the matter stress-energy tensor Tμν . In unimodular coordinates (for which G1 ) the left-hand side of equation () can be written as ∑μν∂gmνTμν∂xμ-12∑∂∂xmgμνTμν . Inserting the core operator (see, e.g., expression () on p. 11R), αβ∂2μν∂xα∂xβ , for the contravariant stress-energy tensor Tμν in expression () and adding an equality sign, one finds the first line of p. 13R: ∑αβμν∂∂xμgmναβ∂2μν∂xα∂xβ-12∑∂gμν∂xmαβ∂2μν∂xα∂xβ . On the next line, Einstein wrote “Third-order derivatives do not appear, if ∑μ∂μν∂xμ0 .” (“Dritte Ableitungen treten nicht auf, wenn … = 0 ist.”). Equation () is the by now familiar Hertz restriction. On the remainder of p. 13R, Einstein rewrote expression () using this restriction. Why was Einstein interested in expression ()? The simplest answer is that he wanted to find what further restrictions, if any, would be needed to guarantee that the field equations αβ∂2μν∂xα∂xβκTμν —understood either as exact or as weak-field equations—be compatible with energy-momentum conservation, i.e., with the vanishing of the covariant derivative of Tμν , or, in unimodular coordinates, the vanishing of expression (). If the field equations () hold, the vanishing of expression () for Tμν implies that expression () for the core operator () must also vanish. It is unlikely, however, that this was the point of Einstein’s considerations on p. 13R. Einstein already knew that the exact field equations cannot be obtained by simply setting the core operator equal to the stress-energy tensor of matter as in equation (). Energy-momentum conservation requires an additional term on the left-hand side that can be interpreted as gravitational energy-momentum density (see p. 9L and the discussion in sec. , especially equations ()–()). In other words, Einstein expected the exact field equations to have the form αβ∂2μν∂xα∂xβ-κtμνκTμν , where the quantity tμν , which represents gravitational energy-momentum, is assumed to be quadratic in first-order derivatives of the metric. In a weak-field approximation, this additional term can be neglected and equations () reduce to equations (). On p. 13R, however, terms quadratic in first-order derivatives of the metric are not neglected. This strongly suggests that Einstein was implicitly using field equations of the form of equation (). This in turn would mean that Einstein expected expression () to be equal, not to zero, but to ∑∂gmμκtμν∂xμ-12∑∂∂xmgμνκtμν . The equality of expressions () and () would guarantee the compatibility of energy-momentum conservation (in the form of the vanishing of expression ()) and field equations of the form (). Presumably, what Einstein tried to do on p. 13R was to rewrite expression () in the form of equation () and identify tμν . For one thing, this would explain his concern with the elimination of third-order derivatives from expression (). Since tμν only contains first-order derivatives of the metric, expression () will contain no derivatives higher than of second order. As we pointed out in sec. , this may also be why Einstein only listed quantities of “first order” (“Erster Ordnung”) on p. 13L. Note that on this reading Einstein must have come to realize that gravitational energy-momentum has a special status. On p. 9L he had still demanded that the quantity tμν representing gravitational energy-momentum satisfy equation () posited for all energy-momentum on p. 5R (cf. footnote 128). In that case the right-hand side of equation () would simply be equal to zero rather than to expression (). When the Hertz restriction () is imposed, the left-hand side of equation () reduces to ∂∂xμgmναβ∂2μν∂xα∂xβ-12∂gμν∂xmαβ∂2μν∂xα∂xβ , which no longer contains any third-order derivatives of the metric. Regrouping terms, one can rewrite this expression in the way Einstein wrote it on the third line of p. 13R: ∑∂gmν∂xμ-12∂gμν∂xmαβ∂2μν∂xα∂xβ+∑μναβgmν∂αβ∂xμ∂2μν∂xα∂xβ . Einstein concentrated on the second summation in this expression. Using ∂∂xμ(∑νgλνμν)0 and the Hertz restriction (), he arrived at: ∑μν∂gλν∂xμ0 . He tried to rewrite the second summation in expression () in such a way that he could apply equation () to the term ∂gmν∂xβμν , but quickly realized that this was not feasible. He began by pulling out the differentiation with respect to xα , thus arriving at:Einstein drew a line connection expression () with the second summation in expression ().,In this equation ∂gμν∂xα should be ∂gmν∂xα . Because of the Hertz restriction there is no term with ∂2αβ∂xα∂xμ . ∂∂xαgmν∂αβ∂xμ ∂μν∂xβ -∂gμν∂xα ∂αβ∂xμ ∂μν∂xβ . He then used relation () to rewrite the first term as: ∂∂xα∂gmν∂xβμν∂αβ∂xμ . Here the calculation breaks off. Einstein did not return to considerations of energy-momentum conservation until p. 19R. At that point he had further deepened his understanding of the special status of gravitational energy-momentum. Interpreting the second term in expression () as the gravitational force density, he tried to rewrite that term as the divergence of the quantity tμν representing the gravitational energy-momentum density (see the discussion following equation () in sec. ). Later in the notebook (on p. 24R and, more systematically, on pp. 26L-R), he used this insight to derive field equations that are automatically compatible both with energy-momentum conservation and with Newtonian theory for static weak fields (see sec.  and sec. ). This led him to the Entwurf field equations. The notion that energy-momentum conservation requires coordinate restrictions over and above the ones needed to recover Newtonian theory for static weak fields stayed with Einstein right up until his introduction of generally-covariant field equations in November 1915.Einstein later compared the Hertz restriction with coordinate restrictions in (Einstein and Grossmann 1914), which not only circumscribe the covariance of the Entwurf theory but also guarantee energy-momentum conservation (see Einstein to Paul Hertz, 22 August 1915 [CPAE 8, Doc. 111]). For further discussion of the role of coordinate restrictions in determining covariance properties, recovering Newtonian theory, and guaranteeing energy-momentum conservation, see “Untying the Knot …” sec. 1.1 (in this volume). Exploration of the Riemann Tensor (14L–25R, 42L–43L) Introduction (14L–25R, 42L–43L) A new stage in Einstein’s search for gravitational field equations began on p. 14L of the notebook with the systematic exploration of the Riemann tensor along the lines of the mathematical strategy.See sec.  for a characterization of the difference between the mathematical and the physical strategy. In the course of this exploration, Einstein considered various gravitational field equations based on the Ricci tensor that he would publish in his communications to the Prussian Academy of November 1915 (Einstein 1915a, b, d). He even considered, albeit only in linear approximation, the crucial trace term that occurs in the final version of the field equations. However, the episode that, from a modern point of view, begins so promisingly on p. 14L with the introduction of the Riemann tensor ends disappointingly on p. 26L with the derivation of the Entwurf field equations along the lines of the physical strategy. What happened on these pages that made Einstein abandon the mathematical strategy and return to the physical strategy? The analysis of pp. 14L–25R and related material on pp. 42L–43L, the last three pages of the part starting from the other end of the notebook, reveals a pattern that holds the key to our answer to this question. Einstein’s starting point invariably is some expression of broad if not general covariance constructed out of the Riemann tensor. To extract from these expressions field equations that reduce to the Poisson equation of Newtonian theory in the special case of weak static fields, Einstein introduced various coordinate restrictions.See sec.  for a discussion of the notion of a coordinate restriction. With the help of these he could eliminate unwanted terms with second-order derivatives of the metric. The left-hand sides of the resulting field equations consist of a term with a core operator (i.e., a term that, in linear approximation, reduces to the d’Alembertian acting on the metric), and terms quadratic in first-order derivatives of the metric, which vanish in linear approximation. Einstein then checked whether these field equations and the coordinate restrictions used in their construction satisfy his other heuristic requirements,See sec.  for a discussion of what we identified as the four major heuristic principles: the correspondence principle, the conservation principle, the relativity principle and the equivalence principle. in particular whether they are compatible with energy-momentum conservation and whether they are covariant under a wide enough class of coordinate transformations (autonomous or non-autonomousSee sec.  for the definitions of the notions of autonomous and non-autonomous and transformations.) to be compatible with the equivalence principle and a generalized relativity principle. All candidates considered by Einstein failed at least one of these tests. Finding coordinate restrictions of sufficiently broad covariance turned out to be particularly difficult. Eventually, Einstein switched back to the physical strategy. He developed the considerations of energy-momentum conservation on p. 13R into a method for generating field equations guaranteed both to have the desired form in linear approximation, and to be compatible with energy-momentum conservation. Field equations constructed in this manner have the same general form as the candidate field equations that Einstein had extracted from the Riemann tensor by eliminating unwanted second-order derivative terms with the help of coordinate restrictions. This suggested that the field equations generated by this new method could also be produced by the mathematical strategy. It remained unclear, of course, from which generally-covariant expression they could be extracted and whether the necessary coordinate restrictions would be of sufficiently broad covariance and would themselves be compatible with energy-momentum conservation. However, it must have been encouraging that the physical strategy yielded field equations, satisfactory on all other counts, of exactly the form that Einstein had come to expect while pursuing the mathematical strategy in his exploration of the Riemann tensor. And as far as the unknown covariance properties of the equations were concerned, the mathematical strategy, for all its promise on this score, had not allowed Einstein to make any real progress either. It thus becomes understandable that Einstein eventually gave up the idea of constructing field equations out of the Riemann tensor and instead adopted the Entwurf equations. General Survey (14L–25R, 42L–43L) At the top of p. 14L Einstein wrote down the fully covariant form of the Riemann tensor. He wrote Grossmann’s name right next to it, suggesting that it was Grossmann who had drawn his attention to this tensor and its importance. Einstein proceeded to form the covariant form of the Ricci tensor by contracting the Riemann tensor over two of its four indices. He looked at the terms involving second-order derivatives of the metric and immediately noticed that in addition to a core-operator term the Ricci tensor contains three other second-order derivative terms that should not occur in the Newtonian limit. So the natural way of extracting a second-rank tensor from the Riemann tensor did not seem to produce a suitable candidate for the left-hand side of the field equations (sec. In this general survey we shall give a reference at the end of each brief summary of a calculation or a line of reasoning in this part of the notebook to the subsection where the calculation or the argument just summarized is discussed in much greater detail.). On pp. 14R–16R, Einstein explored a different way of extracting a two-index object from the Riemann tensor that, if not generally covariant, at least promised to be a tensor under unimodular transformations. In close analogy to his treatment of the Beltrami invariant on p. 9L, he computed the curvature scalar by fully contracting the Riemann tensor. Setting the determinant of the metric equal to unity, he succeeded in rewriting the curvature scalar as the contraction of the metric with a symmetric two-index object denoted (on p. 16L) by Tik . Einstein presumably expected that Tik would turn out to be a tensor under unimodular transformations. Unfortunately, Tik still contains two terms with unwanted second-order derivatives of the metric in addition to a core-operator term (sec. ). On the following pages (pp. 17L–18R), Einstein investigated the relation between the two expressions that he had formed out of the Riemann tensor. Since Tik gives the curvature scalar when contracted with the covariant metric, it is itself a contravariant object. To facilitate comparison between Tik and the covariant Ricci tensor, Einstein (on p. 17L) first formed the contravariant version of the latter. He tried to simplify the resulting expression using that covariance under unimodular transformations is all that matters for the comparison with Tik . This meant that he could assume the determinant of the metric to be a constant. He abandoned this calculation as “too involved.” On p. 17R he formed the covariant version of Tik instead and started to bring the covariant Ricci tensor into a form in which it could be compared with this version of Tik , again assuming the determinant of the metric to be a constant. This calculation also turned out to be complicated, and was abandoned as well (sec. ). Since neither the Ricci tensor nor Tik had the form required by the correspondence principle, Einstein began to investigate the possibility of obtaining suitable candidates for the left-hand side of the field equations by restricting the range of admissible coordinates. On p. 19L he showed that the terms in the Ricci tensor with unwanted second-order derivatives can be eliminated by imposing the harmonic coordinate restriction (sec. ).Considering p. 19L in isolation, one would think that Einstein was simply applying the harmonic coordinate condition in the modern sense to recover the Poisson equation in the limit of weak static fields. This interpretation is even compatible with the whole passage dealing with the Ricci tensor in harmonic coordinates (pp. 19L–21R). The interpretation is incompatible, however, with Einstein’s usage of coordinate conditions elsewhere in the notebook, both on pages preceding and on pages following the examination of the harmonic coordinate condition (cf. especially p. 23L). These other pages suggest that what Einstein had in mind throughout the notebook were coordinate restrictions rather than coordinate conditions in the modern sense. Cf., e.g., our discussion at the end of sec. . He then checked whether these field equations and this coordinate restriction are compatible with his other heuristic requirements. On p. 19R, Einstein examined in linear approximation the harmonic restriction and the field equations constructed with the help of it. Einstein confirmed that the weak-field field equations are compatible with his conservation principle: with the help of these equations, the term giving the gravitational force density in the energy-momentum balance between matter and gravitational field can be written as the divergence of a quantity representing gravitational stress-energy density. To guarantee compatibility between the weak-field equations and energy-momentum conservation, however, Einstein had to impose an additional coordinate restriction, a linear approximation of the Hertz restriction.See sec.  for the introduction of this coordinate restriction. Together, the harmonic restriction and the Hertz restriction imply that the trace of the weak-field metric has to vanish. In turn, this implies that the trace of the stress-energy tensor for matter has to vanish (sec. ). To avoid these consequences, Einstein (on p. 20L) modified the weak-field equations by adding a term proportional to the trace of the weak-field metric. This term was introduced in such a way that Einstein could now use the harmonic restriction to satisfy both the conservation principle and the correspondence principle. From a purely mathematical point of view, the resulting weak-field equations are the Einstein field equations of the final theory of November 1915 in linear approximation. As Einstein checked briefly on p. 20L and more carefully on p. 21L, with these modified field equations the gravitational force density can still be written as the divergence of gravitational stress-energy density (sec. ). The modified weak-field equations, however, do not allow the spatially flat metric that Einstein continued to use to represent static fields.This form of the static metric is also incompatible with the harmonic coordinate restriction, but it is unclear whether Einstein realized that at this point. Given this disparity, Einstein reconsidered his presupposition concerning the form of the metric of weak static fields. On p. 21R he presented a seductive but ultimately fallacious argument that corroborated his prior beliefs on this point. The argument was based on the dynamics of point particles (recapitulated on p. 20R) and Galileo’s principle that all bodies fall with the same acceleration in a given gravitational field. This powerful physical argument seemed to rule out the harmonic restriction (secs. and ). Einstein, however, was not ready to give up his attempt to extract the left-hand side of the field equations from the Riemann tensor. On p. 22R, at the suggestion of Grossmann perhaps, whose name once again appears at the top of the page, he turned to a different coordinate restriction that might help him achieve his goal. First, he noticed that the Ricci tensor can be split into two parts, each of which by itself transforms as a tensor under unimodular transformations. Einstein took one of these as his new candidate for the left-hand side of the field equations. We call this part the November tensor, because it is the left-hand side of the field equations published in the first of Einstein’s four papers of November 1915(Einstein 1915a). (sec. ). The November tensor still contains terms with unwanted second-order derivatives of the metric. Einstein eliminated these by imposing the Hertz restriction. The calculations on p. 19R had shown that the Hertz restriction is compatible with energy-momentum conservation in the weak-field case without the need for an additional trace term in the weak-field equations.Unlike the harmonic restriction, the Hertz restriction as well as the restriction to unimodular transformations are compatible with Einstein’s assumptions concerning the form of the weak-field static metric (cf. footnote 201 above). Given Einstein’s understanding of the status of coordinate restrictions at the time, the logical next step was to investigate the group of transformations allowed by the Hertz restriction. On the facing page (p. 22L) Einstein did indeed derive the condition for non-autonomous transformations leaving the Hertz restriction invariant. Earlier in the notebook (pp. 10L–11L), he had already found that the Hertz restriction rules out uniform acceleration in the important special case of Minkowski spacetime (sec. ).Contrary to what Einstein had concluded on p. 11L (see sec. ), the Hertz restriction also rules out finite transformation to rotating frames in Minkowski spacetime. He nonetheless held on to the Hertz restriction for the time being. The elimination of terms with unwanted second-order derivatives of the metric from the November tensor had left Einstein (at the bottom of p. 22R) with a candidate for the left-hand side of the field equations containing numerous terms quadratic in first-order derivatives. On p. 23L he added another coordinate restriction to the Hertz restriction to eliminate some of these terms. Just as he had obtained the November tensor by splitting the Ricci tensor into two parts each of which transforms as a tensor under unimodular transformations, Einstein (on p. 23L) obtained yet another candidate for the left-hand side of the field equations by splitting the November tensor into various parts, each of which transforms as a tensor under a class of unimodular transformations under which a quantity which we shall call the ϑ -expression transforms as a tensor. Restricting the allowed transformations to such ϑ -transformations, Einstein found that he could eliminate all but one of the terms quadratic in first-order derivatives of the metric from the November tensor. He furthermore discovered that the restriction to ϑ -transformations sufficed to eliminate the terms with unwanted second-order derivatives as well. There was no need to add the Hertz restriction to the ϑ -restriction. Einstein thus abandoned the Hertz restriction and focused on the ϑ -restriction instead. On p. 23R Einstein began to investigate the covariance properties of the ϑ -expression. As with the Hertz restriction, he derived the condition for non-autonomous transformations leaving the ϑ -expression invariant (sec. ). He did not attempt, however, to find the most general (non-autonomous) transformations satisfying this condition. Instead, he focused on the important special case of a vanishing ϑ -expression. The ϑ -expression vanishes, for instance, for the Minkowski metric in standard diagonal form. According to Einstein’s heuristic principles, it should therefore also vanish for the Minkowski metric in accelerated frames of reference (sec. ). It was thus natural for Einstein to investigate what metric fields are allowed by the condition that the ϑ -expression vanish. On pp. 42L–43L,These pages are found toward the end of the part starting from the other end of the notebook. At the beginning of sec.  we address the question of the temporal order of the calculations on pp. 42L–43L at one end of the notebook and those on pp. 23L–24L at the other. Einstein addresses just this problem, further limiting himself to time-independent metric fields and hence to uniformly accelerated frames of reference. The main result of his investigation was both promising and puzzling. He discovered that the ϑ -expression vanishes for a metric, which we shall call the ϑ –metric, whose covariant components are the contravariant components of the rotation metric, i.e., the Minkowski metric in rotating coordinates (sec. ). Einstein tried to come to terms with this result in various ways. First, on the bottom half of p. 42R and again at the bottom of p. 43La, he inserted the suggestive ϑ –metric into the Lagrangian for a point particle moving in a metric field and began to compute the Euler-Lagrange equations. Presumably, the idea was to check whether the components of the ϑ –metric and its derivatives could be given a physical interpretation in terms of centrifugal and Coriolis forces. Einstein broke off this calculation without reaching a definite conclusion (sec. ). A variant of this approach can be found on p. 24L, the page immediately following the introduction of the ϑ -restriction. This time he took the energy-momentum balance between matter and gravitational field as his starting point rather than the Lagrangian for a point particle moving in a metric field (sec. ). Einstein tried a different approach on p. 43La. He replaced the covariant components of the metric in the ϑ -expression by the corresponding contravariant ones. Since the original ϑ -expression vanishes for the rotation metric if only its co- and contravariant components are interchanged, the new version will vanish for the rotation metric itself. This approach did not work either. After three failed attempts to come to terms with the ϑ -metric, Einstein gave up the ϑ -restriction, though not the idea of extracting field equations from the November tensor (sec. ). Although the actual calculation has not been preserved in the notebook, there are strong indications that Einstein at this point did for field equations extracted from the November tensor what he had done earlier (on pp. 19R, 20L, and 21L) for field equations extracted directly from the Ricci tensor. He confirmed that in linear approximation these field equations can be used to write the gravitational force density as the divergence of a quantity representing gravitational stress-energy density. At the top of p. 24R, Einstein wrote down an expression that is most naturally interpreted as the end result of this calculation and noted that it vanishes for the rotation metric. Given some errors in the expression for the rotation metric on p. 24R, one can understand how Einstein reached this conclusion. When these errors are corrected, one sees that the expression, in fact, does not vanish for the rotation metric (see the first half of sec. ). It is the next step that Einstein took on p. 24R that marks the beginning of Einstein’s return to the physical strategy abandoned on p. 13R. So far Einstein had only verified in linear approximation that various candidate field equations allowed him to write the gravitational force density as the divergence of the gravitational stress-energy density. It was not at all clear whether this result would also hold exactly. On p. 24R Einstein introduced an ingenious new approach to this problem. Rather than starting from some candidate field equations and using them to rewrite the gravitational force density as a divergence without neglecting terms of one order or another, Einstein started from the expression for the divergence of the stress-energy density obtained in linear approximation and determined which higher-order terms need to be added to the linearized field equations such that this divergence becomes exactly equal to the gravitational force density. This is the strategy that Einstein used to derive the Entwurf equations (see pp. 26L–R).For a comparison between this method of finding candidate field equations and the method used on pp. 19L–23L, see the introduction to sec. . It is in this context that it was very important for Einstein to check whether the expression at the top of p. 24R would vanish exactly for the rotation metric. This is a necessary condition for the rotation metric to be a solution of field equations constructed with the help of the new strategy. Once he had derived the new field equations, he checked directly whether the rotation metric is a solution and discovered that it is not. However, he also discovered that he had erroneously cancelled two terms in his construction of the new field equations which opened up the possibility that the rotation metric would be a solution of the corrected equations (see the second half of sec. ). On p. 25L Einstein tried to recover his new candidate field equations from the November tensor by imposing an appropriate coordinate restriction. It seems to have been Einstein’s hope at this point that he could derive the same field equations following either the physical or the mathematical strategy (sec. ). Material at the top of p. 25R shows that Einstein considered a variant of the ϑ -restriction, which we shall call the ϑ^ -restriction, to extract field equations from the November tensor. It is unclear whether he was still trying to recover the field equations found on p. 24R in this way. Einstein went back to his calculations on p. 23L, where he had applied the original ϑ -restriction to the November tensor, and indicated the changes that would need to be made if the ϑ -restriction were replaced by the ϑ^ -restriction. Returning to p. 25R, he checked whether the ϑ^ -expression vanishes for the rotation metric, as required by the equivalence and relativity principles. He discovered that it does not, which is probably why he abandoned the ϑ^ -restriction (sec. ). This marks the end of Einstein’s pursuit of the mathematical strategy in the notebook. On the remainder of p. 25R, Einstein started tinkering with the field equations he had found on p. 24R to ensure that the rotation metric would be a solution. Einstein convinced himself that a slightly modified version of the equations satisfies this requirement. These modified equations, however, are not well-defined mathematically (they involve contractions over pairs of covariant indices among other things). It is unclear whether Einstein came to recognize this (sec. ). He abandoned these ill-defined equations when he found that it is not possible to rewrite the gravitational force density as the divergence of gravitational stress-energy density with their help (sec. ). On the next page (p. 26L), Einstein made a fresh start with the physical strategy. A possible indication that he had meanwhile abandoned his hope of recovering field equations found in this manner from the Riemann tensor is that he no longer set the determinant of the metric equal to unity, as he had still done on pp. 24R–25R in the hope perhaps to connect the field equations of p. 24R to the November tensor. The exploration of the Riemann tensor had nonetheless been fruitful (independently of the developments of November 1915). Even though it had failed to produce satisfactory field equations with a well-defined covariance group, it had given Einstein a clear idea of the structure any such field equations would have after unwanted terms in the Ricci tensor or the November tensor had been eliminated by imposing the appropriate coordinate restrictions. He had found a strategy to generate field equations of this form that automatically satisfy the correspondence and conservation principles. Given the difficulties Einstein had run into trying to make sure that the relativity and equivalence principles are satisfied as well, one can understand that he decided to bracket those problems for the time being. First Attempts at Constructing Field Equations out of the Riemann Tensor (14L–18R) On p. 14L the Riemann tensor appears for the first time in the notebook. Einstein immediately considered the Ricci tensor as a candidate for the left-hand side of the field equations, but ran into the problem that it contains unwanted terms with second-order derivatives of the metric. On the following pages (pp. 14R–18R), Einstein extracted another candidate for the left-hand side of the field equations from the curvature scalar and compared the result to the Ricci tensor. Unfortunately, this new candidate also contains unwanted second-order derivative terms. Moreover, its relation to the Ricci tensor remained unclear. Building a Two-Index Object by Contraction: the Ricci Tensor (14L) At the top of p. 14L Einstein wrote down the definitions of the Christoffel symbols of the first kind, [μ νl]12∂gμl∂xν+∂gνl∂xμ-∂gμν∂xl , and of the fully covariant Riemann tensor, iκ,lm12∂2gim∂xκ∂xl+∂2gκl∂xi∂xm-∂2gil∂xκ∂xm-∂2gκm∂xi∂xl +∑ρσρσ[imσ ][ κlρ ]-[ilσ ][ κmρ ]. Next to the Riemann tensor Einstein wrote “Grossmann Tensor vierter Mannigfaltigkeit,” indicating that it was his friend and colleague Marcel Grossmann who introduced him to these mathematical objects. Grossmann’s sources for the Christoffel symbols and the Riemann tensorIt should be emphasized that by calling the expressions () and () “Christoffel symbols” and “Riemann tensor,” we do not mean to suggest that these were the terms used by contemporary authors. were in all probability (Christoffel 1869) and (Riemann 1867).For further discussion of Riemann’s paper, see, e.g., (Reich 1994, sec. 2.1.3). Grossmann was aware of Riemann’s and Beltrami’s work in non-Euclidean geometry at least as early as 1904, as the introductory paragraph of (Grossmann 1904) shows. These are the sources cited for the Riemann tensor in Grossmann’s section of the Entwurf paper,(Einstein and Grossmann 1913, 35). In the case of Riemann the reference is to “Riemann, Ges. Werke, S. 270”. The page number does not make sense for the first edition of 1876 nor for the second edition of 1892. In the first edition Riemann’s “Commentatio” starts on p. 370; in the second on p. 91. It seems probable that Einstein and Grossmann used the first edition and that “270” is a misprint and should be “370.” Among other things, the second edition differs from the first in the German notes to the latin text of the “Commentatio.” Unfortunately, the reference in the Entwurf paper is Einstein’s most explicit reference to Riemann. In (Einstein and Fokker 1914, p. 325), for instance, he simply referred to the “bekannten Riemann-Christoffelschen Tensor.” In (Einstein 1914b, 1053) and in (Einstein 1916, 799), the phrase “Riemann-Christoffel tensor” occurs in (sub)section headings without any reference to the literature. where it appears in almost identical form.In the Entwurf paper, the indices ρ and σ in the second part of equation () are interchanged (Einstein and Grossmann 1913, 35). A comparison of the notation used for the Christoffel symbols and the Riemann tensor by various contemporary authorsSee (Reich 1994, 232) for a survey of the tensor analytic notation employed by various authors. further supports this conjecture.Some indirect confirmation of the conjecture comes from the draft of a letter from Felix Klein to Einstein of 20 March 1918. Commenting on a set of lecture notes he promised to send Einstein, Klein wrote: “You will probably immediately agree with what I have to say about Riemann, Beltrami, and Lipschitz; it seems to me that Grossmann at the time instructed you too much from the point of view of the school of Christoffel more narrowly” (“Was ich von Riemann, Beltrami, und Lipschitz erzähle, wird wohl gleich ihren Beifall haben; es scheint mir, dass Grossmann Sie s. Z. zu einseitig vom Standpunkte der engeren Christoffelschen Schule aus instruiert hat.” CPAE 8, Doc. 487, note 26). The square brackets for the Christoffel symbols were introduced by Christoffel himself, and, according to the survey in (Reich 1994), this notation was taken up again only in 1912 by Friedrich Kottler and then in the Entwurf paper itself, which cites (Kottler 1912) (Einstein and Grossmann 1913, 23 and 30). The notation for the Riemann tensor used in Christoffel’s paper is gkhi , i.e., it does not have the comma, which appears in equation (). The comma was used by Riemann whose notation was ιι′,ι″ι″′ . Also the occurrence of the word “Mannigfaltigkeit,” a phrase used by Riemann that does not appear anywhere in Christoffel’s paper, suggests that Einstein and Grossmann consulted Riemann’s paper in addition to Christoffel’s. There is yet another indication, however, that the actual expressions were directly taken from Christoffel’s paper. In the top right corner, Einstein wrote the expression ∂[i lm]∂xκ-∂[κ lm]∂xi , which gives exactly the four second-order derivative terms of the Riemann tensor in equation (), if the definition of the Christoffel symbols in equation () is inserted. Just this derivation of the Riemann tensor in terms of derivatives and products of the Christoffel symbols was given on the relevant page (p. 54) of (Christoffel 1869), where the Riemann tensor was first published. The Riemann tensor formed a promising starting point for Einstein’s search for gravitational field equations. The first task was to construct from the fourth-rank Riemann tensor an expression that could be used as the left-hand side of field equations with the second-rank stress-energy tensor on the right-hand side. The most natural way to obtain such an expression is to contract over two indices, thus forming the second-rank Ricci tensor. This is exactly what Einstein did on the next line ∑κliκ,lm . He put a question mark next to this expression, presumably to indicate that he wanted to check whether this would be an acceptable candidate for the left-hand side of the field equations It can be seen immediately that there is a problem with the correspondence principle. Upon contraction, the first of the four second-order derivative terms in equation () gives the core-operator term ∑κl∂2gim∂xκ∂xl , but the other three terms should not occur in the Newtonian limit. In fact, at the bottom of the page, underneath a horizontal line, Einstein wrote down these three bothersome terms ∑κ∂2gκκ∂xi∂xm-∂2giκ∂xκ∂xm-∂2gmκ∂xκ∂xi , set them equal to zero, and remarked that they “should vanish” (“Sollte verschwinden”).In deriving this expression, Einstein apparently assumed a weak-field metric with components diag1,1,1,1 to zeroth order, which would explain why the contracting metric does not appear explicitly in the equation. Einstein not only considered the second-order derivative terms in the Ricci tensor, he also started to rewrite the first-order derivative terms. First, he introduced the relationA factor 12 is missing in the expressions following the equality signs. ∑κlκ lρ∑κl∂gκρ∂xl+∂glρ∂xκ-∂gκl∂xρ=-∂lgG∂xρ+2∑κlκl∂gκρ∂xl, where G is the determinant of the metric.The introduction of G into the expression for the Ricci tensor would come to play an important role on p. 22R. He inserted (the corrected version ofSee footnote 214.) this relation into one of the two terms in the Ricci tensor with a product of Christoffel symbols ∑κlρσi mσκ lρ=14∑ρσ∂giσ∂xm+∂gmσ∂xi-∂gim∂xσ-∂lgG∂xρ+2∑κlκl∂gκρ∂xl. In the next step, without writing down their definition, Einstein used the Christoffel symbols of the second kind, i mρρσi mσ12ρσ∂giσ∂xm+∂gmσ∂xi-∂gim∂xσ , again following the notation of Christoffel’s 1869 paper. With the help of equations () and (), and the following relation between Christoffel symbols of the first and the second kind, κlκ mρ12κl∂gκρ∂xm+∂gmρ∂xκ-∂gκm∂xρ=κl∂gκρ∂xm+12κl∂gmρ∂xκ-∂gκm∂xρ-∂gκρ∂xm=κl∂gkρ∂xm-κlm ρκ=κl∂gκρ∂xm-m ρl, he rewrote the relevant part of the Ricci tensor asA factor 12 is missing in the first two terms on the right-hand side (cf. footnote 214). ∑κlρσi mσκ lρ-i lσκ mρ=∑ρ-i mρ∂lgG∂xρ+2∑κlρi mρκl∂gκρ∂xl-∑ρlκi lρ∂gκρ∂xmκl+∑ρli lρρ ml. In summary, p. 14L introduces a set of new invariant-theoretical quantities taken from the mathematical literature that looked promising for constructing gravitational field equations. In his first exploration of these new possibilities, however, Einstein had also hit upon a serious obstacle, viz. the problem of unwanted second-order derivative terms in addition to a core-operator term. Extracting a Two-Index Object from the Curvature Scalar (14R–16R) On the following five pages, Einstein computed the curvature scalar from the Riemann tensor. He simplified the calculation by setting the determinant of the metric equal to unity. As becomes clear toward the end (on p. 16L), the aim of the calculation was to produce an object which transforms as a scalar under unimodular transformations and which is the contraction of the metric and a new two-index object. Einstein presumably expected that this object would transform as a second-rank tensor under unimodular transformations. It would thus be another candidate for the left-hand side of the field equations. Unfortunately, the object, like the Ricci tensor, contains unwanted second-order derivative terms. At the top of p. 14R, Einstein fully contracted the Riemann tensor to form the curvature scalar: ϕimκliκ,lm . Inserting expression () for the fully covariant Riemann tensor on p. 14L into this equation, one arrives at the expression given at the top of p. 14R. The lines and arrows underneath this expression provide a flow-chart for the calculation on the next five pages. They are meant to show at a glance exactly which terms in the expression for ϕ are dealt with on which pages. Einstein first dealt with the second summation in equation (). On p 14R he expanded and simplified the first of the two terms in this summation, ρσimκl[imσ ][κlρ ] . This is equal to 1/4 times the sum of the underlined terms on p. 14R ρσ∂lgG∂xσ ∂lgG∂xρ+3∂lgG∂xσ ∂σα∂xα+∂lgG∂xσ ∂ρα∂xα+4gκρ∂κα∂xα∂ρβ∂xβ . On p. 15L he expanded and simplified the second term, ρσimκl[ilσ ][κmρ ] , the result being 1/4 times the sum of the underlined terms on p. 15L κl∂mρ∂xl∂gmρ∂xκ+im∂κρ∂xi∂gκρ∂xm+ρσ∂κm∂xσ∂gκm∂xρ +gκρ∂∂xσκm∂∂xmρσ+gmρ∂∂xσmκ∂∂xκρσ. Einstein further simplified expressions () and () by setting G1 and by grouping together identical terms in expression (). At the top of p. 15R, he wrote: “The second sum [in equation ()] thus reduces, in the case that one is allowed to set G1 , to” (“Die zweite Summe reduziert sich also in dem Falle, dass G1 gesetzt werden darf, auf”): 14⋅4gκρ∂∂xακα∂∂xβρβ+3κl∂∂xκmρ∂∂xlgmρ+2gκρ∂∂xσκm∂∂xmρσ . With the help of some auxiliary calculations on p. 16R, in which he used that gimdimdlgG0 for G1 , Einstein then rewrote the two terms in the first summation in equation (). Underneath equation () on p.  15R, he wrote: “If the determinant G1 , one has furthermore” (“Wenn Determinante G1 , es ist ferner”): ∑imκl∂2gim∂xκ∂xl∑gimκl∂2im∂xκ∂xl , and ∑imκl∂2gil∂xκ∂xm-∑gil∂∂xκim∂∂xmκl-∑gil∂∂xαlα∂∂xβiβ+∑gilim∂2κl∂xκ∂xm , Einstein erroneously thought that the first term on the right-hand side of equation (), a term that originally had an extra factor 2 , cancelled by the third term of expression (), which originally did not have the factor 14 in front of it. Moreover, in the last term in equation (), xm originally seems to have been xl . All in all, Einstein arrived at the (erroneous) expression 3∑gκρ∂∂xακα∂∂xβρβ+∑gimκl∂2im∂xκ∂xl +∑gilim∂2κl∂xκ∂xl+3∑κl∂∂xκmρ∂∂xlgmρ. for the curvature scalar in coordinates such that G1 .The corrected version is given below as expression (). Einstein proceeded to rewrite this expression as the contraction of gκλ with a new two-index object that would be a candidate for the left-hand side of the field equations. It is immediately clear, however, that these new candidate field equations would suffer from the same problem that Einstein had already encountered on p. 14L with the Ricci tensor. The second term in expression () gives rise to a core-operator term in the field equations, but the third term produces additional second-order derivative terms. Directly underneath the troublesome third term in expression (), Einstein wrote ∑∂2κl∂xκ∂xl0 ? He may have considered imposing an additional constraint to make sure that the offending term vanishes. In passing we note that equation () is a weakening of the Hertz restriction ∂kl∂xl0 , which Einstein had already used on p. 10L–11L, i.e., before he began his exploration of the Riemann tensor. As we have seen, the calculations on p. 15R contain several errors. Einstein corrected some of these, but then made a fresh start at the top of the next page (p. 16L). Using again that gmρdmρdlgG0 if G1 , he rewrote the second term of expression () as 34κl∂∂xκmρ∂∂xlgmρ34gmρκl∂2mρ∂xκ∂xl . Substituting this result into expression () and adding the various terms in expressions ()–(), he arrived at the (correct) expression 14gκραβ∂2κρ∂xα∂xβ-12gκρ∂κα∂xβ∂ρβ∂xα+∂2αβ∂xα∂xβ for the Riemann curvature scalar in coordinates such that G1 . In this expression one only needs to rewrite the last termImmediately above it, the term was restored to the form gκρκα∂2ρβ∂xα∂xβ , in which it was originally written in equation (). Einstein symmetrized this expression in κ and ρ : gκρ12κα∂2ρβ∂xα∂xβ+12ρβ∂2ακ∂xα∂xβ , so that the object left after pulling out the common factor gκρ will be symmetric in κ and ρ . A simpler way to rewrite the last term of expression () as the contraction of gκρ with an expression that is symmetric in κ and ρ is to multiply it by 14gκρκρ (cf. the top of p. 17R, the last term in expression ()). in order to pull out a common factor gκρ and to extract the two-index object Tiκ14αβ∂2iκ∂xα∂xβ-12∂iα∂xβ∂κβ∂xα+12κβ∂2iα∂xα∂xβ+12iα∂2κβ∂xα∂xβ . Einstein apparently assumed that Tiκ is a contravariant tensor under unimodular transformations since it produces a scalar under unimodular transformation when contracted with giκ .To prove that Tiκ is a contravariant tensor (under unimodular transformations), one would, of course, have to show that its contraction with an arbitrary covariant tensor produces a scalar (under unimodular transformations). Notice, however, that Einstein did make sure that Tiκ is symmetric (see the preceding note). Any anti-symmetric contribution to Tiκ would vanish upon contraction with the metric, no matter whether Tiκ is a tensor or not. Unfortunately, Tiκ still contained unwanted second-order derivative terms in addition to a core-operator term. Immediately below the expression for Tik , Einstein wrote down the identity ∂∂iακβ∂xβ∂xαiα∂2κβ∂xα∂xβ+κβ∂2iα∂xα∂xβ+∂iα∂xα∂κβ∂xβ+∂iα∂xβ∂κβ∂xα , probably in an attempt to eliminate the unwanted second-order derivative terms from Tik . He did not pursue this attempt any further. The upshot then is that a promising alternative way of generating a candidate for the left-hand side of the field equations from the Riemann tensor eventually led to the same problem that Einstein had encountered with the Ricci tensor on p. 14L. In deriving these new candidate field equations he had already imposed the condition G1 , but additional constraints would be needed to eliminate unwanted second-order derivative terms. He may have considered one such constraint, equation (), a weakening of the Hertz restriction. Comparing Tiκ and the Ricci Tensor (17L–18R) On pp. 17L–18R, Einstein investigated the relation between the two two-index objects he had constructed out of the Riemann tensor on the preceding pages, the Ricci tensor and the object Tik , which promised to be a tensor under unimodular transformations. On p. 17R he compared the contravariant forms of the two expressions, which meant that he had to raise the covariant indices of the Ricci tensor. On pp. 17R–18R he compared the covariant forms, which meant that he had to lower the contravariant indices of Tik . The resulting expressions quickly became so cumbersome that Einstein abandoned both calculations. On p. 17L, Einstein formed the contravariant Ricci tensor and started to expand the first-order derivative terms. The calculations are very similar to the ones on the preceding pages. Directly underneath the heading “Point tensor of gravitation” (“Punkttensor der Gravitation”), he wrote down the symbol for the fully covariant Riemann tensor introduced on p. 14L, and noted that this is a “plane tensor” (“Ebenentensor”), iκ,lmEbenentensor vierter Mannigfaltigkeit . He then formed the contravariant Ricci tensor, and noted that this is a “point tensor” (“Punkttensor”), ∑iκlmκlipmqiκ,lmPunkttensor . The prefixes “Punkt-” and “Ebene-” were introduced on p. 13L to distinguish between contravariant and covariant indices, albeit only in the context of linear transformations for which G1 .See the discussion of p. 13L in sec. . On p. 13L only the terms “Punktvektor” and “Ebenenvektor” occur explicitly. The terms “Punkttensor” and “Ebenentensor” occur here on p. 17L for the first time, although they were implied on p. 13L by the convention of using dots and dashes to denote contravariant (“Punkt”) and covariant (“Ebene”) indices, respectively, for vectors as well as tensors.On pp. 28L and 29L, the terms “Punkttensor” and “kontravarianter Tensor” are used interchangeably. Einstein did not use this terminology in any of his published writings. Einstein went on to expand and simplify the “point tensor” (). He first wrote down an incorrect expression for the Riemann tensor, iκ,lm∂2gim∂xκ∂xl-∂2gil∂xκ∂xm+∑ρσρσ[imσ ][κlρ ]-[ilσ ][κmρ ] . The four second-order derivative terms in equation () for the Riemann tensor cannot be grouped together in this way. They can, when the Riemann tensor is fully contracted to form the curvature scalar, as was done on p. 14R. This suggests that Einstein read off equation () from expression () for the curvature scalar which formed the starting point of his calculations on pp. 14R–16R. The error does not affect the rest of the calculation, since Einstein did not get beyond rewriting the first-order derivative terms in equation (). The further manipulation of the contravariant Ricci tensor on this page is very similar to that of the covariant Ricci tensor and of the curvature scalar on the preceding pages. He first considered the first term in the second summation in equation () contracted with klipmq (see equation ()), which he expanded to 14κlipmqρσ∂giσ∂xm+∂gmσ∂xi-∂gim∂xσ∂gκρ∂xl+∂glρ∂xκ-∂gκl∂xρ . He then pulled one pair of factors ( ipmq ) inside the first set of parentheses and the other pair into the second. In the resulting terms he used the relation gijikδjk to move the differentiation over from the g ’s to the ’s. He also set the determinant of metric equal to unity again, writing “ lgG set to zero” (“ lgG0 gesetzt ”), in which case the last term in the second set of parentheses in expression () vanishes upon contraction with κl . The next step was to move the factor ρσ from the second set of parentheses to the first. The expression could then be simplified further through contractions of the form gijikδjk . In this way, Einstein rewrote expression () as 14mq∂ρp∂xm+ip∂ρq∂xi-ρσ∂pq∂xσgκρ∂κl∂xl+glρ∂κl∂xκ . Einstein drew a horizontal line, and turned to the second term in the second summation in equation (), proceeding in much the same way as he had with the first term. However, after two lines he broke off the calculation with the comment, written at the bottom of the page, that it was “too involved” (“zu umständlich”). On pp. 17R–18R, Einstein took a slightly different approach. Under the heading “Plane tensor constructed in two different ways” (“Auf zwei Arten Ebenentensor gebildet”), he tried to bring the covariant versions of the Ricci tensor and the object Tiκ of p. 16L (with a minor modification) into a form in which they could be compared to one another. The calculation extends over the following three pages (17R–18R). Again it breaks off before yielding a definite result. Einstein began by considering what he called the “first way” (“1. Art”) of forming a covariant tensor. This refers to the object Tiκ extracted from the curvature scalar on pp. 14R–16R. He presumably went back to p. 16L, to expression () for the curvature scalar under the condition that G1 . Rewriting the last term in this expression in a slightly different way than was done on p. 16L, one arrives atSee footnote 219 above. 14gκραβ∂2κρ∂xα∂xβ-12gκρ∂κα∂xβ∂ρβ∂xα+14gκρκρ∂2αβ∂xα∂xβ , which is 14 times the contraction of gκρ with the expression in square brackets at the top of p. 17R, αβ∂2κρ∂xα∂xβ-2∂κα∂xβ∂ρβ∂xα+κρ∂2αβ∂xα∂xβ . With minor modifications, this is the object Tik defined on p. 16L. Apart from an overall factor of 4 and the labeling of its free indices, it only differs from the object constructed on p. 16L in its last term. The expression at the top of p. 17R gives the covariant version of this object, ∑gκσgρταβ∂2κρ∂xα∂xβ-2∂κα∂xβ∂ρβ∂xα+κρ∂2αβ∂xα∂xβ . On the next two lines, Einstein wrote down equivalent expressions for all three terms in expression (), using the relation gijikδjk and relations that can be derived from it through differentiation.It takes a short calculation very similar to the one rehearsed in the last four lines of p. 16R to show that the first term can be rewritten as -αβ∂2gτσ∂xα∂xβ+2αβκρ∂gρτ∂xβ∂gκσ∂xα . Einstein might have done this calculation on a separate piece of paper. Einstein now turned to the “second way” (“2. Art”) of forming a covariant tensor, which is contracting the Riemann tensor to form the Ricci tensor. Einstein had already started this calculation on p. 14L, but he made a fresh start on this page. Using the expressions for the terms with products of Christoffel symbols on the facing page (p. 17L, cf. expression ()), he wrote the Ricci tensor as 12∑κl∂2gim∂xκ∂xl+∂2gκl∂xi∂xm-∂2gil∂xκ∂xm-∂2gκm∂xi∂xl +14∑klρσκlρσ∂giσ∂xm+∂gmσ∂xi-∂gim∂xσ∂gκρ∂xl+∂glρ∂xκ-∂gκl∂xρ -14∑klρσκlρσ∂giσ∂xl+∂glσ∂xi-∂gil∂xσ∂gκρ∂xm+∂gmρ∂xκ-∂gκm∂xρ. Further simplification of this expression was facilitated by his previous investigation of the Ricci tensor on p. 14L. Without further calculation Einstein noted that the first two terms in the second set of parentheses on the second line are identical and “can be combined” (“vereinigt sich”), and that the third term “vanishes” (“fällt weg”), which indicates that he once again imposed the condition that the determinant G is a constant. These results had explicitly been derived on p. 14L (cf. equation ()). Einstein could therefore immediately rewrite the second line as 12∑klρσκlρσ∂giσ∂xm+∂gmσ∂xi-∂gim∂xσ∂gκρ∂xl12∑∂ρσ∂xρ∂giσ∂xm+∂gmσ∂xi-∂gim∂xσ. His increased facility in handling these expressions is also manifest in his treatment of the third line of expression (). He started to rewrite the contraction of κlρσ with the term in the second set of parentheses, but then he noticed that he could simplify the expression more easily by a symmetry argument. He marked four terms in the expression by wiggly lines, The first two underlined terms are antisymmetric in l and σ , the last two in κ and ρ . Upon contraction with κlρσ , the two remaining terms become symmetric in κ and ρ and l and σ , respectively. So, the product of the underlined terms in one set of parentheses with the non-underlined term in the other produces the contraction of an expression anti-symmetric with an expression symmetric in the same pair of indices. These products thus vanish, and expression () reduces to 14∑κlρσ∂glσ∂xi∂gκρ∂xm-14∑κlρσ∂giσ∂xl-∂gil∂xσ∂gmρ∂xκ-∂gκm∂xρ , the expression written at the bottom of the page. Einstein rewrote the first term as 14∂ρσ∂xi∂gσρ∂xm . At the top of the next page (p. 18L), he inserted the results found on p. 17R (see equation () and expression ()–()) into expression () for the Ricci tensor and arrived atThe expression ∂giσ∂xl-∂glσ∂xi on the third line should be ∂giσ∂xl-∂gil∂xσ , i.e., the expression should be anti-symmetric in l and σ , not in i and l . 12κl∂2gim∂xκ∂xl+∂2gκl∂xi∂xm-∂2gil∂xκ∂xm-∂2gκm∂xi∂xl-12∂σρ∂xρ∂giσ∂xm+∂gmσ∂xi-∂gim∂xσ+14∂ρσ∂xi∂gρσ∂xm14κlρσ∂giσ∂xl-∂glσ∂xi∂gmρ∂xκ-∂gκm∂xρ. On pp. 18L–R, Einstein tried to simplify this expression still further. Thus, in the first two lines of “auxiliary calculations” (“Nebenrechnungen”) on p. 18R, he showed that the condition that the determinant of the metric is constant allows one to rewrite the second term in the first line of expression () as 12∂κρ∂xi∂gκρ∂xm . This expression (without the minus sign) is given on the last line of p. 18L. By the same token, the last term on the second line is equivalent to 14ρσ∂2gρσ∂xi∂xm , the expression written underneath this term, except for the fact that Einstein mistakenly wrote ρσ instead of gρσ . Einstein performed another auxiliary calculation on p. 18R to expand the third line of expression (). Multiplying the end result of this auxiliary calculation by minus 14 , one arrives at the expression -14∂lσ∂xm∂giσ∂xl+∂ρσ∂xi∂gmρ∂xσ-∂lσ∂xm∂glσ∂xi-14κlρσ∂giσ∂xl∂gmρ∂xκ , written directly underneath this term in expression () on p. 18L. At this point Einstein gave up and crossed out his calculations on pp. 18L and 18R in their entirety. The relation between the Ricci tensor and the two-index object Tik , the two candidates for the left-hand side of the field equations that Einstein had considered in his first explorations of the Riemann tensor, still remained unclear. Exploring the Ricci tensor in Harmonic Coordinates (19L–21R) The mathematical strategy which Einstein had been following since the introduction of the Riemann tensor on p. 14L had still not yielded a viable candidate for the left-hand side of the gravitational field equations. The two expressions considered on the preceding pages (14L–18R), the Ricci tensor and the object Tik , both contain, in addition to the desired core-operator term, unwanted second-order derivative terms. On p. 18R Einstein used the restriction to unimodular coordinates, which allowed him to set G1 , to eliminate one of these unwanted terms (cf. eqs. ()-()). On p. 19L Einstein eliminated all unwanted second-order derivative terms through an appropriate choice of coordinates. In this way he reduced the Ricci tensor to the sum of a core-operator term and terms quadratic in the first-order derivatives of the metric. Today these coordinates are called “harmonic coordinates” and the corresponding condition is called the “harmonic coordinate condition.” Let us reiterate that Einstein understood coordinate conditions not in the modern sense of selecting at least one member from each possible equivalence class of metrics,Two metrics are in the same equivalence class if and only if a coordinate transformation exists that maps one onto the other. but as restrictions on the covariance group of the field equations. That is why we adopted the phrase coordinate restriction.For further discussion, see sec. . With the help of the harmonic restriction, Einstein was finally able to extract from the Riemann tensor field equations that satisfy the correspondence principle. On p. 19R, Einstein began to examine these new field equations and the harmonic restriction in a weak-field approximation. He made sure the new field equations were compatible with energy-momentum conservation by checking that the gravitational force on a cloud of dust particles can be written as the divergence of a quantity representing gravitational stress-energy density. To ensure compatibility between the field equations and energy-momentum conservation, Einstein imposed an extra condition on the metric tensor. The combination of this extra condition—a linearized version of the Hertz restriction—and the harmonic restriction leads to the unacceptable result that the trace of the weak-field metric has to be a constant. On p. 20L, Einstein modified the weak-field equations to avoid this implication. By adding a term with the trace of the weak-field metric, he ensured that the requirements of the correspondence principle and the conservation principle are both satisfied by imposing the harmonic restriction. From a purely mathematical point of view, the left-hand side of these modified weak-field equations is, in fact, the linearized Einstein tensor. At the bottom of p. 20L and on p. 21L, Einstein used the new weak-field equations to rewrite the gravitational force as a divergence, thus convincing himself that the addition of the trace term to the field equations does not destroy their compatibility with energy-momentum conservation. There was, however, a different problem. The modified weak-field equations of p. 20L are incompatible with Einstein’s presupposition concerning static gravitational fields.This presupposition is also incompatible with the harmonic coordinate restriction, but it is unclear whether Einstein realized that at this point. It is no coincidence therefore that, on p. 20R, Einstein reexamined the dynamics of point particles moving in a metric field. The purpose of this calculation becomes clear on p. 21R. Einstein checked whether his presupposition concerning static gravitational fields was actually justified. On p. 21R, using elements of his discussion of the dynamics of point particles on p. 20R, he developed what appeared to be a very strong argument in support of his views on the static field. This (fallacious) argument led him to give up the harmonic restriction and the field equations constructed with its help. Extracting Field Equations from the Ricci Tensor Using Harmonic Coordinates (19L) On p. 19L, as is announced in the first line: “Renewed calculation of the plane tensorSee sec. , for a discussion of Einstein’s usage of the term “Ebenentensor” for covariant tensors.” (“Nochmalige Berechnung des Ebenentensors”), Einstein re-calculated the quantity given by the expression 12∂2gim∂xκ∂xl+∂2gκl∂xi∂xm-∂2gil∂xκ∂xm-∂2gκm∂xi∂xl14∑κlρσρσ∂giρ∂xl+∂glρ∂xi-∂gil∂xσρ∂gκσ∂xm+∂gmσ∂xκ-∂gκm∂xσ|κl. This expression gives an incomplete version of the fully covariant Riemann tensor (see, e.g., equation ()) contracted with the contravariant metric kl . In other words, it is an incomplete version of the Ricci tensor, which Einstein had already investigated on p. 14L and on pp. 17R–18R.Expression () differs from the expressions for the Riemann tensor and the Ricci tensor given earlier (cf. equations (), (), and ()) in that the indices ρ and σ in the terms with products of Christoffel symbols (such as the last line of expression ()) have been switched, in accordance with the labeling of these indices in the Entwurf paper (Einstein and Grossmann 1913, 35). This could be an indication that some time elapsed between these earlier calculations and the ones starting on p. 19L. What is missing is another term with products of Christoffel symbols. This was not just an oversight on Einstein’s part. The missing term vanishes if the condition κl[ κl i ]κl2∂gil∂xκ-∂gκl∂xi0 is imposed, which Einstein actually wrote down two lines further down. This is the condition that we today call the harmonic coordinate condition. The point of introducing this condition and the purpose of the whole calculation becomes clear in the line immediately following expression (). On this line, Einstein wrote down the first of the four second-order derivative terms in expression (), 12kl∂2gim∂xκ∂xl , which has the form of a core operator, underlined it, and noted that it would “remain” (“bleibt stehen”). Exactly how this comes about is recapitulated in the calculation that follows. Einstein indicated that he wanted to take the derivative of the condition with free index i in equation () with respect to xm and then do the same with the indices i and m interchanged. Adding the two equations, Einstein gotIn the very last terms in equations () and (), the derivative should be with respect to xm rather than xi . 2κl∂2gil∂xκ∂xm+∂2gκm∂xi∂xl-∂2gκl∂xi∂xm +∂κl∂xm2∂gil∂xκ-∂gκl∂xi+∂κl∂xi2∂gmκ∂xl-∂gκl∂xi0. This relation allowed Einstein to replace the three bothersome second-order derivative terms in expression () by an expression 12κl 14∂κl∂xm2∂gil∂xκ-∂gκl∂xi+∂κl∂xi2∂gmκ∂xl-∂gκl∂xi containing only first-order derivatives. Einstein now turned his attention to the “second term” (“zweites Glied”), i.e., the second line of expression (). He invoked the same symmetry argument as on p. 17R (cf. the discussion following expression () above). He marked the symmetric and anti-symmetric terms in the index pairs ( l , ρ ) and ( κ , σ ) by straight and wiggly lines, respectively, and immediately wrote down the only two non-vanishing contributions, one coming from the symmetric parts, 14ρσ∂glρ∂xi∂gκσ∂xmκl , and one coming from the anti-symmetric parts, 14ρσ∂giρ∂xl-∂gil∂xρ∂gmσ∂xκ-∂gκm∂xσκl=-12ρσκl∂giρ∂xl∂gmσ∂xκ+12ρσκl∂gil∂xρ∂gmσ∂xκ. He now underlined the terms that form the Ricci tensor: the core-operator term in expression (), the right-hand side of equation (), expression (), and the right-hand side of equation (). A short auxiliary calculation showed that expression () and the last term in equation () cancel each other.The two terms only cancel if the final index in equation () is corrected (see the preceding note). Einstein probably realized the index was wrong at this point, although he did not correct it. He added the remaining underlined terms and concluded that “the covariant tensor [the Ricci tensor], multiplied by 2 , thus takes the form” (“Der mit 2 multiplizierte Ebenentensor erhält also die Form”) κl∂2gim∂xκ∂xl-12∂κl∂xm∂gκl∂xi+∂κl∂xm∂gil∂xκ+∂κl∂xi∂gmκ∂xl-ρσκl∂giρ∂xl∂gmσ∂xκ+ρσκl∂gil∂xρ∂gmσ∂xκ. This is the result Einstein wanted on this page: in harmonic coordinates, the Ricci tensor is the sum of a core-operator term, which is the only remaining second-order derivative term, and terms quadratic in first-order derivatives. He seems to have checked this carefully, for at the bottom of the page he wrote: “Result certain. Valid for coordinates which satisfy the equation Δϕ0 ” (“Resultat sicher. Gilt für Koordinaten, die der Gl. Δϕ0 genügen”). These coordinates were well-known and were called “isothermal coordinates” in the contemporary literature.See, e.g., (Bianchi 1910, sec. 36-37) or (Wright 1908, sec. 57). They are now commonly known as “harmonic coordinates.” Einstein’s notation suggests that he (or Grossmann) found the coordinate condition Δϕ0 in the literature.Einstein had already used this condition in a different but equivalent form on p. 6L (see equation ()), but it is unclear whether he recognized the equivalence. Discovering a Conflict between the Harmonic Coordinate Restriction, the Weak-Field Equations, and Energy-Momentum Conservation (19R) On p. 19L, the mathematical strategy had finally born fruit. Einstein had found a way of constructing field equations out of the Ricci tensor that satisfy the correspondence principle. He now had to check whether these field equations and the harmonic restriction used in their construction also satisfy his other heuristic requirements. Unfortunately, in the course of checking the conservation principle on p. 19R, he discovered a problem. The considerations on p. 19R are all in the context of a first-order, weak-field approximation. The metric is assumed to be the sum of a diagonalized Minkowski metric and small deviations from this metric. With the help of an imaginary time coordinate, introduced explicitly further down on the page, the zeroth order metric can be written as diag1,1,1,1. Einstein began by writing down the harmonic restriction (see equation ()) in this weak-field approximation, writing: “For the first approximation our additional condition is” (“Für die erste Annäherung lautet unsere Nebenbedingung”)Einstein’s notation here and in the following is awkward. He used the same summation index κ for two different summations, and did not explicitly distinguish between the diagonal zeroth-order metric and the first-order metric with small deviations from it. Introducing the more explicit notation gμνδμν+g¯μν where δμν is the Kronecker delta and g¯μν≪1 , one can rewrite equation () more        carefully as: δκκ′2∂g¯iκ∂xκ′-∂g¯κκ′∂xi . ∑κκκ2∂giκ∂xκ-∂gκκ∂xi0 . Einstein conjectured that this condition “can perhaps be decomposed into” (“Zerfällt vielleicht in”) the following two conditions,The combination of these two new conditions is, in fact, stronger than the original condition. ∑κκ∂giκ∂xκ0 ,Using the notation introduced in footnote 235, one would write equation () as: δκκ′∂g¯iκ∂xκ′0 . a condition equivalent, at least in first-order approximation, to the Hertz restriction, and ∑κκgκκkonst ,Using the notation introduced in footnote 235, one would write this equation as: δκκ′g¯κκ′0. In other words, the condition is that the trace of g¯iκ vanish. This is the form in which the condition is written at the top of the next page (p. 20L): ∑gκκx0. a condition on the trace of the weak-field metric. As will become clear below, Einstein wanted to ensure energy-momentum conservation by imposing the linearized Hertz restriction (). On p. 19L he had introduced the harmonic restriction () to satisfy the correspondence principle. The combination of these two restrictions implies equation (), which was unacceptable to Einstein. Einstein became aware of these implications in the course of his considerations concerning energy-momentum conservation on the remainder of this page. On the next line, Einstein wrote down the “equations” (“Gleichungen”) ∑κκ∂2gim∂xκ2κρ0dxidsdxmdsgiigmm ,Einstein erroneously wrote dxmdxs instead of dxmds . Using the notation introduced in footnote 235, one would write this equation as: δκκ′∂2g¯im∂xκ∂xκ′κρ0dxi′dsdxm′dsδii′δmm′. which are just the field equations of p. 19L in first-order approximation. The left-hand side is the core-operator term of the reduced Ricci tensor (see expression ()). To first order, this is the only term that contributes. The right-hand side gives the covariant version of the stress-energy tensor for pressureless dust, multiplied by the gravitational constant κ . For Einstein, energy-momentum conservation required that the density of the four-force of the gravitational field on the pressureless dust can be written as the four-divergence of a quantity that can be interpreted as representing gravitational stress-energy density. Einstein checked whether his new field equations would actually allow him to rewrite the gravitational force density in the form of such a divergence.A completely analogous consideration can be found in the Entwurf paper (Einstein and Grossmann 1913, 15). In his second static theory of 1912, Einstein likewise made sure that the (ordinary three-)force density could be written as the divergence of a gravitational stress tensor (Einstein 1912b, 456). The expression for the force density 12G∑∂gμν∂xmTμνb , was introduced on p. 5R (see expression () above). Here Tμνb is the (contravariant) stress-energy tensor for pressureless dust. The force density gives the rate at which four-momentum is transferred from the gravitational field to the pressureless dust. As such it enters into the energy-momentum balance between matter and gravitational field for which Einstein had introduced the equation ∑νn∂GgmνTνn∂xn-12G∑∂∂xmgμνTμν0 on the next line on p. 5R (see equation (); the superscript b was silently dropped). This equation is equivalent to the statement that the covariant divergence of the stress-energy tensor Tμν vanishes. If (minus) the force density can be written as the divergence of gravitational stress-energy density, then equation () can be written as the vanishing of the ordinary divergence of the sum of quantities representing the stress-energy density of matter (pressureless dust in this case) and of the gravitational field, respectively. To find out whether some candidate field equations allow such rewriting of the force density, one substitutes their left-hand side (divided by κ ) for the stress-energy tensor in the second term of equation () and tries to rewrite the resulting expression as a divergence. In the first-order approximation considered on p. 19R, the second term in equation () reduces to 12∂gim∂xσTim . This expression, while not actually written down on p. 19R, provides the link between equation (), giving the field equations in first-order approximation, and the equation written on the next line. Eliminating Tim from expression () with the help of equation () and neglecting a factor 2κ , one arrives at the equation that Einstein did write,Einstein apparently substituted the covariant object on the left-hand side of equation () for the contravariant tensor Tim in expression (). Correcting this and using the notation introduced in footnote 235, one would write equation () as δκκ′∂2¯im∂xκ∂xκ′∂g¯im∂xσδκκ′∂∂¯im∂xκ′∂g¯im∂xσ∂xκ-12∂∂¯im∂xκ′∂g¯im∂xκ∂xσ. ∑κimκκ∂2gim∂xκ2∂gim∂xσ∑κimκκ∂∂gim∂xκ∂gim∂xσ∂xκ-12∂∂2gim 2∂xκ∂xσ . Since the right-hand side of this equation does indeed have the form of a divergence, Einstein concluded that “energy-momentum conservation holds in the relevant approximation” (“Energie- u. Impulssatz gilt mit der in Betr[acht] kommenden Annäherung”). Einstein still had to check whether the harmonic restriction (see equation ()) and the two conditions into which it had tentatively been split (see equations () and ()) are also compatible with energy-momentum conservation. Presumably, the second part of the comment immediately following Einstein’s conclusion that “energy-momentum conservation holds in the relevant approximation” refers to this issue: “uniqueness and additional conditions” (“Eindeutigkeit u. Nebenbedingungen”). As to the first part of this comment, Einstein apparently hoped that his heuristic requirements would uniquely determine the field equations. This may well have been the motive for his (inconclusive) investigation on pp. 17L–18R of the relation between the Ricci tensor and the two-index object extracted from the curvature scalar on pp. 14R–16R. Einstein, however, did not actually address the uniqueness problem on p. 19R. The question regarding the additional conditions appears to have been more pressing. He once again wrote down the linearized field equations, this time in the more compact formThe equation has covariant indices on the left- and contravariant indices on the right-hand side. Since indices are raised and lowered with the Kronecker delta in this approximation, this does not really matter. Using the notation introduced in footnote 235, one could write the equation more carefully as: □¯imκρ0dxidτdxmdτ . Einstein also wrote dxτ instead of dτ again (cf. footnote 239). □gimκρ0dxidτdxmdτ . The considerations on the remainder of the page suggest that Einstein discovered the following problem. In first-order approximation, covariant derivatives become ordinary derivatives and the energy-momentum balance between pressureless dust and gravitational field reduces to the conservation law ∂ρ0dxidτdxmdτ∂xm0 . The easiest way to ensure that the field equations () are compatible with equation () is to impose the linearized Hertz restrictionOr rather (see the preceding note) that: ∂¯im∂xm0 . This relation is equivalent to equation (), which with the help of the notation introduced in footnote 235 can be written as: δmm′∂g¯im∂xm′0 . ∂gim∂xm0 . Equations () and () immediately imply equation (): κ∂ρ0dxidτdxmdτ∂xm∂□gim∂xm□∂gim∂xm0 . As we have seen (cf. equation ()), the linearized Hertz restriction is just one of the two restrictions in the tentative decomposition of the harmonic restriction given at the top of the page. But now a problem arises, which lies neither with the Hertz restriction nor with the harmonic restriction taken by itself, but with the combination of the two. Together these two restrictions imply equation (), which says that the trace of the linearized metric has to be a constant. This was objectionable for two reasons. First, through the field equations, it imposes the condition that the trace of the stress-energy tensor vanish (cf. equation ()), which is clearly violated in the case under consideration, viz. pressureless dust.In the second of his four communications to the Prussian Academy of November 1915, Einstein briefly revived the condition that the stress-energy tensor for matter be traceless (Einstein 1915b, 799) as it is for the electromagnetic stress-energy tensor. At that point he suggested that this constraint can be reconciled with the non-vanishing of the trace of the stress-energy tensor for pressureless dust by assuming that gravity plays an essential role in the constitution of matter. For discussion, see, e.g., “Untying the Knot …” sec. 7 (in this volume). Secondly, the trace of the metric diag±1,±1,±1,c2x,y,z , which Einstein used to represent static fields (see p. 6R and p. 39L), is obviously not a constant. To avoid these problems, Einstein considered giving up the Hertz restriction. That means that it is no longer guaranteed that the divergence of the stress-energy tensor vanishes. The calculations at the bottom of p. 19R suggest that Einstein was prepared to consider the possibility that this divergence is non-vanishing. The result of these calculations, however, convinced him that was not an option. And from this he inferred that the Hertz restriction, which is the natural way of forcing the divergence of the stress-energy tensor to vanish, also had to be retained. Einstein wrote down the “continuity condition” (“Kontinuitätsbedingung”) for a cloud of pressureless dust, with “density of material points” (“Dichte materieller Punkte”) ρ01-q2c2 ,Einstein only wrote an abbreviated form of this equation indicating the last two terms by dots. ∂ic∂tρ0ic1-q2c2∂ρ0qx1-q2c2∂x+∂ρ0qy1-q2c2∂y+∂ρ0qz1-q2c2∂z . To combine the continuity equation with equation (), expressing energy-momentum conservation for a cloud of pressureless dust, Einstein introduced the four-velocityHe wrote wi and wm above the relevant terms on the right-hand side of equation (); and he wrote wx and wu for the expressions x⋅ and ic in the lower right-hand corner of the page. widxidτx⋅i1-q2c2 , in which the relation dτdt1-q2c2 (with qx⋅, y⋅, z⋅ ) between proper time and coordinate time has been used, and in which the dot indicates differentiation with respect to t . With the imaginary time coordinate uict , the wu -component becomes ic1-q2c2 . Using this new notation, Einstein rewrote the continuity equation asIn the notebook the third term is indicated only by a dot. ∂ρ0wx∂x+∂ρ0wy∂y+∂ρ0wz∂z+∂ρ0wu∂u0 . Similarly, the x -component of the divergence of the stress-energy tensor for pressureless dust (see equation ()) can be rewritten as ∂ρOwxwj∂xj∂ρ0wxwx∂x+∂ρ0wxwy∂y+∂ρ0wxwz∂z+∂ρ0wxwu∂u=∂ρ0wx∂x+∂ρ0wy∂y+∂ρ0wz∂z+∂ρ0wu∂uwx +ρ0wx∂wx∂x+ρ0wy∂wx∂y+ρ0wz∂wx∂z+ρ0wu∂wx∂u. The term in square brackets vanishes because of the continuity equation. Bringing the remaining terms over to the left-hand side, one arrives at the final equation of p. 19R,In the notebook the last two terms on both the first and the second line are indicated only by dots. ∂∂xρ0wxwx+∂ρ0wxwy∂y+∂ρ0wxwz∂z+∂ρ0wxwu∂uρ0wx∂wx∂x-ρ0wy∂wx∂y-ρ0wz∂wx∂z-ρ0wu∂wx∂u0. . Einstein noticed that the last four terms on the left-hand side add up to ρDwxDτ . The vanishing of this expression and the corresponding y - and z -components is the condition that the dust particles move on geodesics in what in this first-order approximation is Minkowski spacetime. Looking back at equation (), we thus see that the vanishing of the divergence of the stress-energy tensor follows directly from the continuity equation and the equations of motion for the dust cloud.This cannot have come as a great surprise for Einstein, for it was precisely from the equations of motion for a cloud of pressureless dust that he had derived the equation for energy-momentum conservation (see, e.g., equation ()) in the first place (see p. 5R). It was therefore not an option for Einstein to drop the divergence requirement. The easiest way to guarantee the divergence requirement was to impose the Hertz restriction (see equations ()–()). So, Einstein wanted to hold on to the Hertz restriction to satisfy the conservation principle and at the same time he wanted to hold on to the harmonic restriction to satisfy the correspondence principle. As Einstein put it: “both restrictions are to be retained” (“Beide obige Bedingungen sind aufrecht zu erhalten”). He had hit upon a serious problem: the harmonic restriction plus the Hertz restriction implied the unacceptable condition that the trace of the linearized metric be a constant. Modifying the Weak-field Equations: the Linearized Einstein Tensor (20L, 21L) At the top of p. 20L, Einstein once again wrote down the two conditionsIn the second equation Einstein used the superscript “x” to indicate that he was considering first-order deviations from the flat metric. The same convention was used elsewhere in the notebook (see pp. 41L, 10L, 10R, 12L, 12R). ∑∂∂xκgiκ0, ∑gκκx0. into which he had tentatively decomposed the harmonic restriction at the top of p. 19R (cf. equations ()–()). The harmonic condition was used to reduce the Ricci tensor to the d’Alembertian acting on the metric in the weak-field case. The Hertz restriction was added to make sure that the divergence of the stress-energy tensor vanishes in the weak-field case. The combination of these two restrictions implies that the trace of the metric has to vanish. This is problematic for a couple of reasons. First, if the metric is traceless, the weak-field equations () tell us that the stress-energy tensor be traceless as well. This last inference can be avoided by modifying the weak-field equations so as to make their right-hand side traceless. This is precisely what Einstein did on the next line:Using the notation of footnote 235 along with the d’Alembertian □ and the stress-energy tensor Tiκρ0dxidτdxκdτ (with trace T ), one can rewrite this equation more carefully as: □giκκTi′κ′-14δi′κ′Tδii′δκκ′ . Einstein omitted the gravitational constant κ , and instead of the Kronecker delta δi′κ′ , he wrote “for the same i and κ ” (“für gleiche i u. κ ”) underneath the second term on the right-hand side. ∑σ∂2giκ∂xσ2ρ0dxidτdxκdτ-14ρ0∑dxκdτdxκdτ . With these new weak-field equations, the condition on the trace of the metric tensor no longer implies any condition on the trace of the stress-energy tensor.In the fourth communication of November 1915, Einstein (1915d) similarly added a term with the trace T of the energy-momentum tensor of matter to his field equations to avoid the condition T0 . For discussion, see “Untying the Knot …” sec. 7 (in this volume). Still, Einstein must have found the result unsatisfactory, for he crossed out the two lines with equations ()–(). Eq. () does indeed only solve part of the problem caused by the combination of the harmonic and the Hertz restrictions. It takes care of the problem that a traceless metric would imply a traceless energy-momentum tensor, but it does not address a second problem, namely that a metric of the form Einstein used to represent static fields is not traceless. A more satisfactory way to solve the problems would be to modify the weak-field equations in such a way that one avoids the condition that the metric be traceless altogether. This can be done by modifying the field equations in such a way that the harmonic restriction ensures both the elimination of unwanted second-order derivative terms for the Ricci tensor and the vanishing of the divergence of the stress-energy tensor. The combination of the harmonic and the Hertz restriction is thus replaced by the harmonic restriction alone and the problematic condition that the metric be traceless no longer follows. This is exactly the way in which Einstein took care of the problem in the next line. First he wrote down the harmonic restriction again in first-order approximation (cf. equation () and footnote 235 for a more careful notation), ∑∂giκ∂xκ-12∂gκκ∂xi0 , underlining the left-hand side. He introduced the abbreviation ∑gκκU for the trace of the metric. Instead of adding a term with the trace of the stress-energy tensor to the right-hand side of the weak-field equations as he had done in equation (), he now added a term with the trace of the metric to the left-hand side. He wrote down a few components of these modified “gravitational equations” (“Gravitationsgleichungen”), indicating the remaining components by a dot and three lines of dashesThe Δ used here apparently denotes the d’Alembertian operator Δ∑i14∂2∂xi2 , ( x4ict ) which had been denoted by the □ on the preceding page (p. 19R). The notation □ does not occur anywhere else in the notebook. The use of Δ as the analogue of the Laplace operator or the Laplace-Beltrami operator in four dimensions can also be found at the bottom of p. 19L. For two dimensions it is used on p. 10R, for three dimensions on p. 40L. Δg11-12UT11 Δg12T12 . Δg14T14 _____ ___ ____ ____ _____ _____ _______ _____ _____ ___ ____ ____ _____ _____ _______ _____ _____ ___ ____ ____ _____ _____ _______ _____ In modern notation, using the Kronecker delta, these equations can be written more compactly asStrictly speaking, one should write the right-hand side as Ti′κ′δii′δκκ′ , since Tiκ is a contravariant tensor (cf. footnote ). □giκ-12δiκUTiκ . One can now ensure compatibility between the weak-field equations and the vanishing of the divergence of the stress-energy tensor by imposing ∂∂xκgiκ-12δiκU0 . which is just the harmonic restriction (see equation ()). The calculation for the modified weak-field equations and the harmonic restriction is completely analogous to the calculation for the original weak-field equations and the Hertz restriction (cf. equation ()), ∂Tiκ∂xκ∂□giκ-12δiκU∂xκ□∂giκ-12δiκU∂xm0 . In other words, Einstein’s modification of the weak-field equations removed the need for the Hertz restriction (equation ()), and thereby the need for the troublesome trace condition (equation ()). The modified weak-field equations () have exactly the same form as the weak-field equations for the final theory of November 1915.(Einstein 1915d). The left-hand side is the linearized version of the Einstein tensor Rμν-12gμνR . There is no indication in the notebook that Einstein tried to find the exact equations corresponding to the weak-field equations with trace term. Einstein did write the modified weak-field equations in an alternative form. Taking the trace on both sides of equation (), he found 2ΔU∑Tκκ (the factor 2 on the left-hand side should be 1 ).With the help of this relation, Einstein replaced the term with the trace of the metric on the left-hand side of equation () by a term with the trace of the stress-energy tensor on the right-hand side. He obtained, writing “from this equations” (“Hieraus Gleichungen”) Δg11T11+12∑Tκκ Δg12T12 . Δg14T14 _____ ___ ____ ____ _____ _____ _______ _____ _____ ___ ____ ____ _____ _____ _______ _____ _____ ___ ____ ____ _____ _____ _______ _____ Einstein partly corrected his error in equation (). The plus sign in equation (), however, should be a minus sign. In modern notation, the correct equations can be written more compactly as (cf. equation ()) □gijTij-12δij∑Tκκ . Einstein proceeded to check that the modified weak-field equations (in the form of equation () rather than in the form of equation ()) still allow the gravitational four-force density to be written in the form of a divergence. In first-order approximation, the force density is given by (see expression ()): 12Tiκ∂giκ∂xσ . Using equation () to eliminate Tiκ from this expression, one arrives at 12Δgiκ-12δiκU∂giκ∂xσ12Δgiκ∂∂xσgiκ-12U∂∂xσU On p. 19R, Einstein had already established that the first term on the right-hand side can be written in the form of a divergence (see equation ()). It only remained for him to verify that this is true for the second term as well. He started to rewrite this term at the bottom of p. 20L as Although the final expression still does not have the form of a divergence, Einstein concluded that it is “representable in the required form” (“Darstellbar in der verl[angten] Form”). Since the two terms on the right-hand side of equation () have the exact same structure and since the result had already been established for the first, this conclusion is obvious. Nevertheless, Einstein made a fresh start with this whole calculation on p. 21L, this time considering both terms in equation ().This seemingly redundant calculation may have been done in connection with the calculation at the bottom of p. 21R involving the gravitational stress tensor for Einstein’s 1912 static theory. At the top of p. 21L Einstein wrote down the right-hand side of equation () . He rewrote the sum in the second term as ∂2U∂xν2∂U∂xσ∂∂U∂xν∂U∂xσ∂xν-12∂∂U∂xν2∂xσ , and the one in the first as ∂2giκ∂xν2∂giκ∂xσ∂∂giκ∂xν∂giκ∂xσ∂xν-12∂∂giκ∂xν2∂xσ . Inserting these results into expression (), he wrote the gravitational force density in the required form of a coordinate divergence of gravitational stress-energy density,Notice that he did not take into account the factor 12 in front of expression (), which was probably only added later. ∑iκν∂∂giκ∂xν∂giκ∂xσ∂xν-12∑iκν∂∂giκ∂xν2∂xσ12∑ν∂∂xν∂∂xνU∂∂xσU+14∑∂∂xσ∂∂xνU2. At some point, Einstein deleted the second line and wrote that the trace “ U must vanish” (“ U muss verschwinden”). He probably meant that the derivatives of U must vanish. It is unclear why he resurrected this condition on the trace of the metric tensor. He later deleted the remark about U , but did not rescind the deletion of the second line of expression (). From the fragmentary calculations on the remainder of the page, one can infer that Einstein somehow wanted to produce an exact analogue of his first-order calculation. Exactly how and for what purpose remains unclear. Perhaps he wanted to verify that the exact field equations allow one to rewrite the gravitational force density as the divergence of the gravitational stress-energy density as well; or he wanted to find the exact expression for the quantity representing gravitational stress-energy density on the basis of the approximative expression that could be read off from expression (). Whatever the purpose of these calculations, he drew a horizontal line and wroteNote that the expression is ill-defined since it contracts over pairs of covariant indices. The corresponding approximative calculations, of course, had the same problem, but there it was only a matter of awkward notation (see notes 241–242). ∂giκ∂xσμν∂2giκ∂xμ∂xν∂μν∂giκ∂xσ∂giκ∂xμ∂xν-∂giκ∂xμ∂∂giκ∂xσμν∂xν . Contrary to its linearized analogue (equation ()), the right-hand side of this equation cannot be written as a divergence. Einstein noted that the last term can be written as μν∂giκ∂xμ∂2giκ∂xν∂xσ , if the Hertz restriction, ∑∂μν∂xν0 , is imposed. This still does not make it possible, however, to rewrite the right-hand side of equation () as a divergence. At this point, the calculation breaks off.Einstein drew another horizontal line, started to write down, but then immediately deleted, the transformation law of what he referred to as the “second tensor” (“zweiter Tensor transformiert”). Reexamining the Presuppositions Concerning the Static Field (20R, 21R) On p. 19R Einstein had found that the compatibility between the field equations constructed out of the Ricci tensor on p. 19L and the correspondence and conservation principles required that the trace of the linearized metric be a constant. This condition was problematic for several reasons, one of which was that this requirement is not satisfied by a metric of the form diag1,1,1,c2x,y,z which Einstein used to represent weak static fields. On p. 20L Einstein showed that the condition could be avoided by adding a trace term to the weak-field equations, but then these weak-field equations themselves no longer allow a metric of the form diag1,1,1,c2 as a solution.This is most easily seen when these modified weak-field equations are written in the form of equation (). Consider a weak-field generated by some static mass distribution. The only non-vanishing component of the stress-energy tensor will be the 44 -component. If the weak-field equations are the ones in equation (), the metric of a static weak field will deviate from the Minkowski metric in all its diagonal components, and not just in its 44 -component, as Einstein expected.,A metric of this form is also incompatible with the harmonic restriction with which the field equations of p. 19L were extracted from the Ricci tensor. It is unclear whether Einstein was aware of this problem at this point. Before giving up the promising new field equations in the face of these problems, Einstein reexamined whether his presuppositions concerning the static field were actually justified. On p. 21R he found an argument that convinced him they were. The argument involves the dynamics of point particles in a gravitational field, which Einstein reviewed on p. 20R. Einstein argued that, unless the 44 -component is the only variable component of the metric of a static field, particles with different energy, and hence different inertial mass, fall with different accelerations in such fields. He thus saw himself forced to give up the field equations considered on pp. 19L–20L. The discussion on p. 20R of the mechanics of point particles and continuous matter distributions in a gravitational field is essentially the same as the discussions on p. 5R and in (Einstein and Grossmann 1913, sec. 2 and 4). At the top of the page, Einstein wrote down the line element ∑gμνdxμdxνdτ2 , and introduced the Lagrangian for a point particle of unit mass moving in a given metric field ηdτdtgμνx⋅μx⋅ν . This last equation is written here somewhat more compactly than in the notebook. Though he indicated most terms by dots, Einstein expanded the sum under the square root sign,We use a hybrid summation convention here, as did Einstein in the notebook at various places. writing x,y,z,t for the coordinates xμ . He continued to do so in most of the equations on this page. His argument is easier to follow, however, if the more compact notation is used. On the basis of the Euler-Lagrange equations, ∂η∂xi-ddt∂η∂x⋅i0 , where i1,2,3 , the quantities ∂η∂xi12dtdτ∂gμν∂xidxμdtdxνdt can be interpreted as the components of the force on the particle,In the notebook, the factor 12 on the right-hand side is omitted. and the quantities ∂η∂x⋅i12dtdτ∂∂x⋅igμνx⋅μx⋅νgiμdxμdτ as the components of its momentum. The particle’s energy is given by the Legendre transform η-∑∂η∂x⋅ix⋅idτdt-giμdxμdτdxidt=1dtdτdτ2-giμdxidxμ=1dtdτg4μdx4dxμ=g4μdxμdτ. Equations ()–() show that “minus momentum and energy form a four-vector” (“Negativer Impuls u. Energie bilden Vierervektor”), whose components can be written as gμρdxρdτ . For a particle that does not have unit mass, this expression has to be “multiplied by [its rest mass] m ” (“Noch mit m zu mult[iplizieren]”). Instead of one particle, Einstein now considered a continuous mass distribution. Dividing the four-momentum in expression () (multiplied by m ) by the volume V , which can be written as(Einstein and Grossmann 1913, 10). Einstein had already used this equation on p. 5R (see equation ()). V1GdτdtV01Gdτdtmρ0 , he introduced the four-momentum densityIn the notebook, “momentum density” (“Impulsdichte”) and “energy density” (“Energiedichte”) are introduced separately. gμρdxρdτdx4dτρ0G . Einstein now drew a horizontal line and wrote down the contravariant stress-energy tensor for pressureless dust, or, as he called it, the “tensor of material flow” (“Tensor der materiellen Strömung”) Tiκρ0dxidτdxκdτ . He lowered one index and introduced the notation T′iκ∑igνiTiκ , for the “resulting mixed tensor” (“Hieraus gemischter Tensor”), which he explicitly identified as “stress-energy tensor” (“Sp[annungs]-Energie-Tensor”). The components of the four-momentum density introduced in expression () are simply the μ4 -component of GTμν′ . Finally, Einstein wrote down the divergence of this mixed tensor density ∂GgνiTiκ∂xκ . According to the energy-momentum balance between matter and gravitational field, which Einstein had derived on p. 5R on the basis of considerations closely analogous to those on this page, the sum of this divergence and the force density must vanish. The latter is given by the force per unit mass (see equation ()) multiplied by m and divided by V (see equation ()). The result isIn the expression for the force density or “force per unit volume” (“Kraft pro Volumeinheit”) in the notebook, a factor ρ02 is missing. 12Gρ0∂gκλ∂xνdxκdτdxλdτ12G∂gκλ∂xνTκλ. At the bottom of the page, however, Einstein only wrote down expression (), the first term in the energy-momentum balanced. At the top of p. 21R, he returned to the consideration of force and energy rather than of force and energy densities. He wrote down an expression equivalent to equation (), the first component of which gives the “ x -component of the ponderomotive force” (“ x -Komponente der ponderomotorischen Kraft”) on a point particle of unit mass,A crossed-out factor of ρ0G occurs in the numerator of both this expression and the next, which suggests that Einstein at some point considered force and energy densities. ∑∂gμν∂xσx⋅μx⋅νg44+g11x⋅2+… , and an expression equivalent to equation () for the “energy of the point” (“Energie des Punktes”), g14dxdt+g24dydt+.+g44g44+g11x⋅2+… . Right next to this expression, Einstein explicitly wrote down the metric of a static field 100001000010000c2 , which would allow him to recover his 1912 static theory from a theory based on the metric tensor. He noted that the gi4 -components “definitely vanish in a static field” (“ g14 g24 … verschwinden sicher im statischen Felde”). In the static case, the numerator in expression () for the particle’s energy thus reduces to g44 . Special relativity tells us that energy is proportional to inertial mass. Galileo’s principle, i.e., the principle that the gravitational acceleration is the same for all bodies, tells us that the gravitational force is proportional to inertial mass as well. Expressions () and (), however, imply that, unless all spatial components of the metric are constants, the ratio of force and energy in a static field, ∑∂gμν∂xσx⋅μx⋅νg44 , will depend on the particle’s velocity. As Einstein put it: “If the force is supposed to vary like the energy, then g11 , g22 etc. must vanish for the static field” (“Soll die Kraft sich ändern wie die Energie, so müssen im statischen Felde g11 , g22 etc. verschwinden”).More accurately, the deviations of “ g11 , g22 etc.” from their constant Minkowskian values must vanish in the static case. Although not stated explicitly, Einstein’s conclusion was that the metric of the static field has to be of the form ().Presumably, although this is not made explicit, Einstein only wanted to draw the conclusion that static fields must be of this form in first-order approximation. He did not seem the least bit disturbed when in June 1913, in an attempt to calculate the perihelion advance of Mercury on the basis of the Entwurf theory, he found that the metric field of the sun in second-order approximation is not spatially flat (cf. [p. 6] of the Einstein-Besso manuscript [CPAE 4, Doc. 14]). This argument has a certain prima facie plausibility, but it does not hold up under closer scrutiny. On p. 20R, Einstein had identified a particle’s momentum pi (see equation ()) and the force Fi acting on it (see equation () and expression ()) in such a way that the spatial components of the geodesic equation, the Euler-Lagrange equations for the Lagrangian in equation (), can be written in a form reminiscent of Newton’s second law dpidtFi . For Einstein’s argument on p. 21R to be valid one would have to be able to substitute (modulo a proportionality constant) Ex⋅i (where the energy E is given by equation () or expression ()) for pi in equation (). This substitution, however, is not allowed. In other words, there is no reason to think that the antecedent of Einstein’s conditional (i.e., “the force varies like energy”) is true, and the argument fails. Einstein drew two figures next to expressions ()–(), presumably to illustrate his argument, although their purpose remains unclear. The upshot of Einstein’s considerations, however, is unambiguous. He had found an argument, based on a fundamental postulate of classical mechanics and completely independent of the gravitational field equations, that seemed to show that the metric of static gravitational fields has to be spatially flat. This confirmed his ideas about how to recover both Newton’s theory and his own 1912 theory for static gravitational fields from the metric theory.In Einstein to Erwin Freundlich, 19 March 1915 (CPAE 8, Doc. 63), Einstein once again addressed the question “whether matter at rest can generate any other gravitational field than a g44 -field” (“ob ruhende Materie ein anderes Gravitationsfeld als ein g44 -Feld erzeugen kann”). “It cannot” (“Dies ist nicht der Fall”), he wrote. In support of this claim, he presented a calculation done in first-order approximation and based on the Entwurf field equations (i.e., on weak-field equations without a trace term). He made no reference to any other arguments for his claim. A metric of this form, however, was incompatible with the modified weak-field equations introduced on p. 20L.It is also incompatible with the harmonic restriction (see footnote 261). Einstein therefore gave up the idea of constructing field equations out of the Ricci tensor with the help of the harmonic restriction. Embedding the Stress Tensor for Static Gravitational Fields into the Metric Formalism (21R) For a metric of the form diag1,1,1,c2 Einstein expected his new metric theory to reduce to his 1912 theory for static gravitational fields. That implied that one should also recover the expression for the gravitational stress tensor of the 1912 theory. On the bottom half of p. 21R, Einstein tried to translate the expression for this stress tensor into the language of the metric theory, replacing factors c , c2 , and 1c2 by G , g44 , and 44 , respectively.On p. 39L, Einstein had attempted a similar translation of the field equations of the 1912 theory (see sec. , equations ()–()). Exactly what Einstein hoped to achieve remains unclear. His comments and the fact that he deleted the calculation in its entirety do make clear, however, that he was unhappy with the results. Part of the problem may have been that the expression for the stress tensor of the 1912 theory did not seem to agree with the corresponding components of the quantity representing gravitational stress-energy density constructed on p. 21L. Under the heading “static special case” (“Statischer Spezialfall”), Einstein wrote down the Xx -component of the stress tensor of (the final version of) his 1912 static theory(Einstein 1912b, 456, equation (5)). Xx1c∂c∂x∂c∂x-12cgrad2c . Using the relation ∂c∂x1c∂∂xc22 , he rewrote this equation as 4Xx1c3∂c2∂x∂c2∂x-12c3grad2c2=c1c21c2∂c2∂x∂c2∂x-12c1c21c2grad2c2 He then translated this expression into the language of the metric theory, using that cG , g44c2 , and 441c2 for a metric of the form diag1,1,1,c2 . In this way, he arrived at 4XxG4444∂g44∂x∂g44∂x-124444∑ν∂g44∂xν∂g44∂xν Underneath this expression Einstein wrote: “impossible because of divergence equation” (“Unmöglich wegen Divergenzgleichung”). The “divergence equation” is presumably the equation setting the gravitational four-force density equal to the divergence of gravitational stress-energy density. In the notation of the Entwurf, this equation can be written asCf., e.g., (Einstein and Grossmann 1913, 16–17, equations (12a) and (18)). 12g∂gμν∂xσTμν∂∂xνggσμtμν , where Tμν and tμν denote the contravariant stress-energy tensor for matter and the corresponding pseudo-tensor for the gravitational field, respectively. On pp. 19R, 20L, and 21L, Einstein had checked, in a first-order approximation and using his candidate (weak) field equations to eliminate the stress-energy tensor for matter, whether the gravitational force density can be rewritten as a divergence. The tentative expression for gravitational stress-energy density constructed on p. 21R confronted Einstein with the converse problem, viz. whether the divergence of the gravitational stress-energy density actually gives the gravitational force density. It is not clear how Einstein could tell without further calculation that this is not possible—if that is in fact what he means by his remark “impossible because of divergence equation”—but Einstein may have reached this conclusion on the basis of a comparison of equation () with expression () on p. 21L, which (in linear approximation) gives the gravitational force density in the form of the divergence of gravitational stress-energy density. The translation of equation () into equation () is not unique. Einstein, in fact, gave an alternative translation of the first term in the expression for the stress tensor G∂1c2∂x∂c2∂x∑iκG∂iκ∂x∂giκ∂x . Apparently, this expression was not satisfactory either. Einstein deleted this whole calculation by a diagonal line, and wrote: “special case probably incorrect” (“Spezialfall wahrscheinlich unrichtig”). Synopsis of the Problems with the Harmonic Restriction and the Linearized Einstein Tensor (19L–21R) The calculations on p. 21R mark the end of Einstein’s consideration of field equations extracted from the Ricci tensor with the help of the harmonic restriction, and of the modified weak-field equations that we now recognize as the Einstein equations of the final theory in linearized form. To conclude this section, we summarize the chain of reasoning that produced this unfortunate turn of events. On p. 19L, Einstein showed that the harmonic restriction can be used to eliminate unwanted second-order derivative terms from the Ricci tensor. On p. 19R, examining these new field equations in linear approximation, he found that the natural way to make sure that the weak-field equations are compatible with energy-momentum conservation is to impose a further coordinate restriction, viz. the Hertz restriction. The combination of the harmonic restriction and the Hertz restriction implies that the trace of the linearized metric must be a constant. To avoid this implication, Einstein (on p. 20L) added a trace term to the weak-field equations, effectively changing their left-hand side from the linearized Ricci tensor to the linearized Einstein tensor. This modification obviates the need for the Hertz restriction and thus for the condition on the trace of the weak-field metric. Part of the original problem, however, still persists, albeit in a different guise. One of the difficulties with the restriction on the trace of the metric is that it is not satisfied by a metric of the form diag1,1,1,c2 . Einstein believed that his theory would not have a sensible Newtonian limit unless weak static fields can be represented by a metric of this form. At the same time, a metric of this form would allow him to recover his 1912 theory for static gravitational fields from the new metric theory. It was thus a serious problem that the restriction on the trace of the metric rules out a metric of this form. Unfortunately, with the modification needed to avoid this restriction, the weak-field equations themselves no longer allow a solution with a metric of the form diag1,1,1,c2 . On p. 21R, Einstein therefore reexamined whether his presuppositions about the static field were justified. A fallacious argument convinced him that nothing less than Galileo’s principle that all bodies fall with the same acceleration requires that the metric of static fields does indeed have the form he had been assuming.Einstein checked and confirmed this expectation once more in 1915 (see footnote ). He only realized that a weak static field need not be represented by a spatially flat metric when he calculated the perihelion motion of Mercury in (Einstein 1915c). This sealed the fate of the harmonic restriction and the linearized Einstein tensor. Einstein only returned to field equations including a trace term by a different route in November 1915.For discussion, see “Untying the Knot …” secs. 5–6 (in this volume). Exploring the Ricci Tensor in Unimodular Coordinates (22L–24L, 42L–43L) On p. 22R Einstein considered a new way of extracting a candidate for the left-hand side of the field equations from the Riemann tensor. He started from a new expression for the Ricci tensor, now entirely in terms of the Christoffel symbols and their first-order derivatives. He split this tensor into two parts each of which separately transforms as a tensor under unimodular transformations. Restricting the allowed transformations to unimodular transformations, he took one of these parts as the new candidate for the left-hand side of the field equations and explored it on the following pages. The object returned in the field equations published in the first of Einstein’s four communications to the Berlin Academy of November 1915.(Einstein 1915a). We therefore call it the November tensor. The November tensor still contains terms with second-order derivatives of the metric in addition to a core-operator term. On p. 22R, Einstein imposed the Hertz restriction to eliminate those terms (see secs. –). This coordinate restriction would return as the coordinate condition (in the modern sense) for the November tensor in (Einstein 1915a). The advantage of the Hertz restriction is that it serves two purposes. It can be used to eliminate unwanted terms with second-order derivatives of the metric, and it ensures that the divergence of the matter stress-energy tensor vanishes in a weak-field approximation (see p. 19R). On p. 19R, Einstein had been forced to introduce two separate conditions for these two purposes—the harmonic restriction and the Hertz restriction. The introduction of the Einstein tensor on p. 20L can be seen as an attempt to eliminate the Hertz restriction. The introduction of the November tensor can likewise be seen as an attempt to eliminate the harmonic restriction. The expression extracted from the November tensor with the Hertz restriction contains a large number of terms quadratic in first-order derivatives of the metric. On p. 23L, Einstein added a further addition to eliminate most of these terms. He went back to the original form of the November tensor in terms of the Christoffel symbols and imposed a coordinate restriction with which he could eliminate two of the three terms of the Christoffel symbols. This coordinate restriction is to unimodular transformations under which an expression that we call the ϑ -expression transforms as a tensor. We call this restriction the ϑ -restriction. Einstein discovered that the ϑ -restriction not only eliminates many terms with first-order derivatives of the metric but that it also takes care of the unwanted second-order derivatives that he had eliminated earlier with the Hertz restriction. Einstein thus lifted the Hertz restriction and kept only the ϑ -restriction (p. 23L; discussed in sec. ). He began to investigate which non-autonomous transformations are allowed by the ϑ -restriction. In the end, he abandoned the ϑ -restriction because it does not allow transformation to rotating frames in Minkowski spacetime (pp. 23R–24L, 42L–43L; discussed in secs. –). Extracting the November Tensor from the Ricci Tensor (22R) At the top of p. 22R, Einstein wrote down the covariant form of the Ricci tensor in the formThe summation should be over k and λ rather than over k and l . Til∑kl∂∂xli kk ∂∂xki lk+i kλλ lk-i lλλ kk . Contrary to the expressions for the Riemann and Ricci tensors earlier in the notebook, equation () is written entirely in terms of Christoffel symbols rather than in terms of the metric tensor and its derivatives. Apparently, it was Marcel Grossmann, whose name appears at the top of the page, who suggested this expression to Einstein. Grossmann may also have suggested some of the further manipulations of this expression on p. 22R. Recall that Grossmann’s name was also written next to the first occurrence of the Riemann tensor in the notebook on p. 14L. Two of the four terms in this new expression for the Ricci tensor can be combined to form a quantity that can easily be seen to be a tensor under unimodular transformations. Under unimodular transformations the determinant G of the metric transforms as a scalar. Hence, lgG also transforms as a scalar under such transformations, and the ordinary derivative of this quantity as a vector. Einstein denoted this vector by Ti and wrote that “if G is a scalar, then [...] Ti a tensor of first rank” (“Wenn G ein Skalar ist, dann ∂lgG∂xiTi Tensor 1. Ranges.”).Note that this is probably the first time that vectors are called “tensors of first rank” (also note that this is all in the context of unimodular transformation only). In (Budde 1914), the generalization of tensors to arbitrary dimension and rank was credited to Grossmann’s part of Einstein and Grossmann 1913: “Recently, Mr. Grossmann [...] has proposed a still further reaching generalization. He denotes quantities of arbitrary rank as “tensors,” so that vectors, trivectors, and bitensors are also subsumed under the term “tensor;” the generalization consists in extending his definitions to structures of n th rank in m -dimensional space” (Budde 1914, 246). For further discussion, see the appendix to (Norton 1992) and (Reich 1994). The term “rank” (“Rang”) appears in the notebook only on this page. On p. 14L the fourth rank Riemann tensor was called “Ebenentensor vierter Mannigfaltigkeit.” Using the relation ∂lgg∂xii κκ , one can identify two of the four terms of the Ricci tensor as the covariant derivative of the vector Ti . Einstein regrouped the terms in equation () accordingly, Til∂Ti∂xl-∑i lλTλ-∑kλ∂i lk∂xk-i kλλ lk . Under the first term in parentheses he wrote “tensor of second rank” (“Tensor 2. Ranges”); under the second he wrote “presumed tensor of gravitation Tilx ” (“Vermutlicher Gravitationstensor Tilx ”). Since taking the covariant derivative is a generally-covariant operation and Ti is a first-rank tensor under unimodular transformations, its covariant derivative is a second-rank tensor under unimodular transformations. The second term in parentheses in equation (), the difference between the full generally-covariant Ricci tensor and the covariant derivative of Ti , will therefore also transform as a tensor under unimodular transformations. Einstein took this quantity, Tilx∂i lk∂xk-∑kλi kλλ lk as his new candidate for the left-hand side of the field equations. We shall refer to it as the November tensor. Extracting Field Equations from the November Tensor Using the Hertz Restriction (22L–R) By imposing the Hertz restriction in addition to the unimodularity restriction, one can reduce the November tensor to the required form of a core operator plus terms with products of first-order derivatives of the metric. This is what Einstein confirmed on the next two lines on p. 22R, as the first step in a “further rewriting of the tensor of gravitation” (“Weitere Umformung des Gravitationstensors”). The only terms in Tilx with second-order derivatives of the metric occur in the term with derivatives of the Christoffel symbol. Einstein expanded this term to ∂i lk∂xk12∂kα∂giα∂xl+∂glα∂xi-∂gil∂xα∂xk , and then eliminated all unwanted second-order derivative terms by assuming the Hertz restriction. “We presuppose” (“Wir setzen voraus”), he wrote, that ∑κ∂κα∂xκ0 , adding that “then this [i.e., the right-hand side of equation ()] is equal to” (“dann ist dies gleich”): -12∑kα∂2gil∂xα∂xk-12∑∂kα∂xl∂giα∂xk+∂kα∂xi∂glα∂xk . In the notebook, the factors 12 in front of both terms appear above the summation signs and were probably added later. The core-operator term in expression () comes from the last term on the right-hand side of equation (). The first two terms in equation () turn into the products of first-order derivative terms in expression () with the help of the Hertz restriction and the relation kα∂giα∂xl∂kα∂xlgiα . On the bottom half of p. 22L, we find what appears to be an earlier attempt at eliminating unwanted second-order derivative terms from the November tensor with the help of the Hertz restriction. Relabeling the indices in equation (), one can write the first part of the November tensor as ∂i ml∂xl12∂κl∂gκi∂xm+∂gκm∂xi-∂gim∂xκ∂xl . The first two terms on the right-hand side give rise to unwanted second-order derivative terms 12κl∂2gκi∂xl∂xm+∂2gκm∂xl∂xi . Except for the factor 12 , this is just the expression that Einstein wrote directly underneath the horizontal line on p. 22L. The first term in expression (), multiplied by a factor 2 , can be rewritten as -∂kl∂xm∂gki∂xl+∂kl∂gki∂xl∂xm . In the notebook, the first term, a product of first-order derivatives, is indicated only by a dot (and an expression similar to expression () for the second term in equation () is omitted altogether). The Hertz restriction ensures that the second term, which can be rewritten as ∂∂kl∂xlgki∂xm , vanishes. As Einstein wrote directly underneath the second term in the expression () in the notebook: “suffices, if ∑∂κl∂xl vanishes” (“Gen[ü]gt, wenn … verschwindet”). We now return to p. 22R. On the bottom half of the page, Einstein turned his attention to terms in the November tensor () with products of first-order derivatives of the metric. He began by expanding the term with a product of Christoffel symbols: i kλλ lk14λαkβ∂giα∂xk-∂gik∂xα+∂gαk∂xi∂glβ∂xλ-∂glλ∂xβ+∂gλβ∂xl=-14λαkβ∂giα∂xk-∂gik∂xα∂glλ∂xβ-∂glβ∂xλ+14λαkβ∂gαk∂xi∂gλβ∂xl. As in expression (), the numerical factors were added later. In the second step, Einstein used the same symmetry argument that he had used on pp. 17R and 19L (see the discussion following expression ()). In the following two lines, Einstein noted that the last term can be rewritten as ∂λα∂xi∂gλα∂xl “or” (“oder”) as ∂λα∂xl∂gλα∂xi . Einstein also rewrote the terms with products of first-order derivatives in the first part () of the November tensor. Using the relation (in modern notation) gαβ,igαl,β12gαβ,igαl,β+gβl,α-gαβ,l+12gαβ,igαβ,l=gαβ,iα βl+12gαβ,igαβ,l and relabeling indices, one can rewrite the expression in parentheses in expression () asThis is the first time in the notebook that Einstein introduced the Christoffel symbols to replace ordinary derivatives of the metric. Up to now the calculations in the notebook always proceeded by expanding the Christoffel symbols and by rewriting the tensor expressions in terms of simple derivatives of the metric instead of using the compact Christoffel symbols. Why Einstein proceeded the other way around here is not clear, but it is tempting to speculate that it reflects a first inkling on Einstein’s part of the importance of the Christoffel symbols in his gravitational theory. ∂αβ∂xl∂giβ∂xα+∂αβ∂xi∂glβ∂xα∂αβ∂xlα βi+∂αβ∂xiα βl+∂αβ∂xi∂gαβ∂xl Combining the expressions (), (), and ()–(), one arrives at 2Tilxαβ∂2gil∂xα∂xβ-12αkβλ∂giα∂xβ-∂giβ∂xα∂glk∂xλ-∂glλ∂xk∂∂xiαβα βl+∂∂xlαβα βi+32∂∂xiαβ∂∂xlgαβ for (minus 2 times) the reduced November tensor (i.e., the November tensor in unimodular coordinates satisfying the Hertz restriction). In the corresponding expression at the bottom of p. 22R in the notebook, the second line contains some errors: above the summation sign in front of the first two terms, Einstein added a factor 12 ; and the third term has 14 instead of 32 . Non-autonomous Transformations Leaving the Hertz Restriction Invariant (22L) The field equations based on the reduced November tensor (see equation ()) will be covariant under those unimodular transformations that preserve the Hertz restriction. On the top half of p. 22L, Einstein derived the condition for non-autonomous unimodular transformations leaving the Hertz restriction, and thus the corresponding field equations, invariant.Earlier in the notebook and in a different context (see p. 10L ff.), Einstein had already investigated this question for infinitesimal transformations (see sec. ). The calculation on p. 22L closely follows the one on p. 10L. Note that this calculation precedes the calculations showing that the Hertz restriction can be used to eliminate unwanted second-order derivative terms from the November tensor on the bottom half of p. 22L and on p. 22R. At the top of p. 22L, Einstein began by writing the Hertz restriction in primed coordinates ∑∂′μν∂x′ν0 . Next to it, he wrote the determinant condition on the transformation matrix pμν for a unimodular transformation pμν1 . These two equations can be seen as a cryptic statement of the question being addressed on this page: given a metric field satisfying the Hertz restriction in some (unprimed) coordinate system, what are the unimodular coordinate transformations such that the Hertz restriction will be satisfied in the new (primed) coordinate system as well? The answer to this question takes the form of an equation for the transformation matrix pμν and its inverse μν . This equation involves the components of the metric field in the unprimed system. The transformations preserving the Hertz restriction, the solutions of this equation, will thus depend on which metric field one starts from in the unprimed system. In other words, these transformations are examples of “non-autonomous transformations.”See the discussion in sec. . Other examples of conditions for “non-autonomous transformations” can be found on pp. 7L, 8R, 10L, and 23R. To find the equation for these non-autonomous transformations, Einstein transformed equation () from x′μ - to xμ -coordinatesThe transformation matrices are defined as pαβ∂x′α∂xβ and αβ∂xβ∂x′α (see equations ()–()). ∑νi∂∂xipμαpνβαβ0 . Using the relation νipνβδiβ , he rewrote the left-hand side as pμα∂αi∂xi+αβνi∂pμαpνβ∂xi . The first term vanishes on account of the assumption that forms the starting point of this calculation, viz. that the metric field satisfies the Hertz restriction in the unprimed coordinates. Einstein rewrote the second term as ∑αβν¯ipμα∂pνβ∂xi+pν¯β∂pμα∂xi , and then, on the next line, as ∑αi∂pμα∂xi+∑αβνipμα∂pνβ∂xi . The first term in this last expression was familiar to Einstein from the analogous calculation on p. 10L (cf. the second term in equation ()), which may be why Einstein underlined it. On p. 10L he had found that the vanishing of the first term is the condition for infinitesimal unimodular non-autonomous transformations preserving the Hertz restriction. In the infinitesimal case, the second term vanishes (see footnote 148). This is probably why, on p. 22L, he wrote next to the second term in expression (): “vanishes if funct[ional] det[erminant] = 1.” (“verschwindet, wenn Funkt. Det. = 1.”). For finite transformations, however, this term does not vanish, as Einstein presumably realized, for he included it in an attempt to further simplify the expression in expression () on the next line. The first term in expression () can be rewritten as ∂∂xiαipμα-pμα∂αi∂xi ; the second term as ∂∂xiαβνipμαpνβ-∂∂xiαβνipμαpνβ , which in turn can be rewritten as ∂∂xiαipμα-αβpμαpνβ∂νi∂xi-∂∂xiαipμα . Adding expressions () and (), one arrives at the expression given on the next line in the notebook: ∑∂∂xiαipμα-pμα∂αi∂xi+∂∂xiαipμα-αβpμαpνβ∂νi∂xi-∂∂xiαipμα. As Einstein noted, the second term vanishes (because of the Hertz restriction) and the third term cancels with the fifth. What is left is an expression that has basically the same structure as expression (). On the next line, Einstein reverted to the latter. The upshot then was that the matrix pμν (and its inverse μν ) for some unimodular coordinate transformation from coordinates xμ to x′μ must satisfy αi∂pμα∂xi+αβνipμα∂pνβ∂xi0 to ensure that a metric field that satisfies the Hertz restriction in the xμ -coordinates satisfies the Hertz restriction in the x′μ -coordinates as well. This is a rather complex condition. Examining the simpler version for infinitesimal transformations in which case the second term in equation () vanishes automatically, Einstein had found that it does not allow a transformation to uniformly accelerated frames of reference in the important special case of Minkowski spacetime (see pp. 10L–11L).On p. 11L, Einstein had convinced himself that the Hertz restriction does allow rotations. Rotation, however, is also ruled out by the Hertz restriction (see sec. , especially equations ()–() and notes 163–164) He nonetheless continued to use the Hertz restriction (see p. 23L). Extracting Field Equations from the November Tensor Using the ϑ-Restriction (23L–R) At the bottom of p. 22R, Einstein had arrived at a candidate for the left-hand side of the field equations extracted from the November tensor by imposing the Hertz restriction (see eq. ()). This candidate contains numerous terms with products of first-order derivatives of the metric. Most of these terms come from the product of Christoffel symbols in the second term of the November tensor (). On p. 23L, Einstein returned to the expression for the November tensor in terms of Christoffel symbols and added a new coordinate restriction to the Hertz condition with which he could eliminate two of the three first-order derivative terms in every Christoffel symbol. In this way he could eliminate most of terms quadratic in first-order derivative terms found at the bottom of p. 22R. This additional coordinate restriction limits the range of allowed coordinate transformations to those unimodular transformations under which a quantity denoted by ϑikλ , which is essentially the fully symmetrized version of gik,λ , transforms as a tensor. We call this quantity the “ ϑ -expression,” the unimodular transformations under which it transforms as a tensor “ ϑ -transformations,” and the restriction to such transformations the “ ϑ -restriction.”The Hertz expression, it turns out, transforms as a vector under ϑ -transformations (see the discussion following eq. ()). Einstein’s basic strategy on p. 23L was to subtract terms from the November tensor that by themselves transform as tensors under ϑ -transformations. What was left served as Einstein’s new candidate for the left-hand side of the field equations. In the course of the calculations on p. 23L, Einstein came to realize that with the ϑ -restriction he did not need the Hertz restriction anymore to eliminate terms with unwanted second-order derivatives of the metric. The ϑ -restriction took care of those terms all by itself. Einstein therefore abandoned the Hertz restriction and focused on the ϑ -restriction. On the third line of p. 23L, separated from the first twoAt the top of p. 23L, Einstein began an explicit transformation of the ordinary derivative of the metric: ∂g′ik∂xλλα∂iσkτgστ∂xα ( xλ on the left-hand side should be x′λ ). The calculation breaks off almost immediately but is taken up again on the facing page, p. 23R, in order to find the condition for “non-autonomous transformations” under which the ϑ -expression is invariant (see sec. ). by a horizontal line, Einstein stated the assumption that forms the starting point of the argument on this page, viz. that the quantityIn the notebook, Einstein wrote “ 2 ” above the last plus sign in expression (). Above many of the plus and minus signs in subsequent expressions on this page the opposite has been written (and, in some cases, has been deleted again). These sign changes are related to Einstein’s consideration on p. 25R of a variant of the ϑ -restriction, which we shall call the ϑ^ -restriction (see sec.  for discussion). 12∂gik∂xλ+∂gkλ∂xi+∂gλi∂xk “be a tensor ϑikλ ” (“sei Tensor ϑikλ ”). In other words, Einstein was interested in transformations under which this ϑ -expression would transform according to the transformation law for a fully covariant third-rank tensor. The next line explicitly gives this transformation lawNote that Einstein deviates from the notation pαβ and αβ typically used in the notebook for the transformation matrices ∂x′α∂xβ and ∂xβ∂x′α . ϑ′ikl∑αβϑαβ∂xα∂x′i∂xβ∂x′k∂x∂x′l . Einstein used the ϑ -expression to rewrite the Christoffel symbols of the first and second kind as i lkϑilk-∂gil∂xk and i lkkαϑilα-∂gil∂xα , respectively. He substituted the latter expression into the November tensor (cf. equation ()): Tilx∂kαϑilα-∂gil∂xα∂xk-λαkβϑikα-∂gik∂xαϑlλβ-∂glλ∂xβ . The rationale behind the ϑ -restriction now becomes clear. Any (combination of) term(s) in equation () that transforms as a tensor under ϑ -transformations can be subtracted from the November tensor and what is left will still transform as a tensor under ϑ -transformations. Originally, Einstein probably only meant to eliminate terms quadratic in first-order derivatives of the metric from the November tensor. He initially expected that the Hertz restriction would still be needed to eliminate terms with unwanted second-order derivatives. On the next line he explicitly stated the requirement that the Hertz expression, ∑∂kα∂xk , “be =0 ” (“sei =0 ”). He subsequently added the clause “is not necessary” (“ist nicht nötig”).The evidence for our assumption that this clause was indeed added later is twofold. First, the awkwardness of the syntax of the sentence “ ∑∂kα∂xk be =0 is not necessary” (our emphasis) disappears under the assumption. Second, Einstein uses the Hertz restriction in the equation immediately following this sentence. The end result of the calculation on p. 23L is: ∑∂∂xkkα∂gil∂xα+∑ραkβ∂gik∂xα∂glρ∂xβ . It turns out, as Einstein himself came to recognize, that there is a short-cut to get from eq. (), which gives the November tensor in terms of the ϑ -expression, to eq. (). If one sets ϑikλ0 in eq. (), which amounts to replacing the Christoffel symbols by the truncated Christoffel symbols kα∂gil∂xα (see eq. ()), one arrives at: -∂kα∂gil∂xα∂xk-λαkβ∂gik∂xα∂glλ∂xβ , which is just (minus) Einstein’s expression ().It is tempting to speculate that these considerations are related to a remark Einstein made in 1915 when he published field equations based on the November tensor. Einstein wrote that earlier he had looked upon the quantities “ 12∑μgτμ∂gμν∂xν as the natural expression for the components of the gravitational field although in the light of the formulae of the differential calculus, it is more natural to introduce the Christoffel symbols νστ instead of those quantities. This was a fateful prejudice.” (“… als den natürlichen Ausdruck für die Komponenten des Gravitationsfeldes, obwohl es im Hinblick auf die Formeln des absoluten Differentialkalküls näher liegt, die Christoffelschen Symbole … statt jener Größen einzuführen. Dies war ein verhängnisvolles Vorurteil.” Einstein 1915a, 782). This point is also emphasized in Einstein’s letter to Arnold Sommerfeld of 28 November 1915: “What gave me the key to this solution [the Einstein field equations in their final form] was the insight that not ∑μglα∂gai∂xm but the related Christoffel symbols νστ should be looked upon as the natural expression for the “components” of the gravitational field” (“Den Schlüssel zu dieser Lösung lieferte mir die Erkenntnis, dass nicht …sondern die damit verwandten Christoffel’schen Symbole … als natürlichen Ausdruck für die “Kom­ponente” des Gravitationsfeld anzusehen ist.” CPAE 8, Doc. 153). For further discussion, see sec. 5 of “Untying the Knot …” (in this volume). This shows that the ϑ -restriction takes care of the terms with unwanted second-order derivatives in the November tensor as well. Once Einstein recognized this, he presumably added the clause “is not necessary” to his statement () of the Hertz restriction and made the necessary corrections to the derivation of eq. (). In the original derivation, Einstein used the Hertz restriction in the very first step. He introduced the quantity Tilxx-kα∂2gil∂xk∂xα+kα∂ϑilα∂xk-λαkβ∂gik∂xα∂glλ∂xβ+λαkβ(ϑikα∂glλ∂xβ +ϑlλβ∂∂xαgik), and remarked that this “is also a tensor” (“ist ebenfalls ein Tensor”). What he meant no doubt was: “transforms as a tensor under ϑ -transformations.” The new tensor Tilxx differs from Tilx by two terms: λαkβϑikαϑlλβ and ∂∂xkkα-∂gil∂xα+ϑilα . The first one transforms as a tensor under ϑ -transformations itself. This means that Tilx minus this term will still be a tensor under ϑ -transformations. The second one, expression (), vanishes on account of the Hertz restriction. When Einstein subsequently retracted the Hertz condition, he indicated that kα should be included within the scope of the derivative ∂∂xk in the first term on the right-hand side of equation (). He neglected to do so for the second term. This may simply have been an oversight but the fragmentary calculation at the top of p. 23R suggests that Einstein may have realized that ∂kα∂xkϑilα itself transforms as a tensor under ϑ -transformations. So this term can also be subtracted from Tilx and the result will still be a tensor under ϑ -transformations. On the next line, Einstein wrote down the covariant derivative of the ϑ -expression contracted with kα kα∂ϑilα∂xk-∑k iρϑρlα+k lρϑiρα+k αρϑilρkα . This expression will also transform as a tensor under ϑ -transformations, as Einstein noted: “Likewise [a tensor]” (“Ebenso”). He substituted equation () for the Christoffel symbols into expression () and subtracted terms with products of the ϑ -expression and the metric which are themselves tensors under ϑ -transformations. In this way, he arrived at kα∂ϑilα∂xk+∑kαρβ∂gik∂xβϑρλα+∂gkl∂xβϑiρα+∂gkα∂xβϑilρ , and noted that this would “therefore also” (“also auch”) be “a tensor” (“ein Tensor”) under ϑ -transformations. In modern notation, the last term in expression () can be rewritten as gkαgρβgkα,βϑilρgρβlogg,βϑilρ . The determinant of the metric transforms as a scalar under all unimodular transformation and hence under ϑ -transformations. The derivative of its logarithm therefore transforms as a vector under ϑ -transformations.On p. 22R, it was explicitly noted that this quantity transforms as a vector under unimodular transformations. As Einstein noted, the last term in expression () is therefore “itself a tensor” (“an sich ein Tensor”) under ϑ -transformations. It follows that the remaining three terms in expression () also form a tensor under ϑ -transformations. Precisely these three terms occur in the expression for Tilxx , as Einstein verified by underlining them, both in equation () and in expression (), and by relabeling some of the indices in equation (). After “subtraction” (“Subtraktion”) of these three terms from the right-hand side of equation () and changing the minus signs in front of the two remaining terms to plus signs, he arrived at ∑kα∂2gil∂xk∂xα+∑ραkβ∂gik∂xα∂glρ∂xβ and noted that this still “is a tensor” (“ist Tensor”) under ϑ -transformations. The combination of the Hertz restriction and the ϑ -restriction thus allowed Einstein to extract from the November tensor a candidate for the left-hand side of the field equations that has a remarkably simple form. It is the sum of a core-operator term and a term with a product of first-order derivatives. It is probably at this point that Einstein noticed the short-cut from eq. () to expression (). Expression () only differs from the latter in that kα in the first term is not included within the scope of ∂∂xk . Einstein marked this term to indicate that kα should be included within the scope of ∂∂xk . He did the same with the first term on the right-hand side of eq. (), which is the starting point of the argument that got him from the November tensor to the new candidate for the left-hand side of the field equations (). This amounts to rescinding the Hertz restriction, which explains why Einstein wrote “is not necessary” next to statement () of this restriction. The unwanted second-order derivatives that he had eliminated on p. 22R by imposing the Hertz restriction were absorbed into expression () for the covariant derivative of the ϑ -expression and subtracted from the November tensor. By including kα within the scope of ∂∂xk in eq. (), Einstein had thus extracted the following candidate for the left-hand side of the field equations by imposing the ϑ -restriction alone: ∑∂∂xkkα∂gil∂xα+∑ραkβ∂gik∂xα∂glρ∂xβ . Without the Hertz restriction, expression () inherits one more term involving the ϑ -expression from eq. (), namely ∂kα∂xk ϑilα (cf. the discussion following expression ()). It turns out, however, that this term is a tensor under ϑ -transformations itself, so that expression () without this term is still a tensor under ϑ -transformations.Note that this correction term would only give rise to terms with products of first-order derivatives of the metric anyway, and hence would not affect the result that the ϑ -restriction can be used to eliminate unwanted second-order derivative terms. That ∂kα∂xk —and consequently ∂kα∂xk ϑilα —are indeed tensors under ϑ -transformations can be seen as follows. Consider the contraction of gik and ϑikl : gikϑikl12gikgik,l+gkl,i+gli,k=gik12gik,l+gkl,i=12g,l-gik,igkl. Hence, gik,igkl is the difference between two expressions, gikϑikl and 12g,l , that are both tensors under ϑ -transformations. It is therefore a tensor under ϑ -transformations itself. It follows that gik,i is a tensor under ϑ -transformations as well, which is the result that we wanted to prove. Einstein may in fact have gone through a similar calculation. This is suggested by the calculation at the top of p. 23R, where he wrote down the right-hand side of the first line of equation (), ik∂gik∂xl+∂gkl∂xi+∂gli∂xk , as well as the terms ∑ik∂gli∂xk , ∑gli∂ik∂xk , and ∑∂ik∂xk , which are all involved in the calculation given in equation ().Another possibility is that this calculation was related to the investigation of a modified ϑ -expression on p. 43La (cf. footnote 313). To conclude our discussion of p. 23L, we want to emphasize that the calculation on this page nicely illustrates the usage of coordinate restrictions as opposed to coordinate conditions in the notebook. Thinking in terms of coordinate conditions, one would look upon expression () as representing the left-hand side of some candidate field equations of broad covariance in coordinates satisfying the ϑ -restriction chosen to facilitate comparison with Newtonian theory. It is not at all clear, however, whether expression () actually is the representation of some tensor of broader covariance in the class of coordinate systems determined by the ϑ -restriction. It can be seen as the November tensor in coordinates such that the ϑ -expression vanishes, as follows from the short-cut from equation () to equation (). But the ϑ -restriction only requires that the ϑ -expression transform as a tensor, not that it vanish. Given that Einstein conceived of the restriction to ϑ -transformations as an essential feature of the theory and not as a feature of a particular representation of the theory, it was of no real interest to him to find the tensor of broader covariance corresponding to expression (). The relation between expression () and tensors of broader covariance, such as the November tensor or the Ricci tensor, is important only in that it allowed Einstein to construct field equations that are invariant under a precisely defined class of coordinate transformations. It is important to keep in mind that for Einstein this construction gave field equations, or candidate field equations, in their most general form, not just an expression of field equations of broader covariance in some restricted class of coordinate systems. Non-autonomous Transformations Leaving the ϑ-Expression Invariant (23R) With the exception of the first few lines, which we tentatively identified as a fragmentary version of the calculation given in equation (), the purpose of the calculations on p. 23R is to derive the condition for infinitesimal “non-autonomous transformations” leaving the ϑ -expression and thereby the field equations based on expression () invariant. As in the case of the corresponding condition for the Hertz restriction (see p. 22L and the discussion in sec. ), this condition takes the form of an equation for the transformation matrices pμν and μν involving the components of the metric field in the original coordinate system.Although Einstein presumably was interested only in unimodular transformations preserving the ϑ -expression, he did not explicitly impose the condition pμνμν1 on the determinant of the transformation matrices, as he had on p. 22L in the case of the Hertz restriction (see equation ()). After drawing a horizontal line under the fragmentary calculation at the top of the page, Einstein first derived the condition for non-autonomous transformations under which derivatives of the metric transform as tensors.This condition can also be found on p. 8R (see equation ()). Adding three such equations with cyclically permuted indices, Einstein then derived the corresponding condition for the ϑ -expression. He began by writing down how derivatives of the metric transform under an arbitrary transformation from unprimed to primed coordinates:The transformation matrices are defined as pαβ∂xα′∂xβ and αβ∂xβ∂xα′ (see equations ()–()). Essentially the same equation had been written down on top of p. 23L, which suggests that Einstein began to examine the transformation properties of the ϑ -expression before he extracted eq () from the November tensor with the help of the ϑ -restriction. ∂gik∂xl′lλ∂iιkκgικ∂xλ . If ∂gik∂xl′ is transformed back to the unprimed system on the assumption that the coordinate transformation relating the two coordinate systems is no longer arbitrary but one under which derivatives of the metric actually transform as tensors, one finds ∂gρσ∂xτpiρpkσplτ∂g′ik∂x′l=piρpkσplτlλ∂iιkκgικ∂xλ, where in the second step equation () was used, which is valid for arbitrary transformations. Einstein’s own description of the transition from equation () to the expression on second line of equation () is more cryptic: “transformed back piρ pkσ plτ ” (“zurück transformiert piρ pkσ plτ ”). On the next line, following the comment “in detail” (“ausführlich”), the expression on the second line of equation () is expanded. The relation pijikδjk is used to simplify the resulting three terms ∂gρσ∂xτ+piρ∂iι∂xτgισ+pkσ∂kκ∂xτgρκ . “For infinitesimal transformations” (“Für infinitesimale Transformationen”), one can replace pij in the last two terms by δij . If in addition the summation indices are relabeled, expression () reduces to ∂gρσ∂xτ+∂ρα∂xτgσα+∂σα∂xτgρα . Substituting this result into equation (), one arrives at ∂gρσ∂xτ∂gρσ∂xτ+∂ρα∂xτgσα+∂σα∂xτgρα . It follows that derivatives of the metric transform as tensors under infinitesimal non-autonomous transformations if and only if the last two terms in equation () vanish, i.e., if and only if the matrix ij for such a transformation satisfies the equation ∂ρα∂xτgσα+∂σα∂xτgρα0 for the metric field under consideration. Since ϑρστ is essentially the symmetrized version of ∂gρσ∂xτ , one can derive the condition for infinitesimal non-autonomous transformations under which ϑρστ transforms as a tensor by adding equation () and the two equations obtained from it through cyclic permutation of its indices, ∂gστ∂xρ∂gστ∂xρ+∂σα∂xρgτα+∂τα∂xρgσα∂gτρ∂xσ∂gτρ∂xσ+∂τα∂xσgρα+∂ρα∂xσgτα. As Einstein put it: “Through addition [of the various terms coming] from all three terms [of ϑρστ ] one obtains” (“Durch Addition aus allen drei Termen erhält man”) ϑρστ=ϑρστ+12[gρα∂∂xστα+∂∂xτσα+gσα∂∂xτρα+∂∂xρτα +gτα∂σα∂xρ+∂ρα∂xσ] Einstein only wrote down the right-hand side of this equation, left out the factor 12 in front of the expression in square brackets, and only indicated the last two terms in this expression by a dot. For infinitesimal transformations one can use thatFor infinitesimal transformations, one has ∂σα∂xτ∂pασ∂xτ . Using the definition of pασ and changing the order of differentiation, one can then write: ∂∂xτσα∂pασ∂xτ∂∂xτ∂xα′∂xσ∂∂xσ∂xα′∂xτ∂pατ∂xσ∂∂xστα . Earlier in the notebook, Einstein had already (implicitly) used a very similar relation (see p. 10L and footnote 148). ∂∂xτσα∂∂xστα , in which case the right-hand side of equation () reduces to ϑρστ+gρα∂σα∂xτ+gσα∂τα∂xρ+gτα∂ρα∂xσ . In the notebook, because of the factor 12 omitted in equation (), there is an extra factor 2 in front of the expression in square brackets. It follows that the condition for infinitesimal non-autonomous transformations under which the ϑ -expression transforms as a tensor is that “the [expression in square] brackets [in expression ()] should vanish for all combinations of ρστ ” (“Die Klammer soll für alle Kombinationen von ρστ verschwinden”). In other words, the matrix ij for (the inverse of) such transformations should satisfy the equation gρα∂σα∂xτ+gσα∂τα∂xρ+gτα∂ρα∂xσ0 for the metric field under consideration. The field equations extracted from the November tensor with the help of the ϑ -restriction will be invariant under all (non-autonomous) coordinate transformations that satisfy this condition. Solving the ϑ-Equation (42L–R) No attempt is made in the notebook to find solutions of the condition for infinitesimal non-autonomous ϑ -transformations given in equation (). Einstein adopted a somewhat different approach to get a sense of the range of transformations allowed by the ϑ -restriction. It is clear upon inspection of its definition () that the ϑ -expression vanishes for the Minkowski metric in its standard diagonal form. Any ϑ -transformation will preserve the vanishing of the ϑ -expression. Hence, if transformations to accelerated frames of reference in Minkowski spacetime are ϑ -transformations, the ϑ -expression should also vanish for the Minkowski metric expressed in the coordinates of such accelerated frames. On p. 42L, toward the end of the part starting from the other end of the notebook, Einstein set himself the task of finding the most general form of the metric field gμν satisfying the equation ϑρστ0 , imposing the additional constraints that the metric be time-independent and that its determinant be equal to unity and suppressing one spatial dimension. Given Einstein’s heuristic principles, the general solution should include the Minkowski metric in (uniformly) rotating coordinates. Unfortunately, this turns out not to be the case. What Einstein discovered instead (at the top of p. 42R), is that one does obtain a solution if one simply interchanges the covariant and contravariant components of the rotation metric. We shall call this solution the “ ϑ -metric.” Einstein tried to come to terms with this tantalizing result in various ways. One approach was to derive the equations of motion for a particle moving in a gravitational field described by the ϑ -metric and to identify the various components of the ϑ -metric occurring in these equations in terms of inertial forces in rotating frames of reference just as one would for the ordinary Minkowski metric in rotating coordinates. Einstein made two attempts along these lines. In one case, he derived the equations of motion as the Euler-Lagrange equations for the Lagrangian of a particle moving in a metric field (calculations at the bottom of pp. 42R and 43La; discussed in sec. ). In the other, he derived the equations of motion from the energy-momentum balance between matter and gravitational field (p. 24L; discussed in sec. ). Neither calculation produced a satisfactory result. Neither did a somewhat different approach which Einstein tried at the top of p. 43La (discussed in sec. ). He replaced the covariant components of the metric in the ϑ -expression by contravariant ones, ensuring that the new expression vanishes for the rotation metric without interchanging its co- and contravariant components. Einstein discovered, however, that the modified ϑ -expression could not be used to eliminate terms with unwanted second-order derivatives of the metric from the November tensor. Moreover, the new expression is mathematically ill-defined. In the end, Einstein was thus forced to give up the ϑ -restriction and the promising candidate () for the left-hand side of the field equations constructed with the help of it. The precise temporal order of the calculations on pp. 23L–24L at one end of the notebook and on pp. 42L–43L at the other remains unclear. There are various indications that Einstein switched back and forth between these two sets of pages. As we already noted, for instance, attempts to interpret the ϑ -metric in terms of inertial forces in rotating frames of reference can be found both on p. 42R–43L and on p. 24L.A possible further connection between these two attempts to interpret the ϑ -metric in terms of inertial forces in rotating frames of reference is the derivation on p. 43Lb of the equation of motion for a point mass moving in a metric field from the vanishing of the covariant divergence of the stress-energy tensor for pressureless dust (see sec. ) Several questions remain. Had Einstein already encountered the ϑ -expression in some other context when he used it on p. 23L to extract field equations from the November tensor? If so, it would explain why Einstein’s solution of the equation ϑρστ0 occurs in a different part of the notebook. It would, however, also raise the question why Einstein originally got interested in the ϑ -expression. If the calculation on p. 23L preceded the one on p. 42R,The occurrence in a calculation on p. 43La of Christoffel symbols, which are otherwise absent from the part that starts from the back of the notebook, suggests that at least this calculation is later than pp. 23L–R (cf. footnote 317). this question does not arise, but in that case it is unclear why Einstein turned over the notebook for the calculation on p. 42R instead of simply continuing his entries on p. 24L ff. Given these uncertainties, it is important to emphasize that the order in which we present these calculations may not fully reflect their temporal order. After these introductory and cautionary remarks, we turn to the actual calculation on p. 42L. At the top of the page,In the top-left corner, Einstein wrote the six independent components of this metric field arranging them in a somewhat peculiar way. The reason behind this is unclear. Einstein wrote down the components of the metric suppressing one spatial dimensionUnderneath the 44-component, he wrote “0,” which suggests that Einstein assumed the g44 -component to vanish. This is puzzling but would be consistent with the expression written right next to the metric () for the determinant of the metric. Perhaps, the components of () refer to small deviations of the metric from the flat Minkowski metric diag1,1,1 . This interpretation would be consistent with most of the rest of the calculation on this page, though not with the calculation of the determinant of the metric right next to (). g11g12g14g21g22g24g41g42g44 Einstein then set out to find the most general time-independent solution of a set of linear first-order coupled differential equations for this metric field. In modern notation, this set of equations can be written as gμν,λgμν,λ+gλμ,ν+gνλ,μ0 , where the indices can take on the values 1 , 2 , and 4 . Although the ϑ -expression is not explicitly mentioned on pp. 42L–43L, equation () can also be written as (cf. expression ()) ϑμνλ0 . On p. 42R, Einstein explicitly imposed the additional constraint that the determinant of the metric be equal to unity. This condition was probably part of the original problem that Einstein set himself on p. 42L, as is suggested by the fact that in the top right corner there is an expression for the determinant of the metric (),Einstein either did not finish the calculation or he set g44 equal to zero as he indicated in () (cf. footnote 301). g14g12g24-g22g14 +g24g14g12-g24g12. Einstein then listed the index-combinations for all independent components of the set of differential equations in equation () , He explicitly wrote down the equations corresponding to these index-combinations, with the exception of the last one, a clear indication that Einstein was interested only in time-independent solutions. The equations are grouped together as follows. First, Einstein gave the 111 and 222 components of equation (), i.e., in modern notation, g11,10 , g22,20 . He then gave the components with two indices equal to 4 , followed by the ones with one index equal to 4 . In the first four of these equations, he included but then deleted terms with derivatives with respect to x4 . In the fifth, he omitted this term altogether. Einstein thus arrived at g44,10 , g44,20 , 2g14,10 , 2g24,20 , g14,2+g24,10 . Finally, he wrote down the 112 and 122 components of equation (), 2g12,1+g11,20 , 2g12,2+g22,10 . From equations () and () it follows that g11 can only be a function of x2 and that g22 can only be a function of x1 . Einstein thus wrote g11ϕx2 , g22ψx1 . Substituting these expressions for g11 and g22 into equations () and (), he obtained 2g12,1ϕ′x2 , 2g12,2ψ′x1 . From these last two equations it follows that g12 has to be linear both in x1 and x2 . Terms in g12 of second-order or higher in x1 would make g12,1 dependent on x1 , which is contrary to equation (); terms of second-order or higher in x2 would likewise make g12,2 dependent on x2 , which is contrary to equation (). Hence, g12 has to be of the form g12c0+c1x1+c2x2+αx1x2 , where c0 , c1 , c2 , and α are arbitrary constants. Inserting equation () into equations () and (), Einstein found ϕ′x22c1+αx2 , ψ′x12c2+αx1 Integrating these equations and substituting the results into equations () and (), Einstein foundThe constants κ″ and κ″′ in these equations probably only got these designations after the introduction of the constants κ and κ′ in the expressions for g14 and g24 , which occur immediately below equations ()–() in the notebook (cf. equations ()–() below). g11ϕx22c1x2+α2x22+κ″ , g22ψx12c2x1+α2x12+κ″′ . Expressions for the components g14 and g24 can be found in a similar way. From equations () and () it follows that g14 and g24 can be written as g14ϕ^x2 , g24ψ^x1 , where we introduced the notation ϕ^ and ψ^ to distinguish these functions from the functions ϕ and ψ above. In the notebook, no such distinction is made. Inserting equations () and () into equation (), one finds that ϕ^′x2+ψ^′x10 , which allows the separation ϕ^′x2β , ψ^′x1β , where β is a constant. The notebook has α rather than β at this point, but in subsequent equations Einstein renamed this constant β , presumably to avoid confusion with the constant α introduced in equation (). Integrating equations () and () and substituting the results into equations () and (), one finds g14ϕ^x2βx2+κ , g24ψ^x1βx1+κ′ . An expression for g44 was not explicitly given in the notebook, but from equations () and () it immediately follows that g44 has to be a constant. The most general solution of the equations gμν,λ0 , under the additional constraint that gμν,40 , is thus given by (cf. equations (), ()–(), and ()–())The general solution for the 2+1-dimensional case can trivially be turned into a particular solution for the 3+1-dimensional case by adding gμ3g3μ0,0,1,0 gμν2c1x2+α2x22+κ″c0+c1x1+c2x2+αx1x2βx2+κc0+c1x1+c2x2+αx1x22c2x1+α2x12+κ″′βx1+κ′βx2+κβx1+κ′g44 At the bottom of p. 42L Einstein drew a figure indicating rotation around the z -axis. The relation between the calculation on p. 42L and rotation becomes clear on the next page. At the top of page 42R Einstein wrote down a metric—or a “ g -system” (“ g -Schema”) as he called it here—which is obtained from equation () by setting the integration constants c0 , c1 , c2 , κ , and κ′ equal to zero, κ′′κ″′12 and g441 If gμν represents small deviations from a flat Minkowski metric (cf. footnote 301), one needs to set κ″κ″′0 and g440 instead. gμν1+αx22αx1x2βx2αx1x21+αx12βx1βx2βx11 . This is the solution of the equation ϑμνλ0 that we shall refer to as the ϑ -metric. Einstein now imposed the condition that the determinant G of this metric be equal to unity. For G he wroteWith the help of the fully anti-symmetric Levi-Civita tensor εijk —which is equal to 1 for every even permutation of 1 , 2 , 3 (or, rather, 4 in this case), equal to 1 for every odd permutation, and equal to 0 otherwise—the determinant G can be written as Gεijkg1ig2jg4k . The three terms on the first line of equation () correspond to the even permutations 124 , 421 , and 241 ; the three terms on the second line to the odd permutations 142 , 214 , 421 . This way of evaluating the determinant is known as Sarrus’ rule. G1+αx121+αx22-αβ2x12x22-αβ2x12x22 +1+αx22β2x12-α2x12x22+1+αx12β2x22=1+α+β2x12+α+β2x22 +α2-2αβ2+αβ2+αβ2-α2x12x22. Next to this equation, Einstein wrote down the condition that this determinant be equal to unity, α+β20 . Using this relation between α and β , he inverted the metric in equation () and wrote down the result, μν10βx201βx1βx2βx11+αx12+x22 . which he denoted by , next to “ g -system” at the top of the page. As Einstein noted in a comment that he wrote next to these expressions of the ϑ -metric in its covariant and contravariant form, “the system of the ’s for a rotating body [is] identical to the g -system given here” (“Schema der für rotierenden Körper mit nebenstehendem g -Schema identisch”). This is an intriguing result. The ϑ -restriction does not allow the Minkowski metric in rotating coordinates, but it does allow the ϑ -metric, which is closely related to it. Is that enough to satisfy Einstein’s heuristic requirements? Can the ϑ -restriction be modified in such a way that it does allow the rotation metric without the need to switch its co- and contravariant components? These are the questions that are behind the calculations on pp. 42R, 43La, and 24L that will be discussed in secs. –. Before Einstein began his closer examination of the ϑ -metric, he briefly considered another special case of equation (). He drew a horizontal line and wrote the non-vanishing components of the metric -1-2c1x2 c1x1+c2x2 0c1x1+c2x2 -1-2c2x1 00 0 1. This metric is obtained from equation () by setting the integration constants c0 , α , β , κ , and κ′ equal to zero, setting κ″κ″′12 and g441 .Again, this metric could be obtained by setting κ″κ″′0 and g440 in the case that the gμν were deviations. Taken together both special cases exhaust the general solution (up to constants). This solution, however, does not satisfy one of Einstein’s additional constraints. As he wrote next to the metric (), “the determinant [of this metric field] is not 1 ” (“Determinante ist nicht 1 ”). Einstein then drew another horizontal line and began his closer examination of the tantalizing ϑ -metric of equations () and (). Reconciling the ϑ-Metric and Rotation (I): Identifying Coriolis and Centrifugal Forces in the Geodesic Equation (42R, 43La) On the bottom half of p. 42R, under the second horizontal line, Einstein first gave a short derivation, similar to the one he gave on pp. 12L–R (see sec. ), of the Minkowski metric in rotating coordinates. As in the derivation of the ϑ -metric, with which he wanted to compare the rotation metric, he suppressed one of the spatial dimensions. Consider a Cartesian coordinate system xμx, y, t in 2+1 -dimensional Minkowski spacetime which is rotating clockwise with angular frequency ω with respect to another Cartesian coordinate system xμ′x′, y′, t′ in which the metric has its usual diagonal form: ds2ημνdx′μdx′νv′2-1dt′2 . In this equation, ημνdiag1, 1, 1 , c1 (in accordance with Einstein’s conventions at this point), and the components of v′ are v′idx′idt′x′⋅i ( i1,2 ). The relation between the velocity with respect to the non-rotating frame and the velocity v with respect to the rotating frame is given by (cf. equation ()): v′v+ω→rx⋅+ωy,y⋅-ωx , where ω→r is the cross-product of the vectors ω→0,0,ω ( ω because the rotation is clockwise) and rx,y,0 in three dimensions. Taking the square of equation (), one finds that v′2=x⋅+ωy2+y⋅-ωx2 =x⋅2+y⋅2+2ωyx⋅-2ωxy⋅+ω2r2. The expressions following the equality signs in equation () are actually the only ones Einstein explicitly wrote down before giving the matrix for the rotation metric. As we mentioned above, he had gone through essentially the same derivation on pp. 12L–R. Inserting equation () along with dt′dt into equation (), one finds that the Minkowski line element in rotating coordinates is given by ds2dx2+dy2+2ωydxdt-2ωxdydt-1-ω2r2dt2 , from which one can read off the components of the metric, gμν10ωy01ωxωyωx1+ω2r2 . Einstein read off the components of the metric from the last line of equation (), which is probably why he omitted the term 1 in g44 . As on pp. 12L–R, he also ended up with additional factors of 2 in g14 and g24 .Einstein made the same mistake on pp. 12L–R (see footnote ) and in the Einstein-Besso manuscript (CPAE 4, Doc. 14, pp. [41–42]). When the constant β in equation () is set equal to the angular frequency ω (in which case αω2 according to equation ()), the contravariant form of the ϑ -metric does indeed turn into the covariant form of the rotation metric in equation (), as was noted in the top-right corner of p. 42R.The two equations are still not completely identical because Einstein wrote the flat Minkowski metric as diag1,1,1 in equation () and as diag1,1,1 in equation (). As he had done earlier when he was faced with the conflict between the modified weak-field equations of p. 21R and the static metric (see the discussion in sec. ), Einstein turned to particle dynamics to see whether his mathematical results could be given a physically sensible interpretation. In this case, he apparently wanted to check whether the components of the ϑ -metric and their derivatives can be given the same sort of physical interpretation in terms of inertial forces as the components of the usual rotation metric and their derivatives. Although the relevant calculations—at the bottom of p. 42R and again at the bottom of p. 43La—break off after just a few lines, it seems to be clear that this was their purpose. Einstein inserted the ϑ -metric in its covariant form (see equation ()) into the Lagrangian Hds2dt2gμνx⋅μx⋅ν for a point mass moving in a given metric field. The result is that H-x⋅2-y⋅2+2βyx⋅-2βxy⋅-αy2x⋅2-αx2y⋅2+2αxyx⋅y⋅︸+1.αxy⋅-yx⋅2 Einstein then wrote down the variational principle δ∫Hdt0 , and went through a quick derivation of the x -component of the corresponding Euler-Lagrange equations, writing ∫∂H∂x⋅δx⋅+∂H∂xδxdt0 ,In the notebook the differential dt was omitted. which, upon partial integration, gives -d∂H∂x⋅dt+∂H∂x0 . Einstein now began to compute the first term of this Euler-Lagrange equation for the Lagrangian in equation (). The derivative of H with respect to x⋅ is given by ∂H∂x⋅-x⋅+βy+αxy⋅-yx⋅y-x⋅2-y⋅2+2βyx⋅-2βxy⋅-αxy⋅-yx⋅2+1 . In the notebook, there is a deleted factor of 2 in front of all three terms in the numerator and the denominator was indicated only by a square root sign. On the next line, Einstein took the time derivative of equation () in its uncorrected form ddt∂H∂x⋅-2x⋅⋅+βy⋅+2αyxy⋅⋅-yx⋅⋅+2αy⋅xy⋅-yx⋅ , i.e., with the extra factors of 2 and without the denominator. On the last line of p. 42R, Einstein wrote down the second term of the Euler-Lagrange equation () as well, but did not actually evaluate it for the Lagrangian under consideration. The inclusion of the denominator in equation (), which Einstein had originally omitted, considerably complicates the Euler-Lagrange equations, which may well be why Einstein gave up on this calculation, at least for the time being. At the top of the next page, p. 43La, he tried to resolve the apparent conflict between the ϑ -metric and the rotation metric in a different manner (to be discussed in sec. ). This new approach, however, turned out not to be viable, and at the bottom of p. 43La Einstein briefly returned to the approach he had abandoned at the bottom of p. 42R. Under the heading “dynamics in a symmetric static rotational field” (“Dynamik im symmetrischen statischen Rotationsfeld”), Einstein once again considered the motion of a point mass in the ϑ -metric. The reason he explicitly referred to this case as “symmetric” and “static” may have been that the ϑ -metric is a time-independent solution of an equation in which the fully symmetrized derivative of the metric is set equal to zero (see equation ()). Using the relation α+β20 , which ensures that the determinant of the ϑ -metric is equal to unity (see equation ()), Einstein was able to write the Lagrangian H of equation () more compactly asIn the notebook, the last term of the first line has x⋅y⋅ instead of xy⋅ . H1-x⋅2-y⋅2-2βxy⋅-yx⋅+β2xy⋅-yx⋅2=1-βxy⋅-yx⋅2-x⋅2-y⋅2. However, he only wrote down one term of the Euler-Lagrange equations, ∂H∂x1H⋅βxy⋅-yx⋅y⋅ , before once again breaking off this calculation. This attempt to give physical meaning to the ϑ -metric thus remained inconclusive. On p. 24L, Einstein made another attempt along these lines (see sec. ), but first we shall discuss the calculation at the top of p. 43La. Reconciling the ϑ-Metric and Rotation (II): Trying to Construct a Contravariant Version of the ϑ-Expression (43La) At the top of p. 43La,This is the last page of the part that starts from the back of the notebook (pp. 32L–43L). It contains entries starting from the top and from the bottom of the page (transcribed as 43La and 43Lb respectively), thus suggesting that this is the place where the two parts of the notebook meet. There are, however, six blank pages between p. 43L and p. 31L, the last page of the part that starts from the front of the notebook. The calculation that starts from the bottom of p. 43L is also not a continuation of p. 31L. In this brief calculation, Einstein derived the equation of motion of a particle in a metric field by integrating the energy-momentum balance between matter and gravitational field over the volume of the particle (for discussion see sec. ). Einstein introduced a variant of the ϑ -expression introduced on p. 23L, replacing covariant components of the metric by contravariant ones.The fragmentary calculation at the top of p. 23R could be related to the investigation of the modified ϑ -expression on this page (cf. the discussion following equation () in sec.  for a different interpretation of this calculation). Since the original ϑ -expression vanishes for the ϑ -metric, i.e., the rotation metric with its co- and contravariant components switched, this modified ϑ -expression vanishes for the rotation metric itself. The modified ϑ -restriction, i.e., the restriction to unimodular transformations under which the modified ϑ -expression transforms as a tensor, thus allows transformations to rotating coordinates in the important special case of Minkowski spacetime. However, Einstein found that, unlike the original ϑ -restriction, the modified ϑ -restriction could not be used to eliminate terms with unwanted second-order derivatives of the metric from the November tensor. At that point, he abandoned the modified ϑ -expression. He may also have come to realize that the expression is mathematically ill-defined (because of the way in which it mixes co- and contravariant components). At the top of the page Einstein wrote down the expression gik∂ik∂xl+∂kl∂xi+∂li∂xk . The expression in parentheses is obtained (down to the labeling of the indices) by substituting μν for gμν in definition () of the ϑ -expression. Farther down on the page, Einstein introduced the quantity tλακ which appears to be defined as ( 12 times) this expression tλακ12∂λα∂xκ+∂ακ∂xλ+∂κλ∂xα . Given the convention in the notebook of using Greek and Latin characters to denote covariant and contravariant quantities, respectively,Cf., e.g., on p. 24L where Tik and Θik represent the stress-energy tensor for matter in its contravariant and covariant form, respectively. Note that this convention is just the opposite of the one adopted in (Einstein and Grossmann 1913), where all contravariant quantities are indicated by Greek and all covariant ones by Latin characters. the notation indicates that tλακ is a contravariant version of ϑikl . Solutions of the equation tλακ0 are obtained simply by interchanging co- and contravariant components in solutions of ϑikl0 . Since the ϑ -metric is a solution of ϑikl0 (see equations () and ()), the rotating metric is a solution of tλακ0 . The quantity tλακ , however, is mathematically ill-defined. Covariant indices in one term occur as contravariant ones in another, and there are summations over pairs of covariant indices. Next to expression (), Einstein nonetheless wrote “Tensor,” presumably to indicate that he wanted to consider a variant of the ϑ -restriction, i.e., a restriction to (unimodular) transformations under which tλακ transforms as a tensor.Unfortunately, this interpretation does not explain why, as Einstein noted on the next line, (minus) the ill-defined second term of (), ∑kl∂gik∂xi , should be a vector. Maybe Einstein meant (minus) the unproblematic first term, ∑ik∂gik∂xl , which is equal to ∂logg∂xl and hence a vector under all unimodular transformations. At the beginning of the calculation on p. 23L, the ϑ -expression was used to rewrite the Christoffel symbols (see equations ()–()). If tλακ , the “contravariant” version of the ϑ -expression, is going to be used in a similar manner, one first needs to lower its indices. Einstein indeed wrote down the expression giαgkβ∂ik∂xl+∂kl∂xi+∂li∂xk , which again is mathematically ill-defined, and wrote next to it that this is a covariant or “plane tensor” (“Ebenentensor”The term “Ebenentensor” also appears on pp. 17L, 17R, 19L, and 24R. The term “Ebenenvektor” appears on p. 13L. Cf. the discussion of this terminology following equation () above. ). Einstein rewrote this expression in such a way that, just as the Christoffel symbols, it contains only derivatives of the covariant components of the metric -∂gαβ∂xl-κlgiα∂gκβ∂xi-ligκβ∂giα∂xκ . He then checked whether this modified ϑ -expression can be used to eliminate terms with unwanted second-order derivatives of the metric from the November tensor. He began by writing down the term in the November tensor containing second-order derivativesThis is the first and only occurrence of the Christoffel symbol in the part that starts from the end of the notebook. In the part that starts from the beginning of the notebook, the Christoffel symbols are first introduced on p. 14L. (cf. equation ())The notebook originally had xα instead of xl . ∂lαi κα∂xl . Using the definition of the Christoffel symbol, he rewrote this term asIn the notebook, the Christoffel symbol is indicated only by square brackets and is multiplied by ∂λα∂xα instead of ∂lα∂xl . The first two occurrences of the index l in the second term were originally λ ’s. ∂lα∂xli κα+12lα∂2giα∂xl∂xκ+∂2gκα∂xl∂xi-∂2giκ∂xl∂xα . Einstein deleted this expression and made a fresh start on the next line, writing the expression in parentheses in expression () asEinstein at this point changed his notation for one of the summation indices in expression () from l to λ . 12λα∂giα∂xκ+∂gκα∂xi-∂giκ∂xα . If this expression is inserted into expression (), the first two terms give rise to unwanted second-order derivatives of the metric. Following the strategy on p. 23L, one could try to eliminate these terms by absorbing them into a quantity involving tλακ , transforming as a tensor under transformations under which tλακ itself transforms as a tensor. One can then subtract this quantity from the November tensor without losing invariance under this restricted class of transformations. This appears to be the rationale behind the last two lines of this calculation. First, Einstein underlined the first two terms in expression () and rewrote them as 12giα∂λα∂xκ+gκα∂λα∂xi . Underneath this expression he wroteIn the notebook, the quantity tλακ was actually written as tλακ c . It is unclear what the letter ‘c’ stands for. tλακ-∂κα∂xλ-∂λκ∂xα , On the assumption that Einstein did indeed define tλακ as in equation (), this last expression is equal to ∂λα∂xκ , which is part of the first term in expression (). At this point, it apparently became clear to Einstein that the modified ϑ -expression could not be used to eliminate unwanted second-order derivative terms from the November tensor. Einstein may also have come to realize that tλακ is mathematically ill-defined. In any case, Einstein abandoned this attempt to modify the ϑ -restriction to ensure that it would include transformations to rotating coordinates in Minkowski spacetime. As was already discussed at the end of sec. , he briefly returned to the approach he had tried on p. 42R before giving up on that approach as well. Reconciling the ϑ-Metric and Rotation (III): Identifying the Centrifugal Force in the Energy-Momentum Balance (24L) On p. 24L, Einstein made yet another attempt to give physical meaning to the components of the ϑ -metric. He checked whether the force on a particle at rest in the gravitational field described by the ϑ -metric can be interpreted as the centrifugal force on a particle at rest in a rotating frame of reference. This strategy is similar to the one behind the aborted calculations at the bottom of pp. 42R and 43La (see sec. ). The new element is that in order to find the equations of motion and the expression for the force on the particle Einstein now substituted the ϑ -metric into the energy-momentum balance between matter and gravitational field rather than into the Lagrangian for a point particle in a metric field. The energy-momentum balance can be written as the vanishing of the covariant divergence of either the contravariant or the covariant stress-energy tensor. If the Minkowski metric in rotating coordinates is inserted into the equation which sets the covariant divergence of the contravariant stress-energy tensor equal to zero, one readily establishes that a term, which can be interpreted as the force on a particle at rest in this coordinate system, is equal to the usual centrifugal force (plus correction terms of higher order in the angular frequency of the rotating coordinate system). What Einstein tried to do on p. 24L was to check whether one can establish an analogous result if the ϑ -metric, i.e., the rotation metric with its co- and contravariant components switched, is inserted into the equation which sets the covariant divergence of the covariant stress-energy tensor equal to zero. The result of Einstein’s calculation on p. 24L suggests that one can. The calculation, however, is in error. If the errors are corrected, one sees that one cannot. At the top of p. 24L, Einstein wrote down an equation which expresses the vanishing of the contravariant stress-energy tensor, denoted by Tμν ,We want to remind the reader that the convention used here to distinguish covariant and contravariant quantities is the opposite of the one adopted in (Einstein and Grossmann 1913), where all contravariant quantities are indicated by Greek and all covariant ones by Latin characters. in the special case that the determinant of the metric is equal to unity, ∑∂gmνTνn∂xn-12∂gμν∂xmTμν0 . This equation is obtained (down to the numbering of the indices) from equation () derived on p. 5R by setting G1 . On p. 5R, as in several subsequent publications,(Einstein and Grossmann 1913, sec. 4), (Einstein 1913, sec. 5). this equation was derived as a generalization of the equations of motion that follow from the variational principle δ∫dsdtdt0 for one particle to a cloud of pressureless dust described by the stress-energy tensor Tμνρ0dxμdτdxνdτ , which is explicitly given at the top of p. 24L as well. On p. 43Lb, Einstein went through this derivation in reverse, showing that the equation of motion for one particle can be obtained by integrating equation (), with Tμν given by equation (), over the volume of the particle. This calculation will be discussed in the next subsection (sec. ). Einstein could thus look upon equation () as giving the equation of motion of a particle in a metric field with a determinant equal to unity. In particular, he could look upon the second term, 12∂gμν∂xmTμν , as giving the (density of the) force experienced by a particle in a metric field.The general interpretation of the second term in equation () is that it is “an expression for the effects which are transferred from the gravitational field to the material process [as described by the stress-energy tensor]” (“ein Ausdruck für die Wirkungen, welche vom Schwerefelde auf den materiellen Vorgang übertragen werden;” Einstein and Grossmann 1913, 11). See also the discussion of p. 19R in sec.  (especially expression () and equation ()). Consider a particle at rest with respect to a rotating coordinate system in Minkowski spacetime. In that case, the stress-energy tensor in equation () reduces to Tμνdiag0,0,0,ρ0g44 . For counterclockwise rotation around the z -axis with angular frequency ω , the metric is given by gμν1 0 0 ωy0 1 0 ωx0 0 1 0ωy ωx 0 1-ω2x2+y2 , Inserting equations () and () into the m1 component of expression () for the force on the particle, one arrives at 12g44,1T44ρ0g44ω2xρ0ω2x+Oω4 , which, when terms of order ω4 are neglected, is the x -component of the centrifugal force in ordinary Newtonian theory. On p. 24L, Einstein performed a variant of this calculation starting from the vanishing of the covariant divergence of the covariant rather than the contravariant stress-energy tensor. Einstein wrote the contravariant Tμν in terms of the covariant Θμν , TμνμανβΘαβ , and substituted this expression into to equation (), which then turns into ∑∂nαΘαm∂xn+12∂αβ∂xmΘαβ0 . For the second term Einstein used the result of an auxiliary calculation which appears next to equation (), ∂gμν∂xmμανβΘαβgμν∂μα∂xmνβΘαβ ∂αβ∂xmΘαβ . He then wrote down the covariant version of the stress-energy tensor for pressureless dust in equation () Θαβρ0gμαgνβdxμdτdxνdτ . Under the heading “Force acting on material point at rest n1 ” (“Kraft auf ruhenden materiellen Punkt n1 ”) he then wroteIn the notebook only the 1β - and the 44 -components of Θαβ were written down. ∂αβ∂x100000β0β2αx Θαβ000000000000000ρ0 dxdτ0001g44 The calculation for the ϑ -metric that Einstein indicated in this manner is completely analogous to the calculation for the rotation metric in equations ()–(). Einstein interpreted the second term in equation (), 12∂αβ∂xmΘαβ , with Θμν given by equation (), as the (density of the) force experienced by a particle in a metric field. Whereas expression () for the force density was a contraction of the covariant metric and the contravariant stress-energy tensor, expression () is a contraction of the contravariant metric and the covariant stress-energy tensor. Since the contravariant components of the ϑ -metric are equal to the covariant components of the rotation metric, expression () would give the same result for a particle at rest in the field of the ϑ -metric as expression () for a particle at rest in a rotating frame in Minkowski spacetime, if only the components of Θμν in the former case were equal to those of Tμν in the latter. Contrary to what is suggested by the expression for Θμν in the expressions (), however, this last condition does not hold. For a particle at rest with respect to a given coordinate system, as Einstein explicitly wrote down (see the expressions ()), dxμdτ0,0,0,1g44 , and equation () reduces to Θαβρ0g4αg4βg44 . The covariant components of the ϑ -metric are given by (cf. equation ())In the notebook, both on p. 42R and on p. 24L, the ϑ -metric is given for the 2+1 -dimensional case. The generalization to the 3+1 -dimensional case is trivial (cf. footnote 304) gμν1+αx22αx1x20βx2αx1x21+αx120βx10010βx2βx101 , the contravariant ones by (cf. equation ()) μν100βx2010βx10010βx2βx101+αx12+x22 , where αβ2 (see equation ()). Taking the derivative of equation () with respect to x1 , one arrives at ∂αβ∂x10000000β00000β02αx1 . The notebook has β instead of β (see the expressions ()), but this is only a minor discrepancy. Inserting equation () into equation (), however, one immediately notices that Θμν will be considerably more complicated in this case than diag0,0,0,ρ0 given in the expressions () in the notebook. Inserting the simple expression Θμνdiag0,0,0,ρ0 and equation () into the m1 componentIn the notebook, the free index in the second term in equation () was originally n instead of m . This explain why the header above the expressions () in the notebook has “ n1 ” instead of m1 . of expression (), one arrives at 12∂αβ∂x1Θαβ12∂44∂x1Θ44ρ0αx1 . which is equal to the centrifugal force in Newtonian theory if the identification αω2 is made. This would mean that the ϑ -metric can be interpreted in terms of centrifugal forces just as the rotation metric (cf. equations ()–()). However, when Einstein’s expression for Θαβ is corrected, there will be additional terms which spoil this physical interpretation of the ϑ -metric. It is not entirely clear what conclusion Einstein drew from his calculation on p. 24L. It would seem that initially he felt that he could recover the correct expression for the centrifugal force with the ϑ -metric as well as with the rotation metric. On the next page, however, he made a fresh start, abandoning the idea of the ϑ -restriction that had led him to consider the ϑ -metric in the first place. This suggests that eventually Einstein came to realize that the calculation on p. 24L was in error and that his third attempt at reconciling the ϑ -metric with his heuristic requirements concerning rotation, like the first two (on pp. 42R–43La), had failed. The November tensor, like the Ricci tensor from which it was extracted under the restriction to unimodular coordinate transformations, thus failed to yield acceptable candidates for the left-hand side of the field equations. Einstein had found two different coordinate restrictions, the Hertz restriction and the ϑ -restriction, with which terms containing unwanted second-order derivatives of the metric can be eliminated from the November tensor. He had also discovered, however, that both coordinate restrictions rule out transformations to accelerated frames of reference in the important special case of Minkowski spacetime. Relating Attempts (I) and (III) to Reconcile the ϑ-Metric and Rotation: from the Energy-Momentum Balance to the Geodesic Equation (43Lb) In a brief calculation starting on p. 43Lb, Einstein showed that the vanishing of the covariant divergence of the stress-energy tensor for pressureless dust implies the equations of motion for a point particle in a metric field. On p. 5R, Einstein had proved the converse of this implication. The derivation on p. 43Lb essentially goes through this earlier derivation in reverse.The same pattern can be found in (Einstein and Grossmann 1913, sec. 4). First, Einstein derives the energy-momentum balance between matter and gravitational field from the equations of motion, closely following the calculation on p 5R; then he mentions that the latter can be recovered from the former by “integration over the filament of the flow” (“Integration über Stromfaden”). The calculation may be connected to Einstein’s attempts to interpret the components of the ϑ -metric (see equation ()) in terms of the inertial forces of rotation. On p. 42R and p. 43La he tried to do so starting from the equations of motion (see sec. ), whereas on p. 24R he started from the energy-momentum balance between matter and gravitational field (see sec. ). The point of Einstein’s calculation on p. 43Lb may have been to reassure himself that these two approaches are equivalent. The calculation starts from equation () of p. 5R for the special case that G1 : ∂∂xngmνTνn-12∑∂∂xmgμνTμν0 (cf. equation () on p. 24L). Next to this equation, Einstein drew a line and wrote ∫dxdydz to indicate that he wanted to integrate this equation over three-dimensional space. When the first term of the expression on the left-hand side of equation () is integrated over all of space, the first three terms of the summation over n vanish on account of Gauss’ theorem (and suitable assumptions about Tμν ), while the time derivative in the n4 term commutes with taking the integral. Using xt and Tνt for x4 and Tν4 , respectively, Einstein could thus write the integral as: ∂∫gμνTνtdxdydz∂xt-12∫∂∂xmgμνTμνdxdydz . He then substituted Tμνρdxμdsdxνds , the stress-energy tensor for pressureless dust, and the relation dxdydzVGdsdt between the volume dxdydz in the coordinates used and the rest volume Vdξdηdζ of the particles described by Tμν . This relation follows from the relation G⋅dxdydzdtdξdηdζds written to the far right of equations ()–() in the notebook. Using furthermore that the density ρ is non-vanishing only inside the filament representing the flow of matter, Einstein arrived at ddtgμνρdxνdsddst⋅ddtsVG-12∫∂gμν∂xmρdxμdsddsxν⋅VGdsdt , as is indicated in the line following expression () in the notebook.In the second term, Einstein wrote m instead of xm . The factors G should be set equal to 1: if Einstein had not set G1 in going from equation () to equation (), the factors G would simply cancel at this point. Dividing by ρV , Einstein rewrote expression () as: d1Ggmνdxνdsdt-121G∂gμν∂xmdxμdtdxνdtdtds . Setting x⋅μdxμdt and wdsdt ,The same notation is used in the Einstein-Besso manuscript on the perihelion advance of Mercury (CPAE 4, Doc. 14, e.g., [p. 15], [eq. 105]). he rewrote this expression as d1Ggmνx⋅νwdt-121G∂∂xmgμνx⋅μx⋅νw , and, finally, as d1G∂w∂x⋅mdt-1G∂w∂xm . Setting this expression (with G1 ) equal to zero, one recovers the Euler-Lagrange equations for the Lagrangian Hwdsdt (see p. 5R and equation ()). This shows that the equations of motion of a test particle in a metric field can indeed be derived from the energy-momentum balance between matter and gravitational field. Transition to the Entwurf Strategy (24R–25R) On pp. 19L–23L, Einstein had extracted various candidates for the left-hand side of the field equations from the Ricci tensor and the November tensor by imposing suitable coordinate restrictions. These coordinate restrictions should (a) make it possible to eliminate all unwanted terms with second-order derivatives of the metric, (b) be compatible with energy-momentum conservation at least in linear approximation, and (c) minimally allow transformations to accelerated frames of reference in Minkowski spacetime. In this way, Einstein’s heuristic requirements (the correspondence principle, the conservation principle, the equivalence principle, and the relativity principle) would all at least to some extent be satisfied. All coordinate restrictions he had considered, however, failed on one count or another. On p. 24R, Einstein proposed yet another candidate for the left-hand side of the field equations. However, while the various candidates proposed on pp. 19L–23L had been products of Einstein’s mathematical strategy, the new candidate was a product of the physical strategy.See sec.  for a discussion of these two strategies. Instead of extracting candidate field equations from the Ricci tensor with various coordinate restrictions, Einstein on p. 24R generated field equations starting from the requirement of energy-momentum conservation. That does not mean that Einstein had now given up on the mathematical strategy altogether. The weak-field equations he started from and the restriction to unimodular transformations in all calculations on p. 24R strongly suggest that Einstein hoped to connect the field equations found through considerations of energy-momentum conservation to the November tensor. On p. 25L Einstein explicitly tried find a coordinate restriction with which he could recover field equations found along the lines of the argument on p. 24R from the November tensor. At the top of p. 24R, Einstein wrote down an expression that can be identified as the divergence of a quantity representing gravitational stress-energy density. Although the derivation of this expression is not in the notebook, there are enough clues to give a plausible reconstruction of how Einstein arrived at it. As he had done for other linearized field equations on pp. 19R, 20L, and 21L, Einstein used the linearized version of field equations extracted from the November tensor to rewrite, in linear approximation, the gravitational force density as the divergence of the gravitational stress-energy pseudo-tensor. The expression at the top of p. 24R is the result of this calculation. The gravitational stress-energy pseudo-tensor that one finds in this linear approximation looks like a plausible candidate for the exact expression for this quantity. It is understandable therefore that Einstein proceeded to look for terms that would need to be added to the weak-field equations to make sure that the expression at the top of p. 24R becomes exactly equal to the gravitational force density (see sec. ). Essentially the same method would give him the Entwurf field equations on pp. 26L–R and in (Einstein and Grossmann 1913).Einstein had in effect used this method before to derive the final version of the field equations for his theory for static gravitational fields in 1912. In that case, he had also added terms to the original field equations of the theory to make sure that they be compatible with energy-momentum conservation (see Einstein 1912b, 455–456). Field equations constructed in this manner automatically satisfy both the correspondence principle and the conservation principle. The problem is to determine whether they are covariant under a wide enough class of coordinate transformations to meet the requirements of the relativity and equivalence principles as well. In the case of field equations extracted from the Ricci tensor or the November tensor with the help of coordinate restrictions, this question can, at least in principle, be settled by examining the transformation properties of expressions much simpler than the left-hand side of the field equations, such as the Hertz expression or the ϑ -expression. In the case of the field equations introduced on p. 24R, the construction of the equations is of no help in determining their covariance properties. The construction only guarantees that the equations will be covariant under unimodular linear transformations (unimodular because the determinant of the metric is set equal to unity in all calculations on p. 24R). At the bottom of p. 24R, Einstein checked whether the rotation metric is a solution of the new field equations. According to the first entry on p. 24R, the expression from which they were derived vanishes for the rotation metric, a necessary condition for the rotation metric to be a solution of the vacuum field equations. This was an encouraging result—itself in error, it turns out—but Einstein discovered that the rotation metric is in fact not a solution of his new field equations. He also discovered, however, that he had erroneously cancelled two terms in his derivation of these equations. There would consequently be additional terms quadratic in first-order derivatives of the metric in the field equations. This in turn opened up the possibility that, once the equations were corrected, the rotation metric would be a solution after all. Rather than making these corrections, Einstein (on pp. 25L–R) tried to find a coordinate restriction with the help of which (the corrected version of) these new field equations could be recovered from the November tensor (see sec. ). In other words, he examined whether the field equations found following the physical strategy could also be found following the mathematical strategy. At the bottom of p. 25L he indicated which terms in the November tensor would have to be eliminated by imposing a coordinate restriction and which ones preserved. At the top of p. 25R, Einstein considered an ingenious modification of the ϑ -restriction, which we shall call the ϑ^ -restriction (see sec. ), although it is not clear whether the purpose of the ϑ^ -restriction was to extract (the corrected version of) the candidate field equations of p 24R from the November tensor. In any case, Einstein failed to connect the November tensor to the physically motivated field equations of p. 24R, a connection that would have helped clarify the covariance properties of the latter. It is at this point in the notebook that Einstein abandoned the mathematical strategy completely. On the remainder of p. 25R, Einstein started tinkering with the expression found on p. 24R for the left-hand side of field equations to make sure that it vanishes for the rotation metric (see sec. ). Undeterred by the fact that the resulting expression is mathematically ill-defined, he checked whether this modified expression still allowed him to write the gravitational force density as the divergence of gravitational stress-energy density. He found that it did not, at least not exactly (see sec. ). It may have been because the resulting conflict with energy-momentum conservation already ruled out this ill-defined expression as a candidate for the left-hand side of the field equations, but Einstein made no attempt to connect the modified expression with the November tensor. On p. 25L and at the top of p. 25R, Einstein had used the mathematical strategy to complement the physical strategy that he had used on p. 24R. On the remainder of p. 25R, however, Einstein apparently decided to go exclusively with the physical strategy, which on the very next page gave him the Entwurf equations. Constructing Field Equations from Energy-Momentum Conservation and Checking Them for Rotation (24R) At the top of p. 24R, Einstein wrote down the expressionIn the notebook, the partial derivative with respect to xi in the first term in equation () is written as an ordinary derivative. ∂∂xiiε∂gαβ∂xε∂αβ∂xσ-12∂iε∂gαβ∂xε∂αβ∂xi∂xσ , and noted that it vanishes for a metric field that he wrote down as g10ωy01ωxyωx1-ω2x2+y2 -1+ω2y2ωxyωyωxy-1+ω2x2ωxωyωx1. He explicitly wrote: “The expression [] vanishes for the system []” (“Der Ausdruck … verschwindet für das System …”). The metric in equation () is easily recognized, despite some discrepancies, which are probably due to slips on Einstein’s part, as the rotation metric with one spatial dimension suppressed. It is harder to see what expression () represents. This seems to be one of the few places in the notebook where the calculations are not self-contained. Fortunately, a convincing case can be made for the following reconstruction of how Einstein arrived at expression (). In the field equations extracted from the November tensor with the help of the ϑ -restriction, the term with second-order derivatives of the metric is written as (see, e.g., expression () [p. 23L] and expressions () and () [p. 43La])In the equations extracted from the November tensor with the Hertz restriction (see equation () [p. 22R]) and in those extracted from the Ricci tensor with the harmonic restriction (see equation () [p. 19L]), the term with second-order derivatives is written as αβ∂2gil∂xα∂xβ . The two expressions only differ, of course, by a term quadratic in first-order derivatives of the metric. ∂∂xiiε∂gαβ∂xε , where the labeling of the indices is chosen with a view to expression (), the derivation of which we want to reconstruct. In linear approximation this is the only non-negligible term on the left-hand side of these field equations. The linearized version of these equation thus becomes ∂∂xiiε∂gαβ∂xεκΘαβ , where Θαβ is the covariant stress-energy tensor in the notation that Einstein used in the notebook.See, e.g., equation (). In our reconstruction of the derivation of equation (), we follow the notation of the notebook. Energy-momentum conservation required that these linearized field equations can be used to rewrite the gravitational force density as the divergence of a quantity representing gravitational stress-energy density.Cf. the discussion following equation () in sec. . Einstein had checked this for the linearized field equations considered on p. 19R (see equation ()) and on p. 20L (see equation ()). A natural explanation of how he arrived at the expression at the top of p. 24R is that he did the same for the linearized field equations (). The expression for the force density can be read off from the energy-momentum balance between matter and gravitational field in either its covariant or its contravariant form. On p. 24L, the page immediately preceding the one under consideration here, Einstein had actually considered both possibilities in his attempt to find a physical interpretation for the ϑ -metric (see sec. ). These two alternative expressions for the force density are 12∂gαβ∂xσTαβ , where Tαβ is the contravariant stress-energy tensor, and 12∂αβ∂xσΘαβ (cf. expressions () and (), respectively). Note that these expressions are correct only in linear approximation. In the exact version there will be another factor g (cf. expression ()), which in linear approximation can be set equal to unity. On pp. 19R, 20L, and 21L, Einstein had used expression () for the force density but had substituted the left-hand side of the covariant linearized field equations for the contravariant stress-energy tensor (cf. footnote 241). Using expression () instead and eliminating the stress-energy tensor with the help of the linearized field equations (), one arrives at 12κ∂αβ∂xσ∂∂xiiε∂gαβ∂xε . Ignoring the factor 2κ (as Einstein did in the corresponding calculations on pp. 19R, 20L, and 21L), one can rewrite this expression as ∂∂xiiε∂gαβ∂xε∂αβ∂xσ-iε∂gαβ∂xε∂2αβ∂xi∂xσ . The first term is identical to the first term in expression () written at the top of p. 24R. The second term can be rewritten in the form of the second term in expression () if terms of third power in derivatives of the metric are neglected. Such third-power terms would correspond to quadratic terms in the field equations. Since this whole calculation is based on the linearized field equations, such terms can indeed be neglected. One can thus write the second term in expression () as iε∂gαβ∂xε∂2αβ∂xi∂xσ∂∂xσiε∂gαβ∂xε∂αβ∂xi-iε∂2gαβ∂xσ∂xε∂αβ∂xi . Once again neglecting terms of third power in derivatives of the metric, one easily establishes that the last term on the right-hand side is equal and opposite to the term on the left-hand sideFarther down on p. 24R, Einstein initially cancelled these two terms with one another in a calculation that is supposed to be exact. He later rescinded these calculations. We prove the approximate equality of these two terms in modern notation. Using that 0gμρgρν,αgμρ,αgρν+gμρgρν,α , one can write gαβ,εσg ,iαβgαμgβνg ,εμν,σgακgβλgκλ,i . If terms of third power in derivatives of the metric are neglected, this reduces to gαβ,εσg ,iαβg ,εσαβgαβ,i . Contracting both sides with giε and switching i and ε on the right-hand side, one arrives at giεgαβ,εσg ,iαβgiεg ,iσαβgαβ,ε , which is what we set out to prove. iε∂2gαβ∂xσ∂xε∂αβ∂xiiε∂gαβ∂xε∂2αβ∂xi∂xσ . equation () can thus be rewritten as iε∂gαβ∂xε∂2αβ∂xi∂xσ12∂∂xσiε∂gαβ∂xε∂αβ∂xi . Substituting this expression for the second term in expression (), one recovers expression () written at the top of p. 24R. The reconstruction given here of how Einstein arrived at this expression also seems to fit with the deleted phrase on the very first line of p. 24R: “divergence of a plane tensor Θik ” (“Divergenz eines Ebenentensors Θik ”).The term “plane tensor” (“Ebenentensor”) is used in the notebook for a covariant tensor (cf. p. 17L and the discussion in sec. ). The expression in equation () is, in fact, the divergence of ( 2κ times) the covariant gravitational stress-energy pseudo-tensor of the Entwurf theory.Relabeling indices to make it easier to compare the expression in the notebook with the corresponding expressions in the Entwurf paper, we can rewrite expression () as: ∂∂xνμν∂gτρ∂xμ∂∂xστρ-12gμσαβ∂gτρ∂xα∂∂xβτρ . The quantity in square brackets is equal to 2κtμσ as defined in (Einstein and Grossmann 1913, p. 16, equation (14)). Substituting this definition into the expression above and dividing by 2 , one arrives at the left-hand side of equation (12b) of the same paper for the special case that g1 . Equation (12b) is obtained from equation (12), the left-hand side of which reduces to expression () if one sets g1 . The notation Θik suggests that Einstein was referring to the stress-energy tensor of matter rather than to the stress-energy pseudo-tensor of the gravitational field, but energy-momentum conservation requires the divergence of the latter to be equal and opposite to the divergence of the former. The deleted phrase at the top of p. 24R thus seems to provide additional support for our reconstruction of the derivation of expression (). If there is no matter, energy-momentum conservation requires that the divergence of the stress-energy pseudo-tensor of the gravitational field vanishes. It follows that expression () should vanish for vacuum solutions of the field equations, at least in linear approximation. As becomes clear on the bottom half of p. 24R, Einstein tried to find the exact field equations corresponding to the linearized field equations () on the assumption that expression () is exactly equal to the gravitational force density. In that case, the expression should vanish exactly for vacuum solutions of the field equations. We can thus understand why it was important for Einstein to check whether the expression vanish exactly, for instance, for the rotation metric, which should be a vacuum solution of any acceptable candidate field equations according to Einstein’s heuristic principles. Contrary to Einstein’s claim at the top of p. 24R, expression () does not vanish for the rotation metric. In the 2+1 -dimensional case, the covariant components of the rotation metric are given by gμν10ωy01ωxωyωx1-ω2x2+y2 , and the contravariant components by μν-1+ω2y2ω2xyωyω2xy-1+ω2x2ωxωyωx1 . These expressions differ slightly, but significantly as it turns out, from the expressions () given in the notebook. Consider the σ1 component of expression () for the metric in equations ()–(). One easily verifies that the first term only contributesThere will be identical contributions for α2,β4 and α4,β2 . 2∂21∂g24∂x1∂24∂x1∂x22∂21∂x2ω22ω4x , whereas the second term only contributesThere will be identical contributions for α1,β4 and α4,β1 . ∂22∂g14∂x2∂14∂x2∂x1 ∂22∂x1ω22ω4x . The σ1 component of expression () thus gives 4ω4x for the metric in equations ()–(). The σ2 component likewise gives 4ω4y .The non-vanishing contributions to the σ2 component of expression () for this metric are 2∂∂x112∂g14∂x2∂14∂x2-∂11∂g24∂x1∂24∂x1∂x22∂12∂x1ω2-∂11∂x2ω24ω4y . Only the σ4 component vanishes, since the metric is time-independent. A comparison between the correct expressions for the components of the metric in equations ()–() and Einstein’s faulty expression () suggests that it was because of the sign errors in 12 and 21 that Einstein came to believe that expression () vanishes for the rotation metric. Inserting ω2xy instead of ω2xy for 21 in equation (), the contribution coming from equation () would cancel the contribution coming from equation () and the σ1 component of equation () would vanish for the rotation metric.A similar cancellation would occur in the σ2 component of expression () (see the preceding note). It is interesting to note in this context that the sign error in 12 is the only one of the errors in expression () that Einstein repeated on p. 25R where he once again wrote down the components of the rotation metric. What may have happened is that Einstein read off the contravariant components of the rotation metric from the expression for the covariant components of the ϑ -metric on p. 42R (see equation ()), which would give 12αxy , and then set α equal to ω rather than to ω2 . Whatever happened, Einstein somehow convinced himself that expression () vanishes exactly for the important special case of the rotation metric.In passing we note that Einstein thus missed an early opportunity to discover that the rotation metric is not a solution of the Entwurf equations. As was pointed out in footnote , expression () is the divergence of the gravitational stress-energy pseudo-tensor of the Entwurf theory in the special case that g1 . The rotation metric has a determinant equal to unity. The vanishing of expression () for this metric is therefore a necessary condition for it to be a solution of the Entwurf field equations. It expresses energy-momentum conservation in this case. For further discussion of Einstein’s struggles with rotation, see (Janssen 1999; 2005, 68–71), and “What Did Einstein Know …” sec. 3 (in this volume). It now made sense for him to look upon equation (), which he presumably found as the result of a calculation in linear approximation, as the exact expression for the divergence of the gravitational stress-energy density,Note the close structural similarity between the expression for gravitational stress-energy density one reads off from expression () (see the expression in square brackets in footnote ) and the gravitational stress tensor of Einstein’s 1912 static theory, the 11 -component of which Einstein had tried to translate into a gravitational stress-energy (pseudo-)tensor in his metric theory at the bottom of p. 21R (see sec. ). This similarity may have been another factor leading Einstein to adopt expression () as the exact expression for the divergence of the gravitational stress-energy density. at least for metric fields with a determinant equal to unity. It is not entirely clear whether Einstein was aware of this last complication at this point. Perhaps he erroneously continued to use expression () for the gravitational force density in linear approximation, even though the calculation on the bottom half of p. 24R was supposed to be exact. It is also possible that he consciously set g1 to facilitate comparison of the result of his calculations with the November tensor, which is a tensor only under unimodular transformations. Einstein would include the factors g that were omitted on p. 24R in the derivation of the Entwurf equations on p. 26L. Once again it is not clear whether that was because he realized that he should have used the exact expression for the gravitational force density in a calculation that is supposed to be exact or because he was no longer interested in trying to recover his new candidate field equations from the November tensor. If expression () is exactly equal to the divergence of the gravitational stress-energy density and the determinant of the metric is set equal to unity, one can find the exact field equations by going through the derivation of expression () given in equations ()–() in reverse, this time without neglecting any terms. More specifically, one can rewrite expression () in the form ∂αβ∂xσ αβ and take the as yet unknown expression in parentheses as the left-hand side of the field equations. After all, substituting the right-hand side of these candidate field equations—the stress-energy tensor Θαβ —for the as yet unknown expression in parentheses above, one recovers (except for an immaterial factor of 12 ) expression () for the gravitational force density ∂αβ∂xσΘαβ . These new field equations thus automatically and exactly satisfy energy-momentum conservation. They guarantee that the gravitational force density is equal to the divergence of the gravitational stress-energy density, in which case the energy-momentum balance between matter and gravitational field can be written as the vanishing of the divergence of the total stress-energy density. This is exactly Einstein’s line of reasoning on p. 24R. With the comment “the above expression yields” (“obiger Ausdruck liefert”), he began to rewrite expression () in the form (): ∂αβ∂xσdiε∂gαβ∂xεdxi+iε∂gαβ∂xε∂2αβ∂xi∂xσ-iε∂2gαβ∂xε∂xσ∂αβ∂xi-12∂iε∂xσ∂gαβ∂xε∂αβ∂xi In the notebook, Einstein initially (and erroneously) cancelled the second term in this expression against the third.These two terms cancel only if terms of third power in derivatives of the metric are neglected (see equation () and footnote 337). Relabeling the summation indices in the last term, he read off the left-hand side of the field equations from the remaining two terms. “This suggests” (“Hierdurch nahe gelegt”), he wrote, ∂∂xiiε∂gαβ∂xε-12∂giε∂xα∂iε∂xβ . Einstein’s next step was to check whether expression () vanishes for the rotation metric as it should if this metric is to be a vacuum solution of these new field equations. It is not, as Einstein noted on the last two lines of p. 24R: “Tried for the case of a rotating body[.] αβ1 gives ω2 ” (“Probiert am Fall des rot[ierenden] Körpers αβ1 liefert ω2 ”). Inserting either our equations ()–() or Einstein‘s expressions ()The errors in () do not matter in this calculation. for the components of the rotation metric into the 11 -component of expression (), one readily verifies this result. The only non-vanishing contributions come from the second term in expression () ∂g24∂x1∂24∂x1ω2 . Einstein’s new candidate field equations are therefore unacceptable as they stand. However, Einstein also discovered that the two terms that he had cancelled with one another in his derivation of expression () for the left-hand side of his new field equations do in fact not cancel. To the right of expression () in the notebook, Einstein did a short calculation to check whether these two terms are equal to one another. He started with the fairly self-evident relationIn modern notation, the proof runs as follows (cf. footnote 337): gαβ,σg ,iαβgαμgβνg ,σμνgακgβλgκλ,iδμκδνλg ,σμνgκλ,ig ,σκλgκλ,i . ∂gαβ∂xσ∂αβ∂xi∂αβ∂xσ∂gαβ∂xi . This equation expresses that one can simply switch co- and contravariant components of the metric in contractions of two first-order derivatives of this form. One might expect that this is also true for similar contractions of a first-order derivative and a second-order derivative. Differentiating equation () with respect to xε , however, as Einstein did on the next line, ∂2gαβ∂xε∂xσ∂αβ∂xi+∂gαβ∂xσ∂2αβ∂xε∂xi∂2αβ∂xε∂xσ∂gαβ∂xi+∂αβ∂xσ∂2gαβ∂xε∂xi , one sees that this is not the case. At this point, Einstein probably rescinded his cancellations in expression () with the proof readers’ stet mark that he typically used for this purpose. On the face of it, these two terms in expression () would contribute additional terms with unwanted second-order derivatives of the metric to the field equations in equation (). However, as we already noted in our reconstruction of the derivation of equation (), the two terms only differ by an expression of third power in first-order derivatives of the metric (see footnote 337). They thus only give rise to another term quadratic in first-order derivatives of the metric in the field equations. Einstein did not add such a term to expression (), but he probably realized that his new candidate field equations only needed to be corrected by terms quadratic in first-order derivatives rather than by second-order derivative terms. Otherwise it becomes hard to understand his calculations on the next page. On p. 25L he looked for a new coordinate restriction to extract field equations from the November tensor, using as his guide that these equations should at least include the two terms in expression () and no additional terms with second-order derivatives of the metric. All in all, Einstein had made important progress on p. 24R. He had found a method to construct exact field equations out of linearized ones by demanding exact compliance with energy-momentum conservation. The first result of this method, however, was problematic. He found that the rotation metric is not a solution of his new field equations. Einstein also realized, however, that he had made an error along the way. It was thus at least conceivable that the correct application of his new method would yield field equations that do allow the rotation metric as a solution. Trying to Recover the Physically Motivated Field Equations from the November Tensor (25L) On p. 25L, Einstein tried to recover the field equations he had found on p. 24R from the November tensor, not just the two terms explicitly given at the bottom of p. 24R (see expression ()) but also the additional terms coming from the erroneously cancelled terms in expression () for the gravitational force density from which he had read off these new field equations. At the top of p. 25L, Einstein wrote down the November tensor (see equation ()) ∂α βk∂xk-α λμβ μλ . A first indication that his purpose was to recover his new candidate field equations from this object is that he changed the free indices i and l in the original expression for the November tensor to α and β , the free indices in expression () for the left-hand side of the new candidate field equations. Using the definitions of the Christoffel symbols, Einstein rewrote ( 2 times) expression () as ∂kλ∂gαλ∂xβ+∂gβλ∂xα-∂gαβ∂xλ¯∂xk-12μσλτ∂gασ∂xλ-∂gαλ∂xσ∂gβτ∂xμ-∂gβμ∂xτ12μσλτ∂gλσ∂xα∂gμτ∂xβ¯ From the way the product of Christoffel symbols in expression () was rewritten in expression (), it is clear that Einstein once again used the symmetry argument that he had already used several times before in this calculation (on pp. 17R, 19L, and 22 R; see the discussion following expression ()). The two terms that Einstein underlined in expression () are easily recognized as (minus) the two terms of the new candidate field equations in equation (). For the first term this is just a matter of relabeling indices. For the second term it is shown by the calculation immediately following expression () in the notebook. Using that λτ∂gλσ∂xα ∂λτ∂xαgλσ and that μσgλσ∂gμτ∂xβ∂gλτ∂xβ , Einstein rewrote ( 2 times) the last term in expression () as μσλτ∂gλσ∂xα∂gμτ∂xβ ∂λτ∂xα∂gλτ∂xβ ∂λτ∂xβ∂gλτ∂xα . In the notebook, the relations () and () are given underneath the relevant terms on the left-hand side of equation (). Dividing the right-hand side of equation () by 2 , one recovers the second term in expression (). This was a promising start. The next task would be to identify those terms in the November tensor that still need to be added to the field equations based on expression () because of the erroneous cancellation of two terms in expression () for the gravitational force density. These terms would have to come from the second of the three terms in expression (). This is the term that Einstein turned to next. It can be rewritten as 12μσλτ∂∂xλgασ∂∂xμgβτ-∂∂xτgβμ+12μσλτ∂∂xσgαλ∂∂xμgβτ-∂∂xτgβμ . The second term in this expression is equal and opposite to the first, as one easily verifies by relabeling indices ( λσμτ→σλτμ ). Expression () can thus be rewritten as -μσλτ∂gασ∂xλ∂gβτ∂xμ+μσλτ∂gασ∂xλ∂gβμ∂xτ , which is the expression written on the next line in the notebook. To facilitate comparison with the two cancelled terms in expression (), Einstein contracted expression () with ∂αβ∂xi , which appears next to it separated by a vertical line. If the terms in expression () were part of the left-hand side of the field equations, this contraction would give their contribution to the gravitational force density. Einstein rewrote the first term in expression () as ∂μσ∂xλ∂λτ∂xμgασgβτ , which, upon contraction with ∂αβ∂xi gives +∂μσ∂xλ∂λα∂xμ∂gασ∂xi . The calculation was not pursued any further. Einstein drew a horizontal line and schematically rewrote (parts of) the three terms in expression () for the November tensor at the top of the pageIn the notebook, some of the indices in the first term in expression () are barely legible. They have been transcribed on the assumption that Einstein copied this term from the corresponding term in expression (). The first term in equation () is the last of the three terms with second-order derivatives of the metric in equation () and the only one that occurs in the candidate field equations on p. 24R (see equation ()). The problem is to find a coordinate restriction with the help of which the other two second-order derivative terms in the November tensor can be eliminated. Application of this coordinate restriction will also eliminate some of the terms quadratic in first-order derivatives that are schematically indicated in the last two terms in expression (). A suitable coordinate restriction, however, should preserve the last term in its entirety (i.e., the second underlined term in expression ()) as well as the part of the second term corresponding to the erroneously cancelled terms in expression (). In this way the field equations of p. 24R could be extracted from the November tensor. Labeling the three terms in expression () for the November tensor a , b , and c , respectively, and using a prime to distinguish parts that should be preserved from parts that should be eliminated, one can schematically write expression () asMore explicitly, a′∂kλ∂gαβ∂xλ∂xk and c′12∂λτ∂xβ∂gλτ∂xα . a-a′-b+b′+c′ . The signs of a′ and c′ reflect that the signs with which the corresponding expressions occur in the November tensor are the opposite of the signs with which they occur in the field equations of p. 24R. Stated in terms of expression (), the problem is to find a coordinate restriction such that a and b can be eliminated and the left-hand side of the field equations becomes: a′-b′-c′ . Since Einstein did not pursue the calculation on p. 25L any further it is hard to interpret the material at the bottom of the page, but the reasoning leading to equations () and () above at least provides a plausible interpretation of the two lines at the bottom the p. 25L, a-a′a′ -b+b′b′ +c′c′. It remains unclear why next to this expression he wrote down the terms 2μσλτ∂gασ∂xλ∂gμτ∂xβ+∂∂xμgβτ∂∂xαgλσ. These terms can be identified as coming from the expression -12μσλτ∂gλσ∂xα∂gβτ∂xμ-∂gβμ∂xτ+12μσλτ∂gασ∂xλ-∂gαλ∂xσ∂gμτ∂xβ , which Einstein neglected in his expansion of (twice) the product of Christoffel symbols in equation () on the basis of the symmetry argument that he had come to use routinely in this calculation. Both terms vanish identically, since they are contractions of a part that is symmetric and a part that is anti-symmetric in the same index pair. Perhaps Einstein wanted to include these terms because the application of the coordinate restriction he was looking for at this point would preserve some of the terms in equation () while eliminating others. Even on this interpretation, however, it remains unclear what special appeal the two terms in equation () had for Einstein or why they appear with a factor 2 rather than with a factor 12 as in equation (). Despite these uncertainties, it seems clear that the purpose of Einstein’s calculations on p. 25L was to find a way of extracting the physically motivated field equations of p. 24R from the November tensor.This is similar to Einstein’s attempt on pp. 9L–9R to connect the physically motivated core operator to the mathematically well-defined second Beltrami invariant. The material on the bottom half of the page strongly suggests that Einstein hoped to achieve this goal by finding a coordinate restriction that would allow him to eliminate all terms from the November tensor that do not occur in these new field equations. At the top of the next page, p. 25R, Einstein considered a variant of the ϑ -restriction, with the help of which he had eliminated unwanted terms from the November tensor on p. 23L. There is no indication, however, that Einstein specifically introduced or used this restriction to recover the field equations of p. 24R and its correction terms from the November tensor. The J^ -Restriction (25R, 23L) The fragmentary material at the top of p. 25R—various arrays of numbers, the components of the rotation metric, and several equations—can all be understood as part of a variant of the calculations on p. 23L and pp. 42L–R involving what we have called the ϑ -restriction. We shall call this variant the ϑ^ -restriction. This interpretation of the material on p. 25R also explains the alternate signs in many of the expressions on p. 23L. Einstein, it seems, went back to p. 23L and indicated what would need to be changed in his earlier calculation if the ϑ -restriction were replaced by the ϑ^ -restriction. Most of what is actually written down at the top of p. 25R is aimed at determining whether the ϑ^ -restriction allows transformations to rotating frames of reference in Minkowski spacetime. It does not, which is probably why the ϑ^ -restriction was quickly abandoned. The basic idea of the ϑ -restriction (see sec. ) was to absorb terms that need to be eliminated from the November tensor into the so-called ϑ -expression. The November tensor was then split into various parts that separately transform as tensors under ϑ -transformations, i.e., those unimodular transformations under which the ϑ -expression transforms as a tensor. Subtracting the parts containing the terms that need to be eliminated, one arrives at candidate field equations that are invariant under ϑ -transformations. Looking at the first term of the November tensor, which Einstein had just reexamined on p. 25L (cf. expressions ()–() on p. 25L), ∂α βk∂xk∂∂xkkλ12∂gαλ∂xβ+gβλ∂xα-gαβ∂xλ , one sees that the unwanted terms with second-order derivatives of the metric come from the first two terms in the Christoffel symbol α βλ12∂gαλ∂xβ+gβλ∂xα-gαβ∂xλ . These terms can be absorbed into the ϑ -expression, defined as (see expression ()) ϑαβλ12gαλ∂xβ+gβλ∂xα+gαβ∂xλ . With the help of the ϑ -expression, the Christoffel symbols of the first kind can be written as (see equation ()) α βλϑαβλ-∂gαβ∂xλ . The two unwanted terms can also be absorbed into a slightly different expression, which we shall call the ϑ^ -expression, and which we define as ϑ^αβλ12gαλ∂xβ+gβλ∂xα-2gαβ∂xλ . With the help of the ϑ^ -expression, the Christoffel symbols of the first kind can be written as α βλϑ^αβλ+12∂gαβ∂xλ . The ϑ^ -expression was not written down explicitly in the notebook. In the upper right corner of p. 25R, however, we find the schematic array of numbers +1 +1 +112 . Note that this is essentially a matrix with three rows and three columns. The first two columns just have +1 on all three rows. Comparison of the matrix () with equations (), (), and () suggests the following interpretation of these numbers. The three rows may represent the coefficients of the three terms in the ϑ -expression, the Christoffel symbol, and the ϑ^ -expression, respectively. One can use either the ϑ -expression or the ϑ^ -expression in combination with the corresponding coordinate restriction to eliminate unwanted terms from the November tensor. The pluses and minuses written above many of the signs in expressions on p. 23L suggest that Einstein actually went back to p. 23L to see what would need to be changed in his earlier calculations if the ϑ -expression were replaced by the ϑ^ -expression. He began with expression (), the definition of the ϑ -expression (see also equation ()), where he wrote 2 above the last plus sign, . In this way the expression turns into the definition of the ϑ^ -expression (see equation ()). Einstein then wrote plus signs above the minus signs in equations () and () to indicate that he wanted to express the Christoffel symbols of the first and the second kind in terms of this new ϑ^ -expression, , . Comparing equation () to equation (), reading ϑ^ for ϑ , one sees that there is a factor 12 missing in front of the derivative of the metric in equation (). Equation () inherits this error from equation (). This may be why Einstein subsequently deleted the plus signs in these two equations. He went through the rest of the calculation on p. 23L, however, on the assumption that the Christoffel symbols are related to the ϑ^ -expression according to equations ()-(). He added pluses and minuses to equations () and (), and to expression (), making only one minor error. He neglected to change the minus sign in the first term in equation () to a plus sign. Because of these errors, he found that the field equations extracted from the November tensor with the help of the ϑ^ -restriction are exactly the same as those extracted with the help of the ϑ -restriction (see expression ()). Einstein had abandoned these field equations because of problems with the ϑ -restriction (see secs. –). Perhaps these problems could be avoided with the ϑ^ -restriction. In particular, it would be interesting to know whether the ϑ^ -expression, unlike the ϑ -expression, vanishes for the rotation metric. In that case the ϑ^ -restriction, unlike the ϑ -restriction, would at least allow transformations to rotating coordinates in the important special case of Minkowski spacetime (cf. the discussion at the beginning of sec. ). At the top of p. 25R, Einstein once again wrote down the covariant and contravariant components of the rotation metric in the 2+1 -dimensional case (cf. equation ()–()) 10ωy01ωxωyωx1-ω2x2+y2-1+ω2y2ω2xyωyω2xy-1+ω2x2ωxωyωx1. He also wrote down several components of an equation that can more compactly be written as 2ϑ^αβλ2gαβ∂xλ-gαλ∂xβ-gβλ∂xα0 . The choice of index combinations for which equation () was examined on p. 25R confirms that the purpose of Einstein’s calculations at this point was to check whether the metric () satisfies equation (), i.e., whether the ϑ^ -expression vanishes for the rotation metric. Notice that expression () for the components of the metric is still not completely accurate. Comparing expression () to expression () for the rotation metric, one sees that most of the errors on p. 24L have been corrected on p. 25R, but that 12 and 21 still have the wrong signSee the discussion following equation () in sec.  for a possible explanation of why Einstein made this particular error. (as does 44 which was given correctly on p. 24L). To the left of expression (), Einstein schematically wrote down equation () for the index combinations 112 and 121 : 2112-121-121 , 2121-121-112 . Both components give the same equation, ∂g12∂x1-∂g11∂x20 . This equation is obviously satisfied by the metric () as are all components of equation () involving only the indices 1 and 2 . Only components in which at least one index is equal to 4 need to be examined. To the right of equation (), Einstein wrote down equation () for three such components, corresponding to the index combinations 124 , 224 , and 441 .To the left of equation (), Einstein wrote 124 to indicate that he was considering the 124 -component of equation (). 2∂g12∂x4-∂g14∂x2-∂g24∂x10 ∂g22∂x4-∂g24∂x20 , ∂g44∂x1-∂g14∂x40 . The metric () satisfies the first and the second equation, but not the third. Equation (), the last of the five components of equation () given in the notebook, gives ∂∂x1g44-∂∂x4g142ω2x . The ϑ^ -expression therefore does not vanish for the rotation metric. At this point, Einstein seems to have abandoned the ϑ^ -expression and the corresponding ϑ^ -restriction. In the calculation on the next two lines of p. 25R, the metric that we have called the ϑ -metric briefly resurfaces (for discussion, see sec. ). Recall that Einstein had found the ϑ -metric on p. 42R as a solution of the equation ϑαβλ0 , the analogue of equation () for the original ϑ -restriction. The ϑ -metric is obtained by interchanging co- and contravariant components of the rotation metric (see sec. ). Despite considerable effort (see pp. 42R, 43La, 24L and the discussion in secs. and ), Einstein had been unable to find a satisfactory physical interpretation for this metric. A possible explanation for the reoccurrence of the ϑ -metric on p. 25R is that Einstein checked whether the ϑ^ -expression, like the original ϑ -expression, vanishes for the ϑ -metric. Since g441 for the ϑ -metric, equation (), the 441 -component of the equation ϑ^αβλ0 which was not satisfied by the rotation metric, is trivially satisfied by the ϑ -metric. The ϑ -metric, however, does not satisfy equation (), the 121 -component of the equation ϑ^αβλ0 ,Einstein’s sign error in g12 changes the last two steps in equation () to ω2y-2ω2yω2y . ∂∂x1g12-∂∂x2g11-ω2y-2ω2y3ω2y . So the ϑ^ -expression does not vanish for the ϑ -metric either. There would thus be no reason for Einstein to resume his efforts to make sense of this peculiar metric. But if Einstein, as we conjectured, did return to the ϑ -metric in this context, it may have given him the idea for another calculation involving the ϑ -metric, which can be found on the next two lines of p. 25R and which we shall turn to below. Tinkering with the Field Equations to Make Sure That the Rotation Metric Is a Solution (25R) Underneath the material at the top of p. 25R discussed in the preceding subsection, in the two lines above the first horizontal line on p. 25R, Einstein once again wrote down the candidate for the left-hand side of the field equations that he had found on p. 24R (see equation ()) ∂∂xεεi∂gαβ∂xi-12∂λτ∂xα∂gλτ∂xβ . At the bottom of p. 24R he had noted that the 11 -component of this expression does not vanish for the rotation metric (see equation ()). He now inserted the ϑ -metric, i.e., the rotation metric with its co- and contravariant components switched, into the 11 -component of equation ().For a possible connection between this calculation and the calculations at the top of p. 25R, see the discussion at the end of sec. . This gives ∂∂xεε2∂g11∂x2-∂24∂x1∂g24∂x1 . The second term gives ω2 for the ϑ -metric as it does for the rotation metric. The first term vanishes for the rotation metric since g111 , but it does not for the ϑ -metric which has g11-1+ω2y2 . In the case of the ϑ -metric, the first term in equation () can thus be written as ∂∂xεε2 2ω2y∂∂x112 2ω2y+∂∂x222 2ω2y . Einstein initially confused co- and contravariant components of the ϑ -metric and substituted ω2xy for 12 and -1+ω2x2 (or rather -1+ωx2 ) for 22 . He subsequently corrected these errors and substituted 0 and 1 for 12 and 22 , respectively. In the notebook, equation () was thus written as . Even with Einstein’s corrections, this equation still contains some minor errors.On the left-hand side, ω should be ω2 and in the last term on the right-hand side, a closing bracket is missing. Einstein also seems to have dropped the minus sign in the second term on the right-hand side at this point. This can be inferred from the fact that he clearly was under the impression that the 11 -component of expression () would vanish for the ϑ -metric, if only the coefficient of the second term of the expression were changed from 12 to 1 .This in turn can be inferred from his comment “ α1 β1 correct” (“ α1 β1 stimmt”) farther down on p. 25R (see the paragraph following expression ()). Inserting the ϑ -metric into this modified version of expression () and substituting 2ω2 instead of 2ω2 for its first term, one arrives at ∂∂xεεi∂g11∂xi-∂λτ∂x1∂gλτ∂x12ω2-2ω20 . Einstein’s sign error can easily be corrected. Rather than changing the coefficient 12 of the second term in expression () to 1 , Einstein should have changed it to 1 . If an expression vanishes for the ϑ -metric, it is only a matter of interchanging co- and contravariant components of the metric to obtain an expression that vanishes for the rotation metric.Einstein had used this same insight to construct a modified ϑ -expression on p. 43La (see sec. ) On the next line of p. 25R, after drawing a horizontal line, Einstein therefore wrote down the expression ∂∂xεgεi∂αβ∂xi-∂λτ∂xα∂gλτ∂xβ , which is obtained by interchanging co- and contravariant components of the metric in expression () and changing the factor 12 in the second term to 1 . Einstein assumed that the 11 -component of this new expression vanishes for the rotation metric. This expression thus seemed to be a promising new candidate for the left-hand side of the field equations.On the face of it, it may seem somewhat dubious to simply change the coefficient of the second term in expression () to make the expression vanish for the ϑ -metric. Einstein knew, however, that there should be additional terms quadratic in first-order derivatives of the metric in expression () coming from the terms that he had neglected in its derivation. Einstein’s hope at this point may have been that these correction terms would only result in a change of the coefficient of the second-term in expression () as it stood. Mathematically, however, it is ill-defined: α and β appear as contravariant indices in the first term and as covariant ones in the second, and the summations over i and ε are summations over pairs of covariant indices. Einstein had concocted expression () to meet the requirement that the rotation metric be a vacuum solution of the field equations and had done so at the expense of basic mathematical requirements. Einstein still had to check whether all components of expression () vanish for the rotation metric. So far, he had only convinced himself that the 11 -component does. In principle, one would have to check six index combinations: 11 , 12 , 14 , 22 , 24 , and 44 .In the 3+1 -dimensional case, the components 13 , 23 , 33 and 34 trivially vanish. All other components are equal to the corresponding ones in the 2+1 -dimensional case. Upon inspection of expression (), however, one easily sees that the 24 -component will be equal to the 14 -component for this particular metric, and that the 22 -component will be equal to the 11 -component. It thus suffices to check whether the remaining four components vanish. Einstein first considered the index combination “ α4 β1 .” The second term in expression () vanishes because the metric is time-independent. The first term can only contribute for “ i2 ,” as Einstein noted, since otherwise ∂41∂xi0 . Even for i2 , however, there will be no contribution, as Einstein also noted, ∂∂xεgε2ω0 . Einstein then turned to the index combination “ α4 β4 .” The second term in expression () again vanishes because the metric is time-independent. Consider the first term ∂∂xεgεi∂44∂xi . Since 441 , this term also vanishes. Einstein initially seems to have inserted 441-ω2x2+y2 instead, which would explain the expressions i12ω2x, i2 which he wrote down and deleted. He then simply wrote: “vanishes” (“verschwindet”). For the next index combination, “ α1 β1 ,” which he wrote down after drawing another horizontal line, Einstein just wrote: “correct” (“stimmt”). After all, he had constructed expression () so that its 11 -component would vanish for the rotation metric (see equation ()). As we already pointed out, the 11 -component does in fact not vanish, ∂∂xεgεi∂11∂xi-∂λτ∂x1∂gλτ∂x1g22∂211∂x22-2∂24∂x1∂g24∂x14ω2 . Finally, Einstein considered the index combination “ α1 β2 .” One easily sees that the second term in equation () once again vanishes. Given the sign error in the 12 -component of μν in expression (), Einstein wrote the possible contributions coming from the first term in expression () as i1∂∂xεgε1 ω2y i2∂∂xεgε2 ω2x. Both these terms vanish. Einstein could thus concluded that the 12 -component of equation () also “vanishes” (“verschwindet”) for the rotation metric. Einstein had now checked all independent components of equation () and concluded: “Equation satisfied” (“Gleichung erfüllt”). Making the right errors in the right places, he had convinced himself that the ill-defined expression in equation () finally gave him field equations with the rotation metric as a vacuum solution, a test that so many other promising candidates had failed. Testing the Newly Concocted Field Equations for Compatibility with Energy-Momentum Conservation (25R) In the last four lines of p. 25R, Einstein checked whether the ill-defined new field equations based on expression (), which looked promising from the point of view of the equivalence principle, would satisfy the conservation principle as well. As he had done for several other candidate field equations,See p. 19R, 20L, 21L, 24L and 25L, and the discussion following equation () in sec. . he checked whether these field equations could be used to write the gravitational force density as the divergence of gravitational stress-energy density. A few lines of calculation show that this can easily be done in linear approximation, but that the equality cannot hold exactly. Einstein thereupon abandoned these field equations. In the course of this short calculation, Einstein may also have come to realize that the equations were mathematically unacceptable to begin with. In linear approximation, the gravitational force density can be written as the contraction of a derivative of the metric with the left-hand side of the linearized field equations. If one uses the covariant field equations, one has to use the contravariant metric; if one uses the contravariant field equations, one has to use the covariant metric (cf. expressions () and ()). Einstein, however, wrote the force density as a contraction of the first term in expression (), which has contravariant free indices, and the derivative of the contravariant metric, ∂αβ∂xσ∂∂xεgεi∂αβ∂xi . He thereby further compounded the problem that his new field equations are ill-defined. In expression () the summations over α and β are over pairs of contravariant indices, while the summations of over ε and i are over pairs of covariant ones. Oblivious to these problems, it seems, Einstein proceeded to rewrite expression () in the form of a divergence. First he rewrote it asIn the notebook, the expressions in parentheses in equations () and () as well as the first expression in parentheses in equation () are indicated by pairs of parentheses only. ∂∂xε∂αβ∂xσgεi∂αβ∂xi-∂2αβ∂xε∂xσgεi∂αβ∂xi The first term has the required form of a divergence, so Einstein could focus on the second term, which he rewrote as -∂∂xσ∂αβ∂xεgεi∂αβ∂xi+∂αβ∂xε∂αβ∂xi∂gεi∂xσ+∂αβ∂xεgεi∂2αβ∂xi∂xσ . The first term again has the required form of a divergence. The second term is of third power in derivatives of the metric and vanishes in linear approximation. Rather than deleting this term, however, Einstein underlined it, a clear indication that he wanted to make this seemingly approximative calculation exact. But first he turned his attention to the third term in expression (), which he rewrote asOne of the derivatives ∂∂xε in the last term should be ∂∂xi . ∂∂xi∂αβ∂xεgεi∂αβ∂xσ-∂αβ∂xσ∂∂xεgεi∂αβ∂xε . He underlined both terms. This is the end of the calculation. Einstein was clearly dissatisfied with the result, for in the lower left corner of p. 25R, to the left of equations () and (), he wrote: “impossible” (“unmöglich”). The reason for Einstein’s dissatisfaction must have been that the calculation in equations ()–() convinced him that his new candidate field equations are incompatible with the exact equality of the gravitational force density and the divergence of gravitational stress-energy density. In linear approximation the calculation seems to show that the equations are compatible with this equality.In fact, the calculation shows nothing of the sort, since both the field equations and the expression that was used for the gravitational force density in this calculation are mathematically ill-defined. Einstein, however, appears to have been unaware, at least initially, of these problems. The last term in expression () is equal and opposite to expression () that Einstein started from. Bringing this term to the left-hand side and dividing both sides by a factor 2 , Einstein would have succeeded in writing his ill-defined expression for the gravitational force density as a sum of three terms that all have the required form of a divergence (i.e., the first terms in expressions (), (), and ()) and a term of third power in derivatives of the metric that vanishes in linear approximation (i.e., the second term in expression ()). The equality would hold exactly if this last term could be interpreted as (minus) the contraction of ∂αβ∂xσ with the term quadratic in first-order derivatives of the metric in the field equations. Comparison of the second term in expression () to the second term in expression (), the left-hand side of the candidate field equations under consideration, shows that the former cannot be interpreted in this way,It may be interesting to note that such an interpretation would have been possible had Einstein contracted ∂∂xεgεi∂αβ∂xi with ∂gαβ∂xσ , as he should have done, rather than with ∂αβ∂xσ (cf. the discussion in the paragraph with equation () above). In that case, the second term in equation () would change to ∂gαβ∂xε∂αβ∂xi∂gεi∂xσ∂gλτ∂xα∂λτ∂xβ∂gαβ∂xσ , where the term in parentheses is indeed equal to minus the second term in equation (). With this modification of the calculation in equations ()–(), however, the last term in equation () would change to ∂αβ∂xσ∂∂xεgεi∂αβ∂xi and would no longer be equal to the expression in equation (). Consequently, the equality of gravitational force density and divergence of gravitational stress-energy density would not even hold in linear approximation. It seems unlikely that Einstein considered this alternative. ∂αβ∂xε∂αβ∂xi∂gεi∂xσ≠∂αβ∂xσ∂gλτ∂xα∂λτ∂xβ . This is probably why Einstein concluded that it was “impossible” (“unmöglich”) to write the gravitational force density as the divergence of gravitational stress-energy density. Einstein’s remark may also refer, at least in part, to the more fundamental problem that expressions ()–() as well as the new candidate field equations themselves are mathematically ill-defined. Whatever the case may be, Einstein at this point abandoned these ill-defined field equations and on the next page (p. 26L) made a fresh start with the method for generating field equations automatically satisfying energy-momentum conservation that he had introduced on p. 24R. Conclusion: Cutting the Gordian Knot (19L–25R) Of all attempts to find suitable gravitational field equations recorded in the notebook, the one on p. 25R was clearly the most desperate. As was shown in detail in secs. –, the attempt is riddled with errors. One can, however, also look upon Einstein’s calculations on this page in a more positive way. They reveal very clearly which of the various requirements that had to be satisfied by putative field equations weighed most heavily for Einstein at this point. Looking at p. 25R from this perspective, one is struck by the fact that, even though many of the expressions considered by Einstein were not even well-defined mathematically, he continued to adhere strictly to three requirements. First and foremost, there was the correspondence principle. The field equations should consist of a core operator, i.e., a term with second-order derivatives of the metric that reduces to the d’Alembertian acting on the metric in linear approximation, and terms quadratic in first-order derivatives of the metric that vanish in linear approximation. Then there was the conservation principle in the very specific form that the field equations should make it possible, not just in linear approximation but exactly, to write the gravitational force density as the divergence of gravitational stress-energy density. Finally, there was the demand that the rotation metric be a vacuum solution of the field equations. Einstein was willing, it seems, to weaken, if only temporarily, the much stronger demands of his relativity and equivalence principles that the field equations be invariant under (autonomous or non-autonomous) transformations to arbitrarily accelerated frames of reference. The focus on these three requirements, which constrain even the very problematic calculations on p. 25R, made the task of finding suitable field equations much more manageable. The strategy that Einstein had used on pp. 19L–23L and again on p. 25L and at the top of p. 25R of extracting field equations from expressions of broad covariance by imposing coordinate restrictions had failed several times. With hindsight, one can see that most of the problems come from Einstein using coordinate restrictions rather than coordinate conditions in the modern sense to make sure that the field equations satisfy the correspondence principle. Einstein had found three different coordinate restrictions (the harmonic restriction, the Hertz restriction, and the ϑ -restriction) with the help of which the correspondence principle could be satisfied. The covariance properties of these coordinate restrictions, however, proved to be intractable. Since Einstein used coordinate restrictions rather than coordinate conditions, this meant that the covariance properties of the field equations themselves became intractable as well. It thus remained unclear whether these equations satisfy the demands of the equivalence and relativity principles. Moreover, Einstein had not been able to confirm that the coordinate restrictions and the associated field equations would be compatible with the exact validity of energy-momentum conservation. In this situation something had to give. Einstein, it seems, cut the Gordian knot by weakening two of his heuristic principles. He weakened the equivalence principle to the requirement that the field equations at least allow the rotation metric. He weakened the relativity principle to the obvious minimal requirement that the field equations at least have well-defined transformation properties. On p. 24R Einstein had found candidate field equations imposing the correspondence and conservation principles and bracketing the problem of satisfying the remaining two principles. On pp. 26L–R, he turned this derivation into a powerful method for generating field equations that automatically meet the requirements of the correspondence and conservation principles. What probably made this option all the more appealing was that Einstein believed (mistakenly as was shown in sec. ) that the expression that forms the starting point of the calculation on p. 24R vanishes exactly for the rotation metric, a necessary condition for the resulting field equations to satisfy the weakened version of the equivalence principle. Derivation of the Entwurf Equations (26L–R) On pp. 26L–R, Einstein derived the identity that is at the heart of the derivation of the Entwurf field equations in (Einstein and Grossmann 1913). His approach on these pages is very similar to his approach on p. 24R (cf. the discussion following expression () in sec. ). He substituted the left-hand side of some linearized field equations for the stress-energy tensor Tμν in the expression ∂gμν∂xmTμν for the gravitational force density and tried to rewrite the resulting expression as the divergence of the gravitational stress-energy density. In addition to divergence terms, he found terms that are contractions of ∂gμνxm and terms quadratic in first-order derivatives of the metric. By adding the latter to the linearized field equations, Einstein arrived at exact field equations that guarantee that the gravitational force density is exactly equal to the divergence of the gravitational stress-energy density. The calculation on pp. 26L–R is thus very similar to the calculation on p. 24R, but differs from it in two important respects. First, Einstein no longer set the determinant of the metric equal to unity, which can be seen as an indication that he had meanwhile given up hope to recover the new candidate field equations from the November tensor as he had still tried to do on p. 25L for the field equations of p. 24R. Second, Einstein did not start, as he had done on p. 24R (see expression ()) from a specific expression for (the divergence of) the gravitational stress-energy pseudo-tensor. Einstein read off both the expression for the left-hand side of the field equations and the expression for the gravitational stress-energy pseudo-tensor from the identity obtained by rewriting expression () after substitution of the left-hand side of the linearized field equations for the stress-energy tensor. At the top of p. 26L, Einstein began by writing down the energy-momentum balance between matter and gravitational field,In the first term of equation (), gμν should be gmν (see equation ()). ∂∂xnGgμνTνn-12G∂gμν∂xmTμν0 , or, as he called it, the “system of the equations for matter” (“System der Gleichungen für Materie”) and, on the next line, the (contravariant) stress-energy tensor for pressureless dust, Tμνρdxμdτdxνdτ . Equation () shows that the exact expression for the gravitational force density is 12G∂gμν∂xmTμν , but in the calculations on the remainder of the page Einstein ignored the factor 12G , a simplification that is easily corrected for at the end of the calculation. Under the heading “derivation of the gravitation equations” (“Ableitung der Gravitationsgleichungen”), Einstein now substituted the left-hand side of linearized field equations based on a core operator, ∂∂xααβG∂μν∂xβκTμν , for the stress-energy tensor in the crude expression () for the force density, ∂gμν∂xm∂∂xααβG∂μν∂xβ , also ignoring, as he had done on several earlier occasions (see pp. 19R, 20L, 21L, 24R, and 25R), the gravitational constant κ . Einstein set out to rewrite expression () as a sum of terms of two kinds, divergence terms (marked ‘+’ and ‘¥’ on pp. 26L–R) and terms that are contractions with ∂gμνxm (marked ‘o’ on pp. 26L–RExcept in one case (see expression () below) which Einstein apparently forgot to mark.). Einstein first rewrote expression () as ∂gμν∂xm| ∂∂xααβG∂μν∂xβ∂∂xααβG∂μν∂xβ∂gμν∂xm-Gαβ∂μν∂xβ∂2gμν∂xm∂xα. The first term on the right-hand side is a divergence term. It is underlined and marked ‘+.’ Einstein proceeded to rewrite the second term as -∂∂xmαβG∂μν∂xβ∂gμν∂xα+∂gμν∂xα∂∂xmGαβ∂μν∂xβ . Once again, the first term is a divergence term, which is underlined and marked ‘¥.’ The second term gives three terms: 12∂gστ∂xmστG∂gμν∂xα∂μν∂xβαβ+∂αβ∂xmG∂gμν∂xα∂μν∂xβ+∂gμν∂xαGαβ∂2μν∂xm∂xβ . This expression extends beyond the right margin of p. 26L and is continued on the facing page, p. 26R. The first term in expression () is obtained with the help of the relation ∂G∂xm12G∂G∂xm12GGστ∂gστ∂xm , which Einstein had encountered several times before (see pp. 6L, 8R, and 9R, and equation ()). By relabeling indices ( στμν→μνστ ), Einstein could show that this term can be written as a contraction of ∂gμνxm and a term quadratic in first-order derivatives of the metric, 12∂gμν∂xmμνG∂gστ∂xα∂στ∂xβαβ . The term is underlined, but Einstein for some reason neglected to mark it ‘o’ as he did with other terms of this kind. The second term in expression () is of the same kind, although it takes a little more work to show this. Substituting ∂∂xαgασα′σ∂∂xα′ and ∂∂xβgβτβ′τ∂∂xβ′ , Einstein rewrote this term as ∂αβ∂xmgασgβτ | α′σβ′τ G∂gμν∂xα′∂μν∂xβ′ , which, with the help of the relation ∂αβ∂xmgασgβτ∂gστ∂xm , he then rewrote as ∂gστ∂xmGασβτ∂gμν∂xα∂μν∂xβ . So, this term does indeed also have the form of a contraction of ∂gμν∂xm and a term quadratic in first-order derivatives of the metric. It was underlined and marked ‘o’ accordingly. At the bottom of p. 26R, Einstein turned to the third term in expression (). This term contributes one ‘+’-term and two ‘o’-terms. Einstein began by rewriting it as ∂∂xβ∂μν∂xm∂gμν∂xαGαβ-∂μν∂xm∂∂xβGαβ∂gμν∂xα . The first term is a divergence term. It is underlined and marked ‘+’. Rewriting it as ∂∂xα∂gμν∂xm∂μν∂xβGβα , one sees that it is equal to this term in equation (). The second term in expression () is a contraction with ∂μνxm rather than with ∂gμνxm . Einstein therefore rewrote it as (cf. expression ())In expression (), gμν should be gμ′ν′ . ∂μν∂xmgμσgντ|μ′σν′τ∂∂xβGαβ∂gμν∂xα , and then set ∂μν∂xmgμσgντ equal to ∂gστ∂xm . His next move was to bring μ′σ and ν′τ within the scope of the differentiation ∂∂xβ in order to turn the derivative of gμν into a derivative of μν (as in expression ()). Expression () thus turns intoExpression () originally had μ′σν′τ , but the primes were crossed out. ∂gστ∂xm | ∂∂xβμσντGαβ∂gμν∂xα-Gαβ∂gμν∂xα∂μσντ∂xβ . He then rewrote the two terms to the right of the vertical line asFor the first term, he used the relation μσντ∂gμν∂xα∂στ∂xα . The second term can be rewritten as a sum of two identical terms . Gαβ∂gμν∂xα∂μσντ∂xβ=-Gαβ∂gμν∂xα∂μσ∂xβντ-Gαβ∂gμν∂xα∂ντ∂xβμσ =Gαβ∂ντ∂xα∂μσ∂xβgμν+Gαβ∂μσ∂xα∂ντ∂xβgμν. -∂∂xβ∂στ∂xααβG +2Gαβgμν∂μσ∂xα∂ντ∂xβ . He underlined both terms and marked them ‘o.’ The first term is (minus) the core operator that formed the starting point of this whole calculation (cf. equations ()–()). Einstein now collected the ‘o’-terms on the left-hand side and the ‘+/¥’-terms on the right-hand side. The ‘o’-terms come from the left-hand side of equation () (with a factor 2 because of the identical contribution from the first term in expression ()), from expressions () and (), and from the second term in expression () (contracted with ∂gστ∂xm ). These terms add up to ∂gμν∂xm[2∂∂xααβG∂μν∂xβ 12Gμναβ∂gστ∂xα∂στ∂xβ +Gαμβν∂gστ∂xα∂στ∂xβ-2Gαβgστ∂μσ∂xα∂ντ∂xβ]. The ‘+/¥’-terms come from the first term on the right-hand side of equation () (with a factor 2 because of the identical contribution from expression ()) and from the first term in expression (): 2∂∂xαGαβ∂στ∂xβ∂gστ∂xm-∂∂xmGαβ∂gστ∂xα∂στ∂xβ . Regrouping the ‘o’-terms in expression (),The physical reasoning behind this regrouping will become clear below. setting them equal to the ‘+/¥’-terms in expression (), and dividing both sides by 2, one arrives at the identity ∂gμν∂xm[∂∂xααβG∂μν∂xβGαβgστ∂μσ∂xα∂ντ∂xβ +12Gαμβν∂gστ∂xα∂στ∂xβ-12μναβ∂gστ∂xα∂στ∂xβ]=∂∂xαGαβ∂στ∂xβ∂gστ∂xm-12∂∂xmGαβ∂gστ∂xα∂στ∂xβ, which Einstein wrote down at the bottom of pp. 26L–R as the “summary” (“Zusammenfassung”) of his calculations. Underneath this identity, he wrote: “This is the contra-form” (“Dies ist die Kontra-Form”). It is from this identity—given as equation (12) in Einstein’s part of (Einstein and Grossmann 1913) and derived in sec. 4.3 of Grossmann’s part—that both the contravariant form of the field equations and the contravariant form of the gravitational stress-energy pseudo-tensor of the Entwurf theory can be read off. To identify the exact expressions for the field equations and the pseudo-tensor we need the exact relations between the left-hand side of the field equations, the gravitational force density, and the gravitational stress-energy pseudo-tensor. We can no longer afford to ignore factors G . Consider once again equation (), the vanishing of the covariant divergence of the mixed tensor density GgmνTνn . This equation can be written as the ordinary divergence of the sum of the stress-energy density of matter and gravitational field if 12G∂gμν∂xmTμν∂∂xnGgmνtνn , where tμν is the contravariant form of the gravitational stress-energy pseudo-tensor. Multiplying both sides by 2κ , one arrives at G∂gμν∂xmκTμν∂∂xnGgmν2κtνn . The identity () guarantees that this equation holds if both the left-hand side of the field equations (to be substituted for κTμν in equation ()) and ( 2κ times) the pseudo-tensor tμν are suitably chosen. Comparing the left-hand sides of equations () and (), one sees that one must choose the expression 1G∂∂xααβG∂μν∂xβαβgστ∂μσ∂xα∂ντ∂xβ +12αμβν∂gστ∂xα∂στ∂xβ-12μναβ∂gστ∂xα∂στ∂xβ, i.e., the expression in square brackets in expression () divided by G , as the left-hand side of the field equations. This is indeed the left-hand side of the Entwurf field equations. To bring the right-hand side of equation () in form that can be compared with the right-hand side of equation (), one has to relabel the summation index α by n , substitute ∂∂xmgmναν∂∂xα in the first term, and substitute ∂∂xmgmνnν∂∂xn in the second term. Comparing the resulting expression, ∂∂xnGgmνnβαν∂στ∂xβ∂gστ∂xα-12nναβ∂gστ∂xα∂στ∂xβ , to the right-hand side of equation (), one sees that one must choose the expression ανβn∂gστ∂xα∂στ∂xβ-12νnαβ∂gστ∂xα∂στ∂xβ as the gravitational stress-energy pseudo-tensor 2κtνn . This is the definition of this quantity given in equation (13) of (Einstein and Grossmann 1913).In the paper, the contravariant form is denoted by ϑμν , and the covariant form by tμν . Notice that the expression in parentheses on the second line of expression () is equal to 2κtμν . If we now introduce the notation Dμν for the expression on the first line,This is in keeping with (Einstein and Grossmann 1913), where this same quantity is introduced in equation (15) as Δμν . the left-hand side of the field equations can be written as Dμν-κtμν , and the field equations themselves as DμνκTμν+tμν , which is the form in which the Entwurf field equations are given in equation (18) of (Einstein and Grossmann 1913). In (Einstein and Grossmann 1913), they are written as ΔμνκΘμν+ϑμν . Commenting on this equation (ibid., p. 17), Einstein pointed out that any acceptable field equations must be such that the stress-energy of matter and the stress-energy of the gravitational field enter the equations in the exact same way.A similar result holds in the final version of Einstein’s earlier theory for static fields (see Einstein 1912b, 457) and in general relativity in its final form (see Einstein 1916, 807–808). Before pp. 26L–26R, this requirement had not explicitly played a role in Einstein’s search for suitable field equations. But the way in which Einstein grouped the terms on the left-hand side of the identity ()—with the terms in the first set of parentheses giving Dμν and those in the second giving 2κtμν — suggests that Einstein did consider this requirement when he wrote down the calculations on pp. 26L–26R. One can understand why Einstein would have been pleased with these new candidate field equations. They satisfied the correspondence principle and they satisfied the conservation principle, not just in linear approximation but exactly. Since the new field equations were not extracted from some quantity of broad covariance with the help of a suitable coordinate restriction, it remained unclear whether they satisfy the relativity principle and the equivalence principle. The one encouraging result on this score (erroneous as it turns out) was that the divergence of the gravitational stress-energy pseudo-tensor vanishes for the rotation metric, a necessary condition for this metric to be a vacuum solution of the Entwurf field equations.See p. 24R, expression () and the discussion in sec.  following expression (). More importantly, Einstein had not been able to find any acceptable field equations along the lines of his mathematical strategy that satisfied the relativity principle and the equivalence principle. And he had at best been able to show in linear approximation that these candidates satisfied the conservation principle. It is thus not surprising that Einstein gave up his attempt to construct field equations out of the Riemann tensor and decided to publish, in his joint paper with Marcel Grossmann, the field equations found along the lines of his physical strategy. Acknowledgments The first draft of this commentary was prepared in the early 1990s by a working group of the Arbeitsstelle Albert Einstein (see preface). The working group—joined early on in its work on the notebook by John Norton and John Stachel and later by Michel Janssen—did not have to start from scratch. It had John Norton’s “How did Einstein find his field equations?” (Norton 1984) to go on, as well as drafts of the annotation of the notebook prepared for Vol. 4 of The Collected Papers of Albert Einstein, both the footnotes added to the transcription (CPAE 4, Doc. 10) and the editorial note, “Einstein’s Research Notes on a Generalized Theory of Relativity” (CPAE 4, 192–199). The joint work was continued in a number of workshops hosted by the Max-Planck-Institut für Bildungsforschung and later by the Max-Planck-Institut für Wissenschaftsgeschichte. On the basis of all this material, Michel Janssen and Jürgen Renn wrote the final version of the commentary (with the exception of sec. 4.5.8, which was written by Tilman Sauer). They produced the first half in close collaboration, partly at the University of Minnesota, partly (courtesy of Bernhard Schutz) at the Max-Planck-Institut für Gravitationsphysik (Albert Einstein Institut) in Golm. 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