Einstein II: From 1948 on

9 Einstein II: From 1948 on

In the meantime, Einstein had gone on struggling with his field equations and, in a letter to M. Solovine of 25 November 1948, had become less optimistic ([160 *], p. 88):

“Scientifically, I am still lagging because of the same mathematical difficulties which make it impossible for me to affirm or contradict my more general relativistic field theory […]. I will not be able to finish it [the work]; it will be forgotten and at a later time arguably must be re-discovered. It happened this way with so many problems.”¹¹⁵

In his correspondence with Max Born during the second half of the 1940s, Einstein clung to his refusal of the statistical interpretation of quantum mechanics. According to him, physics was to present reality in space and time without, as it appeared to him, ghostly interactions at a distance. In a letter of 3 March 1947, he related this to UFT:

“Indeed, I am not strongly convinced that this can be achieved with the theory of my continuous field although I have found for it an – until now – apparently reasonable possibility. Yet the calculatory difficulties are so great that I shall bite the dust until I myself have found an assured opinion of it. […]” ¹¹⁶

In spite of such reservations, Einstein carried on unflagging with his research. In his next publication on UFT [148 *], he again took a complex (asymmetric) metric field. In order to justify this choice in comparison to Schrödinger’s who “has based his affine theory […] on real fields […]”,¹¹⁷ he presented the following argument: Just by multiplication and the use of a single complex vector $Ai$ a Hermitian tensor $− AiAk$ can be constructed. By adding four such terms, the Hermitian metric tensor $− gik = Σ c A iA k κ κ κ κ$ can be obtained. “A non-symmetric real tensor cannot be constructed from vectors in such close analogy” ([148 *], p. 39). Nevertheless, in Einstein’s future papers, the complex metric was dropped.

The field equations were derived from the Lagrangian

i.e., from the Lagrangian (205 *) without the multiplier terms. In order to again be able to gain the weak field equations, an additional assumption was made: the skew-symmetric part of the metric (density) $[ik] ˆg$ be derived from a tensor “potential” $ˆgikl$ anti-symmetric in all indices. Thus, in the Lagrangian, $ˆgik$ is replaced by $ˆgik = ˆg(ik) + ˆgikl,l$ . The motivation behind this trick is to obtain the compatibility equation (30 *) from $δℋ ∕δΓ m = 0 ik$ , indirectly. The skew-symmetric part of $δℋ ∕δΓ m = 0 ik$ is formed and a trace taken in order to arrive at $1 [ik] (ik) s 2ˆg ,l + ˆg Γ [ks] = 0$ . Introduction of $il ils ˆg ,l = ˆg ,ls$ then shows that $Γ i = 0$ holds. With the help of this equation, $δℋ ∕δΓimk = 0$ finally reduces to (30 *).

The field equation following directly from independent variation with regard of $Γ m ,ˆg(ik) ik$ and $ˆgikl$ are: ¹¹⁸

i+k− ˆg ||l = 0, (249 ) P(ik) = 0, (250 ) P[ik],l + P [kl],i + P[li],k = 0. (251 )

In addition, the equations hold:

As in [179 *], Einstein did not include homothetic curvature into the building of his Lagrangian with the same unconvincing argument: from his (special) field equations and (252 *) the vanishing of the homothetic curvature would follow.

In his paper, Einstein related mathematical objects to physical observables such that “the antisymmetric density $ikl ˆg$ plays the role of an electromagnetic vector potential, the tensor $ˆg[ik],l + ˆg[kl],i + ˆg[li],k$ the role of current density.” More precisely, the dual object $s sikl j ∼ 𝜖 (ˆg[ik],l + ˆg[kl],i + ˆg[li],k)$ with vanishing divergence $js,s = 0$ is the (electric) current density ([148 *], p. 39).

Einstein summed up the paper for Pauli on one page or so and concluded: “The great difficulty lies in the fact that we do not have a method for deriving exact solutions free of singularities, which are the only ones of physical interest. The few things we were able to calculate strengthened my confidence in this theory.” ([489 *], p. 518)¹¹⁹ In his answer three weeks later, Pauli was soft on “whether a mathematically unified combination of the electromagnetic and gravitational fields in a classical field theory is possible […]”, but adamant on its relation to quantum theory:

“[…] that I have another opinion than you on the question, mentioned in your letter, of the physical usability of singularity-free solutions of classical field equations. To me it deems that, even if such solutions do exist in a suitably chosen field theory, it would not be possible to relate them with the atomic facts in physics in the way you wish, namely in a way that avoids the statistical interpretation, in principle.” ([489 *], p. 621.) ¹²⁰

9.1 A period of undecidedness (1949/50)

With two sets of field equations at hand (the “strong” and “weak” versions), it cost some effort for Einstein to decide which of the two was the correct one. As will be seen in Section 9.2, early in 1949 he had found a new way of deriving the “weak” field equations, cf. [149 *].¹²¹ In a letter of 16 August 1949 to his friend Besso, who had asked him to tell him about his generalized field equations, Einstein presented these “weak” equations and commented:

“Now you will ask me: Did God tell this into your ear? Unfortunately, not. But the way of proceeding is: identities between the equations must exist such that they are compatible. […] For their compatibility, i.e., that continuation from a [time-] slice is possible, there must be 6 identities. These identities are the means to find the equations. […]” ([163 *], p. 410).¹²²

Six weeks later, on 30 September 1949, Einstein had changed his mind: he now advocated the “strong” version (200 *) – (202 *) of Section 7.2.

“I recently found a very forceful derivation for this system; it shows that the equations follow from the generalized field as naturally as the gravitational equations from the postulate of the symmetric field $g(ik)$ . The examination of the theory still meets with almost unsurmountable mathematical difficulties […]” ([163 *], p. 423).¹²³

Consistent with Einstein’s undecidedness are both, his presentation of UFT in Appendix II of the 3rd Princeton edition of The Meaning of Relativity [150 *], and another letter to Besso of 15 April 1950 [163 *]. In both, he had not yet come to a final conclusion as to which must be preferred, the “weak” or the “strong” equations. To Besso, he explained that the “weak” equations could be derived from a variational principle and thus are “compatible”.

“On the other hand, one is pushed to the stronger system by formal considerations […]. But the compatibility for this stronger system is problematic; i.e., at first one does not know whether the manifold of its solutions is sufficiently large. After many errors and efforts I have succeeded in proving this compatibility” ([163 *], p. 439).¹²⁴

At first, Einstein seems to have followed a strategy of directly counting equations, variables, and identities. However, early in 1952 he seems to have had a new idea: the $λ$ -transformations. He wrote to Besso on 6 March 1952 that he had made “very decisive progress (a couple of weeks ago).” The field equations, hitherto not uniquely determined theoretically, now were known:

“ Apart from [coordinate-] transformation invariance, invariance also is assumed for the transformations of the non-symmetric ‘displacement field’ $Γikl$ : $(Γikl )∗ = Γ ikl+ δliλk$ , where $λk$ is an arbitrary vector. In this extended group, the old gravitational equations are no longer covariant […].”([163 *], p. 465)¹²⁵

We will come back to his final decision in Section 9.2.3.

9.1.1 Birthday celebrations

Einstein’s seventieth birthday was celebrated in Princeton with a seminar on “The Theory of Relativity in Contemporary Science”, in which E. P. Wigner, H. Weyl, and the astronomers G. M. Clemence and H. P. Robertson lectured. UFT was left aside [183 ]. Weyl, in his lecture “Relativity Theory as a Stimulus in Mathematical Research”, came near to it when he said:

“The temptation is great to mention here some of the endeavors that have been made to utilize these more general geometries for setting up unified field theories encompassing the electromagnetic field beside the gravitational one or even including not only the photons but also the electrons, nucleons, mesons, and whatnot. I shall not succumb to that temptation.” ([693 ], p. 539.)

As it suited to a former assistant of Einstein, in his article celebrating his master’s 70th birthday, Banesh Hoffmann found friendly if not altogether exuberant words even for Einstein’s struggle with UFT [281 *]. For 25 years Einstein had devoted his main scientific work to the problem of the structure of matter and radiation. He tried to gain an insight:

“[…] by abstract reasoning from a few general assumptions. In this he is following the heroic method that proved so successful […] in the theory of relativity. Unfortunately there are many possible approaches, and since each requires a year or more of intensive computation, progress has been heartbreakingly slow.”

That Hoffmann himself was a little outside of mainstream physics can be seen from his remark that quantum theory, now dominating physics, “has developed a stature comparable to that of the theory of relativity.” ([281 ], p. 54/55.) Hoffmann was also one of the contributors to the special number of Reviews in Modern Physics “in commemoration of the seventieth birthday of Albert Einstein” issued in September 1949. Possibly, the best remembered paper among the 38 articles is Gödel’s “new type of cosmological solutions”, with local rotation and closed timelike world lines, now just named “Gödel’s solution” [227 ]. Only E. Straus wrote an article about UFT: “Some results in Einstein’s unified field theory” [592 *]. The others, big names and lesser known contributors except for the mathematician J. A. Schouten, shunned this topic. Schouten’s contribution surveyed classical meson theories in view of their making contact with the conformal group [539 ]. In connection with Yukawa’s prediction of a meson and with Hoffmann’s similarity geometry (cf. Section 3.1), he boldly stated: “[…] the conformal field theory failed to ask for a meson field, but the meson field came and asked for a conformal theory!” (ibid., p. 423.) Einstein’s oldest son Hans Albert reported on “Hydrodynamic Forces on a Rough Wall” [180 ].

Belatedly, toward the end of 1949, some sort of a “Festschrift” for Einstein appeared with 25 contributions of well-known physicists and philosophers, among them six Nobel prize winners [536 ]. Most interesting is Einstein’s own additional contribution, i.e., his “Autobiographical Notes”, written already in 1946. He described his intentions in going beyond general relativity and essentially presented the content of his paper with E. Straus [179 *] containing the “weak field equations” of UFT. His impression was:

“that these equations constitute the most natural generalization of the equations of gravitation. The proof of their physical usefulness is a tremendously difficult task, inasmuch as mere approximations will not suffice. The question is: ‘What are the everywhere regular solutions of these equations?’ ” ([153 ], p. 93–94.)¹²⁶

9.2 Einstein 1950

9.2.1 Alternative derivation of the field equations

As we have seen, one of Einstein’s main concerns was to find arguments for choosing a quasi-unique system of field equations for UFT. His first paper of 1949 opened with a discussion, mostly from the point of view of mathematics, concerning the possibilities for the construction of UFT with a non-symmetric fundamental tensor. According to Einstein: “The main difficulty in this attempt lies in the fact that we can build many more covariant equations from a non-symmetric tensor than from a symmetric one. This is due to the fact that the symmetric part $g(ik)$ and the antisymmetric part $g[ik]$ are tensors independently” ([149 *], p. 120). As the fundamental tensor is no longer considered symmetrical, the symmetry of the connection (as in Riemannian geometry) must also be weakened. By help of the conjugate quantities of Section 2.2.2 (Hermitian, transposition symmetry), Einstein’s constructive principle then is to ask “that conjugates should play equivalent roles in the field-equations.” According to him, this necessitates the introduction of the particular form (30 *) for the compatibility condition. In fact, for the conjugate:¹²⁷

while

Einstein seemingly was not satisfied with the derivation of the field equations from a variational principle in “both previous publications” ([147 *, 179 *]), because of the status of equations (252 *). To obtain them, either Lagrangian multipliers or a restriction of the metric (“must be derivable from a tensor potential”) had to be used. Now, he wanted to test the field equations by help of some sort of Bianchi-identity such as (cf. Section 2.3.1, Eq. (68 *), or Section 2.1.3 of Part I, Eq. (30)):

After a lengthy calculation, he arrived at: ¹²⁸

and by further trace-forming¹²⁹

In Eq. (257 *), two contractions of $i K− jkl$ were introduced: $l − K− jk := K− jkl$ and $jk i Σml = − gmig K− jkl$ . Neither $Σ ml$ nor $K − jk$ are Hermitian: We have $−K&tidle; = Σ . − lm ml$ The anti-Hermitian part of $K − jk$ is given by

If vector torsion is absent, i.e., $s Sk = L [ks] = 0$ , then $K− jk$ becomes Hermitian, and $− K− lm = Σml$ . Equation (257 *) then can be written as

Therefore, Einstein demanded that the contribution of $K− [jk]$ to the Eq. 259 *) be in general:

Schrödinger had derived (260 *) before by an easier method with the help of the Lie-derivative; cf. [556 *]. We will meet (260 *) again in Section 10.3.1. A split of (259 *) into symmetric and skew symmetric parts (inside the bracket) would give the equation:

At best, as a sufficient condition, the vanishing of the symmetric and skew-symmetric parts separately could take place. Besides (260 *) the additional equation would hold:

As we will see, by a further choice (cf. (265 *), this equation will be satisfied. Einstein first reformulated (260 *) into:

and then took

as its solution and as part of the field equations. He then added as another field equation:

by which (262 *) is satisfied. Thus, in [149 *], with a new approach via identities for the curvature tensor and additional assumptions, Einstein had reached the weak field equations of his previous paper [148 *]. No physical interpretation of the mathematical objects appearing was given by him.

9.2.2 A summary for a wider circle

In Appendix II of the third Princeton edition of his book The Meaning of Relativity, Einstein gave an enlarged introduction on 30 pages into previous versions of his UFT. The book was announced with fanfare in the Scientific American [151 *]:

“[…] Einstein will set forth what some of his friends say is the long-sought unified field theory. The scientist himself has given no public hint of any such extraordinary development, but he is said to have told close associates at the Institute for Advanced Studies that he regards the new theory as his greatest achievement” ([564 ], p. 26).

It was Princeton University Press who had used Einstein’s manuscript for this kind of advertising much to his distress; a page of it even “appeared on the front page of The New York Times under the heading ‘New Einstein Theory Gives a Master Key to the Universe’.” ([469 *], p. 350.) Einstein’s comment to his friend M. Solovine, on 25 January 1950, was:

“Soon I will also send you the new edition of my little book with the appendix. A few weeks ago, it has caused a loud rustling noise in the newspaper sheets although nobody except the translator had really seen the thing. It’s really drole: laurels in advance” ([160 *], p. 96).¹³⁰

In the book, the translator is identified to have been Sonja Bargmann, the wife of Valentine, who also had translated other essays by Einstein.

In Appendix II, with the assumption that

all equations remain unchanged with respect to simultaneous substitution of the $g ik$ and $l Γik$ by $g&tidle;ik$ and $&tidle; l Γ ik$ (transposition invariance),
all contractions of the curvature tensor (54 *) vanish,
that (30 *) hold,

Einstein arrived at the field equations¹³¹ (30 *), (201 *), and (202 *):

g ik∥l := gik,l − grkLilr− girLlkr= 0, (266 ) +− m Sj (L ) := L [im] = 0, (267 ) Kjk (L) = 0. (268 )

In place of (266 *) the equivalent equations $ik g+−∥l = 0$ , or $ik ˆg+−∥l = 0$ with $∘ -------- ˆgik = det(gik) gik$ can be used. Moreover, if in addition (267 *) is taken into account, then also

is satisfied. This is due to a relation to be met again below [(cf. Eq. (276 *)]:

([179 ], Eq. (3.4), p. 733.)

For Einstein, this choice (“System I”) “is therefore the natural generalization of the gravitational equation” ([150 *], p. 144). A little later in the Appendix he qualified his statement as holding “from a formal mathematical point of view […]” ([150 *], p. 150) because the manifold of solutions of “System I” might be too small for physical purposes. Moreover, “System I” could not be derived from a variational principle. He then set up such a variational principle¹³²

with the Hermitian Ricci tensor $Pik$ . As Einstein wanted to again get (267 *), he introduced another connection $Γ ∗$ by $Γ k= Γ ∗ k− 2Γ ∗δk ij ij 3 [ij]$ which does not satisfy $Γ ∗ = Γ ∗ m = 0 j [im ]$ , such that just independent components could be varied. The result is his “System Ia”:

i+k− 1- i k k i ˆg ∥l − 3(ℳ δl − ℳ δl ) = 0, (272 ) Li = 0, (273 ) Pik = 0, (274 )

with $[i+l−] ℳi := ˆg ||l$ . The following identity holds:¹³³

In order to make vanish $i ℳ$ , with the help of a Lagrange multiplier $li$ , he introduced the term $[ir] li ˆg ,r$ into the Lagrangian and arrived at ‘System Ib’, i.e., the weak field equations:

ik ˆg+− ||l = 0, Li = 0, P(ik) = 0, (277 ) P[ik],l + P [kl],i + P[li],k = 0. (278 )

In all three systems, equations (249 *), (269 *), (272 *) are to be used for expressing the components of the (asymmetric) connection by the components of the (asymmetric) metric. The metric then is determined by the remaining equations for the Ricci tensor.

The only remarks concerning a relationship between mathematical objects and physical observables made by Einstein at the very end of Appendix II are:

(269 *) shows that there is no magnetic current density present (no magnetic monopoles),
the electric current density (or its dual vector density) is represented by the tensor $g[ik],l + g[kl],i + g[li],k)$ .

In order to obtain these conclusions, a comparison to Maxwell’s equations has been made (cf. (210 *) and (211 *) of Section 7.3). As for all of Einstein’s papers in this class, $g (ik)$ describes the gravito-inertial, and $g[ik]$ the electromagnetic fields.

For the first time, Einstein acknowledged that he had seen Schrödinger’s papers without giving a reference, though: “Schrödinger, too, has based his affine theory […]”. Max Born, in his review of Einstein’s book, bluntly stated: “What we have before us might therefore be better described as a program than a theory.” ([41 ], p. 751.) According to Born, Einstein “tries to find a theory of the classical type of such refined structure that it contains the essential features of atomic and quantum theory as consequences. There are at present few physicists who share this view.” The review by W. H. McCrea reflected his own modesty. Although more cautious, he was very clear:¹³⁴

“Nevertheless, what has been written here shows how much of the subsequent formulation appears to be entirely arbitrary and how little of it has received physical interpretation. It is clear that a tremendous amount of investigation is required before others than the eminent author himself are enabled to form an opinion of the significance of this work” ([420 ], p. 129).

The “eminent author” himself confessed in a letter to Max Born of 12 December 1951:

“Unfortunaletly, the examination of the theory is much too difficult for me. After all, a human being is only a poor wretch!” ([168 *], p. 258).¹³⁵

Also, W. Pauli commented on this 3rd edition. A correspondent who inquired about “the prospects of using Einstein’s new unified field theory as an alternative basis for quantum electrodynamics” obtained a demoralizing answer by him in a letter of 4 July 1950:

“Regarding Einstein’s ‘unified’ field theory I am extremely skeptical. It seems not only arbitrary to add a symmetrical and an anti-symmetrical tensor together but there is also no reason why Einstein’s system of equations should be compatible (the counting of identities between these equations given in the appendix of the new edition turned out to be incorrect). Certainly no work on similar lines will be done in Zürich.” ([490 *], p. 137–138)

Einstein’s former assistant and co-author Leopold Infeld sounded quite skeptical as well when he put the focus on equations of motion of charges to follow from “the new Einstein theory”. By referring to the 3rd Princeton edition of The Meaning of Relativity he claimed that, in 1st approximation, “the equations of motion remain Newtonian and are uninfluenced by the electromagnetic field.” But he offered immediate comfort by the possibility “that this negative result is no fault of Einstein’s theory, but of the conventional interpretation by which it was derived” [303 ].

9.2.3 Compatibility defined more precisely

In a long paragraph (§7) of Appendix II, of this 3rd Princeton edition, Einstein then asked about the definition of what he had termed “compatibility”. This meant that “the manifold of solutions” of the different systems of field equations “is extensive enough to satisfy the requirements of a physical theory” ([150 *], p. 150), or put differently, the field equations should not be overdetermined. In view of the “System I”’s containing four more equations, i.e., 84, than the 80 unknowns, this might become a difficulty. Starting from the Cauchy problem, i.e., the time development of a solution off an initial hypersurface, he counted differential equations and the variables to be determined from them.¹³⁶ To give an example for his method, he first dealt with general relativity and obtained the result that the general solution contains four free functions of three (spacelike) coordinates – “apart from the functions necessary for the determination of the coordinate system” ([150 *], p. 155). The corresponding results for “Systems Ia, (I)” according to him turned out to be: 16, (6) arbitrary functions of three variables, respectively. In case “System I” should turn out to be too restrictive to be acceptable as a physical theory, Einstein then would opt for the “weak field equations” (“System Ib”). “However, it must be admitted that in this case the theory would be much less convincing than if system (I) can be preserved” ([150 *], p. 160).

This discussion calls back into memory the intensive correspondence Einstein had carried on between 1929 and 1932 with the French mathematician E. Cartan on an equivalent problem within the theory of teleparallelism, cf. Section 6.4.3 of Part I. At the time, he had asked whether his partial differential equations (PDEs) had a large enough set of solutions. Cartan had suggested an “index of generality” $s0$ for first-order systems in involution which, essentially, gave the number of arbitrarily describable free data (functions of 3 spacelike variables) on an initial hypersurface ( $t = t0$ ). He calculated such indices, for Maxwell’s equations with currents to be $s0 = 8$ , and without $s0 = 4$ , for Einstein’s vacuum field equations $s0 = 4$ , (in this case 4 free functions of 4 variables exist¹³⁷ ), and of course, for Einstein’s field equations in teleparallelism theory. Note that Maxwell’s and Einstein’s vacuum field equations according to Cartan exhibit the same degree of generality. It had taken Cartan a considerable effort of convincing Einstein of the meaningfulness of his calculations also for physics ([116 ], pp. 114, 147, 174). Already in the 3rd Princeton edition of The Meaning of Relativity, in Appendix II [150 *], Einstein tried to get to a conclusion concerning the compatibility of his equations by counting the independent degrees of freedom but made a mistake. As mentioned above, W. Pauli had noticed this and combined it with another statement of his rejection of the theory. Compatibility was shown later by A. Lichnerowicz [369 *] (cf. Section 10.5).

It is unknown whether Einstein remembered the discussion with Cartan or had heard of Pauli’s remark, when he tackled the problem once more; in the 4th Princeton edition of his book The Meaning of Relativity, Appendix II [156 *] Cartan’s name is not mentioned. In the meantime, Mme. Choquet-Bruhat, during her stay at Princeton in 1951 and 1952, had discussed the Cauch-problem with Einstein such that he might have received a new impulse from her. To make the newly introduced concept of “strength” of a system of PDE’s more precise, he set out to count the number of free coefficients of each degree in a Taylor expansion of the field variables; if all these numbers are non-negative, he called the system of PDE’s “absolutely compatible”. He then carried out a calculation of the number of coefficients $Ωn$ remaining “free for arbitrary choice” for the free wave equation, Maxwell’s vacuum equations, the Einstein vacuum equations, and particular field equations of UFT. Let us postpone the details and just list his results.¹³⁸ For the wave equation $(n + 3) Ωn = 6n n$ . According to him “the factor $6n$ gives the fraction of the number of coefficients (for the degree $n ≫ 1$ ), which remain undetermined by the differential equation” ([156 *], p. 152). Similarly, he found for the Maxwell vacuum equations, $( ) n + 3 12 Ωn = n n$ . Einstein noted that by introducing the vector potential $A i$ , and taking into account the Lorentz gauge, i.e., by dealing with

the counting led to $( ) Ω = n + 3 18 n n n$ . He ascribed the increase in the number of freely selectable coefficients (loss of strength) to the gauge freedom for the vector potential. For the Einstein vacuum equations, he obtained $(n + 3) Ωn = 15n n$ . In applying his method to the “weak” field equations of Section 9.2.2, Einstein arrived at $( ) Ω = n + 3 45 n n n$ . In comparing this to the calculation for other field equations in UFT, he concluded that “the natural generalization of the gravitational equations in empty space” is given by the “weak” field equations ([156 *], p. 164).

Obviously, Einstein was not satisfied by his calculations concerning the “strength” of PDE’s. In the 5th Princeton edition of his book The Meaning of Relativity, Appendix II [158 *],¹³⁹ he again defined a system of PDE’s as “absolutely compatible” if, in a Taylor expansion of the field variable $Φ$ , the number of free $n$ -th order coefficients $----∂n----- ∂x1 ∂x2...∂xnΦ |P$ , at a point $P$ does not become negative. He then gave a name to the number of free coefficients calculated before: he called it “coefficient of freedom”. The larger this coefficient is, the less acceptable to him is the system of PDE’s. Let $p$ denote the number of field variables, $s$ the number of field equations of order $q$ , and $w$ the number of identities among the field equations in the form of PDE’s of order $q′$ . Then, writing $z$ in place of the previous $Ωn$ , his formulas could be condensed into:

( ) [ ( ) ( ′) ] z = p n + 3 − s n + 3 − q − w n + 3 − q (280 ) n n − q n − q′ ( ) { z z } = n + 3 a + -1 + -2-+ ... , (281 ) n n n2

with $! a = p − s + w ≥ 0$ required for absolute compatibility; $′ z1 = 3(qs − q w)$ is the “coefficient of freedom.”¹⁴⁰ Again, Einstein calculated $z1$ for several examples, among them Maxwell’s vacuum field equations in flat space-time:

with the identities

Here $p = 6,s = 8,q = 1,q′ = 2,w = 2$ . In agreement with the result from the previous edition, the calculation led him to $a = 0, z = 12 1$ . In contrast, it turned out that the “coefficient of freedom” for the gravitational vacuum field equations in general relativity, in the 2nd calculation became smaller, i.e., $z1 = 12$ ([158 *], p. 139). Fortunately, now both equations have the same “index of generality” (Cartan) and the same “coefficient of freedom” (Einstein). Likewise, Einstein found $z1 = 42$ instead of the previous $z1 = 45$ for the “weak”, and $z1 = 48$ for a concurring system with transposition invariance such that he again adopted the “weak” one as before. (280 *) with its values for $a,z 1$ is not yet the correct formula. Such a formula was derived for involutive, quasi-linear systems of PDEs by a group of relativists around F. Hehl at the University of Cologne at the end of the 1980s ([595 *], Eqs. (2.9), (2.10), [596 ]). By their work, also the relation between the Cartan coefficient of generality and Einstein’s coefficient of freedom has now been provided. According to M. Sué ([595 ], p. 398), it seems that Einstein’s coefficient of freedom is better suited for a comparison of the systems investigated than Cartan’s degré d’arbitraire. In mathematics, a whole subdiscipline has evolved dealing with the Cartan–Kähler theorem and the Cartan-Characters for systems of PDE’s. Cf. [57 ].

The fact that Einstein had to correct himself in his calculations of the “coefficient of freedom” already may raise a feeling that there exists a considerable leeway in re-defining field variables, number and order of equations etc. Moreover, he did not prove the independence of the relative order of “strength” for two PDE’s from mathematical manipulations affecting the form of the equations but not their physical content: remember (279 *), (282 *), (283 *). Regrettably, it is to be noted that Einstein’s last attempt to gain a reliable mathematical criterion for singling out one among the many possible choices for the field equations in UFT remained unconvincing.¹⁴¹

9.2.4 An account for a general public

Following an invitation by the editors of Scientific American to report on his recent research, Einstein made it clear that he would not give

“[…] a detailed account of it before a group of readers interested in science. That should be done only with theories which have been adequately confirmed by experiment.” ([152 *], p. 14.) ¹⁴²

He then talked about the epistemological basis of science, men’s curiosity and passion for the understanding of nature before touching upon problems connected with a generalization of general relativity. Two questions were very important, though not yet fully answered: the uniqueness of the field equations and their “compatibility”. He then sketched the three systems of field equations obtained, here denoted $E3$ (System I), and $E1, E2$ (Systems Ia, Ib). He again stressed that $E3$ (System I) “is the only really natural generalization of the equations of gravitation”. However, it was not a compatible system as were the other two.

“The skeptic will say: ‘It may well be true that this system of equations is reasonable from a logical standpoint. But this does not prove that it corresponds to nature.’ You are right, dear skeptic. Experience alone can decide the truth. Yet we have achieved something if we have succeeded in formulating a meaningful and precise question.” ([152 *], p. 17.)

Painstaking efforts and probably new mathematics would be required before the theory could be confronted with experiment. The article is illustrated by a drawing of Einstein’s head by the American artist Ben Shahn ([152 ], p. 17).

There were not only skeptics but people like Dr. C. P. Johnson in the Chemistry Department of Harvard University who outrightly criticized “Dr. Albert Einstein’s recent unified field theory” [311 ]. He pointed out that the theory permits a class of similarity solutions, i.e., with $gik(xl)$ also $′gik(kxl)$ solves the field equations. For a system of two charged and one uncharged massive bodies he qualitatively constructed a contradiction with Coulomb’s law. Einstein replied with a letter printed right after Johnson’s by stating that if solutions depending upon a continuous parameter exist, “then the field equations must prevent the coexistence within one system of such elementary solutions pertaining to arbitrary values of their parameters.” The underlying reason was that “for a system of field equations to be acceptable from a physical point of view, it has to account for the atomic structure of reality.” This would entail that regions of space corresponding to a ‘particle’ have discrete masses and charges. The coexistence of similar solutions “in one and the same world” would make the theory unacceptable [161 ]. As we shall see, the situation of Einstein’s UFT was worse: it did not lead to Coulomb’s law – at least not in the lowest approximations. See Sections 9.3.3, 9.6, and Section 10.3.2.

Nevertheless, Einstein remained optimistic; in the same letter to Max Born, in which he had admitted his shortcoming vis-a-vis the complexity of his theory, he wrote:

“At long last, the generalization of gravitation from a formal point of view now is fully convincing and unambiguous – unless the Lord has chosen a totally different way which no one can imagine.”¹⁴³

His Italian colleague Bruno Finzi was convinced that the final aim had been reached:

“[…] all physical laws laws of the macrocosm reduce to two geometrical identities […]. Therefore, the game is over, and the geometric model of the macrocosm has been constructed.”([200 ], p. 83)¹⁴⁴

However, at the end of his article, Finzi pointed out that it might be difficult to experimentally verify the theory, and thought it necessary to warn that even if such an empirical base had been established, this theory would have to be abandoned after new effects not covered by it were observed.

9.3 Einstein 1953

In the fourth edition of Einstein’s The Meaning of Relativity, invariance with regard to $λ$ -transformations was introduced as a new symmetry principle (cf. (52 *) of Section 2.2.3). Also transposition invariance is now claimed to be connected to the “indifference of the theory” (UFT) with respect “to the sign of electricity” ([156 *], p. 144). This interpretation rests on Einstein’s identification of the electric current density with $g + g + g [ik],l [kl],i [li],k$ . Einstein still grappled with the problem of how to set up a convincing system of field equations. As in the previous edition, he included (269 *) as an “a priori condition” in his variational principle by help of a (1-form)-multiplier $σi$ . However, he renounced using $Γ i = 0$ . Without specification of the Lagrangian $ℋ$ , from $∫ ∫ δ ℋd τ = (Vˆikδ Γ l + Wik δˆgik)dτ = 0 l ik$ the field equations follow – without use of the multiplier-term – to be:

– with use of the multiplier-term –

Elimination of the multiplier $σi$ led to the equations (named “System II” by Einstein)

A paragraph then was devoted to the choice of the proper Lagrangian. Einstein started from (196 *) and removed a divergence term in $Her K ikˆgik −$ . After variation (with inclusion of the multiplier term) the ensuing field equations, Einstein’s “system IIa”, were:

ˆg = 0, (287 ) i+k−||l Γ i = 0, (288 ) K− (ik) = 0, (289 ) K− [ik],l + K− [kl],i + K− [li],k = 0, (290 )

i.e., a version of the Einstein–Straus weak field equations. The road to the weak field equations (287 *) – (290 *) followed here still did not satisfy Einstein, because in it the skew-symmetric parts of both the metric and the connection could also be taken to be purely imaginary. In order to exclude this possibility and work with a real connection, he introduced $λ$ -transformations and presented a further derivation of the field equations. He set up a variational principle invariant under the $λ$ -transformation and arrived at the same system of field equations as before. The prize payed is the exclusion of a physical interpretation of the torsion tensor.

In a discussion covering twelve pages, Einstein again took up the question of “compatibility” from the previous edition and introduced the concept of the “strength” of a system of differential equations in order to bolster up his choice of field equation. A new principle applying to physical theories in general is put forward: “The system of equations is to be chosen so that the field quantities are determined as strongly as possible” ([156 *], p. 149). In Section 9.2.3, a detailed discussion of this new principle has been given such that we need not dwell on it. The paucity of physical input into Einstein’s approach to UFT becomes obvious here. May it suffice to say that according to the new principle the weak field equations (277 *), (278 *) are called “stronger” than the strong field equations (268 *). However, this has lead to the misleading labeling of the system II as the “strong system” [704 *]. The relation of geometrical objects to physical observables remained unchanged when compared to the 3rd edition ([150 *]). Einstein saw a close relationship to Maxwell’s theory only in the linear approximation where “the system decomposes into two sets of equations, one for the symmetric components of the field, and the other for the antisymmetric components.[…] In the rigorous theory this independence no longer holds.” ([156 *], p. 147.)

Both, the concept of “strength” of a system of differential equations and the concluding §5 “General remarks concerning the concepts and methods of theoretical physics” point to Einstein’s rather defensive position, possibly because of his feeling that the particular field equations of unified field theory for which he strove so hard, rested on flexible ground. This was due not only to the arbitrariness in picking a particular field equation from the many possibilities, but also to the failure of the theory to include a description of concepts forming an alternative to quantum theory. Einstein stuck to the classical field and rejected both de Broglie’s “onde pilote”, and Bohm’s attempt away from the statistical interpretation of the wave function. At the very end of his Meaning of Relativity he explained himself in this way:

“[…] I see in the present situation no possible way other than a pure field theory, which then however has before it the gigantic task of deriving the atomic character of energy. […] We are […] separated by an as yet insurmountable barrier from the possibility of confronting the theory with experiment. Nevertheless, I consider it unjustified to assert, a priori, that such a theory is unable to cope with the atomic character of energy.” ([156 *], p. 165.)

An indirect answer to this opinion was given by F. J. Dyson in an article on “field theory” in the Scientific American. He claimed “that there is an official and generally accepted theory of elementary particles, known as the ‘quantum field theory’.” According to him, while there still was disagreement about the finer details of the theory and its applications:

“The minority who reject the theory, although led by the great names of Albert Einstein and P. A. M. Dirac, do not yet have any workable alternative to put in its place.” ([137 ], p. 57.)

Such kind of sober judgment did not bother The New York Times which carried an almost predictable headline: “Einstein Offers New Theory to Unify Law of the Cosmos.” ([469 ], p. 350.)

Privately, in a letter to M. Solovine of 28 May 1953, Einstein seemed less assured. Referring to the appendix of this 4th edition of “The Meaning of Relativity”, he said: “[…] Of course, it is the attempt at a theory of the total field; but I did not wish to give the thing such a demanding name. Because I do not know, whether there is physical truth in it. From the viewpoint of a deductive theory, it may be perfect (economy of independent concepts and hypotheses).” ([160 *], p. 96).¹⁴⁵

9.3.1 Joint publications with B. Kaufman

In the Festschrift for Louis de Broglie on the occasion of his 60th birthday (15.8.1952) organized by M.-A. Tonnelat and A. George, Einstein again summarized his approach to UFT, now in an article with his assistant Bruria Kaufman [172 *]. In a separate note, as kind of a preface he presented his views on quantum theory, i.e., why he still was trying “[…] to solve the quantum riddle on another path or, to at least help for preparing such a solution.” ([155 *], p. 4.)¹⁴⁶ He expressed his well-known epistemological position that something like a “real state” of a physical system exists objectively, independent of any observation or measurement. A list of objections to the majority interpretation of quantum theory was given. At the end of the note, a link to UFT was provided:

“My endeavours to complete general relativity by a generalization of the gravitational equations owe their origin partially to the following conjecture: A reasonable general relativistic field theory could perhaps provide the key to a more perfect quantum theory. This is a modest hope, but in no way a creed.” ([155 ], p. 14.)¹⁴⁷

As in the 4th edition of his book [156 *], the geometrical basics were laid out, and one more among the many derivations of the weak field equations of UFT given before was presented. At first, it looked weird, but in referring to a result of the “researches of E. Schrödinger” (without giving a reference, though) Einstein & Kaufman took over Schrödinger’s “star”-connection:

introduced in ([552 ], p. 165, Eq. (10)). For it, $∗Γ k = 0$ holds which leads to simplifications. Under a $λ$ -transformation the Ricci curvature is not invariant [cf. (87 *)]. In order to make the variational principle invariant, due to
$∫ ∫ ∫ [il] δ d4x ˆgikK ik(Γ ) = δ[ d4x ˆgikK ik(∗Γ ) + 2 d4x ˆg ,lλi ] − −$ , as an ad-hoc- (or as Einstein & Kaufman called it, an a priori-) condition is needed:

The further derivation of the field equations led to the known form of the weak field equations:

g = 0, (293 ) ik ||∗l K = 0, (294 ) ∗ (ik) K [ik],l + K [kl],i + K [li],k = 0. (295 ) ∗ ∗ ∗

Here, the covariant derivative refers to the connection $∗Γ$ and $K ≡ K (∗Γ ). ∗ (ik) − ik$

In addition, a detailed argument was advanced for ruling out the strong field equations. It rests partially on their failure to guarantee the possibility to superpose weak fields. The method used is a weak-field expansion of the metric and the affine connection in a small parameter $𝜖$ :

After expanding the field equations up to 2nd order, the authors came to the conclusion that the “strong equations” strongly constrain “the additivity of symmetric and antisymmetric weak fields. It seems that by this any usefulness of the ‘strong system’ is excluded from a physical point of view.” ([172 *], p. 336.) ¹⁴⁸

In an appendix to the paper with title “Extension of the Relativistic Group” [172 *], Einstein combined the “group” of coordinate transformations with the $λ$ -transformations to form a larger transformation group $U$ . (cf. the letter to Besso mentioned in Section 9.1.) He then discussed the occurring geometric objects as representations of this larger group and concluded: “The importance of the extension of the transformation group to $U$ consists in a practically unique determination of the field equation.”([172 *], p. 341.)¹⁴⁹

The next paper with Bruria Kaufman may be described as applied mathematics [173 *]. Einstein returned to the problem, already attacked in the paper with E. Straus, of solving (30 *) for the connection in terms of the metric and its derivatives. The authors first addressed the question: “What are necessary and sufficient conditions for constant signature of the asymmetric metric-field to hold everywhere in space-time?” At first, it was to be shown “that the symmetric part $g(ik)$ of the tensor $gik$ is a Riemannian metric with constant signature”. For a proof, the conditions $det(g (ik)) ⁄= 0$ and a further algebraic inequality were needed. In addition, the connection $Γ$ , calculated from $ˆg ik||l = 0 +−$ , had to be finite at any point and “algebraically determined”. This is meant in the sense of interpreting $ˆgi+k−||l = 0$ as an inhomogeneous linear equation for the components of $Γ$ . The situation was complicated by the existence of the algebraic invariants of the non-symmetric $gik$ as well as by the difficulty to solve for the connection as a functional of the metric tensor. Although not necessary for a solution of the field equations, according to the authors it is “of interest to give a closed expression for the $Γ$ as a function of the $gik$ and its first derivatives.” This problem had been addressed before and partial results achieved by V. Hlavatý [258 , 260 *], and S. N. Bose [52 *].¹⁵⁰ The papers by M.-A. Tonnelat published earlier and presenting a solution were not referred to at all [622 *, 623 *, 630 *, 629 *].

In the sequel, $g (ik)$ is given a Lorentz signature. By using special coordinates in which $gki = ρi δki$ (no summation on the index i), it can be seen that “ $ρ1 = 1- ρ2$ is on the unit circle from which the point $− 1$ has been excluded”; the other two roots $-1 ρ3 = ρ4$ are positive. It is shown in the paper that among the three algebraic invariants to be built from $gk = gksgsi i$ only two are independent:

As a “sufficient condition for regularity and unique algebraic determination of $l Γ ik$ ” the authors derive ([173 *], p. 237):

In a lengthy calculation filling six pages, a formal solution to the compatibility equation (30 *), seen as an algebraic equation for $Γ$ is then presented: “[…] it is cumbersome, and not of any practical utility for solving the differential equations” ([173 ], p. 238).

9.3.2 Einstein’s 74th birthday (1953)

Einstein agreed to let his 74th birthday be celebrated with a fund-raising event for the establishing of the Albert Einstein College of Medicine of Yeshiva University, New York. Roughly two weeks later, according to A. Pais The New York Times carried an article about Einstein’s unified field theory on the front page [471 ]. It announced the appearance of the 4th Princeton edition of The Meaning of Relativity with its Appendix II, and reported Einstein as having stated that the previous version of 1950 of the theory had still contained one important difficulty. According to him: “[…] This last problem of the theory now finally has been solved in the past months.”¹⁵¹ Probably, this refers to Einstein’s new way of calculating his “coefficient of freedom” introduced for mirroring the “strength” of partial differential equations. In a letter to Carl Seelig of 14 September 1953, Einstein tried to explain the differences between the 3rd and the 4th edition of The Meaning of Relativity:

“A new theory often only gradually assumes a stable, definite form when later findings allow the making of a specific choice among the possibilities given a priori. This development is closed now in the sense that the form of the field laws is completely fixed. – The theory’s mathematical consistence cannot be denied. Yet, the question about its physical foundation still is completely unsettled. This follows from the fact that comparison with experience is bound to the discovery of exact solutions of the field equations which seems impossible at the time being.”¹⁵² ([570 ], p. 401–402)

9.3.3 Critical views: variant field equation

Already in 1950, Infeld had pointed to the fact that the equations of motion for particles following from Einstein’s UFT (weak field equations), calculated in the same way as in general relativity, did not lead to the Lorentz equations of motion [304 *]. This result was confirmed by Callaway in 1953. Callaway identified the skew part of the fundamental tensor with the electromagnetic field and applied a quasistatic approximation built after the methods of Einstein and Infeld for deriving equations of motion for point singularities. He started from Einstein’s weak field equations and showed that (208 *) could not influence the equations of motion. His conclusion was that he could reduce “Einstein’s new unified field theory to something like Maxwell’s equations in a sufficiently low approximation”, but could not obtain the Lorentz equation for charged particles treated as singularities in an electromagnetic field [69 *].

In order to mend this defect, Kursunŏglu modified the Einstein–Straus weak field equations by beginning with the identity (257 *) and adding another identity formed from the the metric $gij = aij + ikij$ and its 1st derivatives only [342 , 343 *]:¹⁵³

Kursunŏglu’s ensuing field equations were:

ˆg [il],l = 0, (300 ) 2 K− (ik) + p (hij − bij) = 0, (301 ) 2 K− [ik],l + K− [kl],i + K− [li],k + p (k[ik],l + k[kl],i + k [li],k) = 0, (302 )

with $p$ real or imaginary, and¹⁵⁴

where $bij$ is the inverse of $bij$ . $Iikl := k[ik],l + k[kl],i + k [li],k$ is connected with the electrical 4-current density $r J$ through $r Iikl := 𝜖iklrJ$ .

In fact, as Bonnor then showed in the lowest approximation (linear in the gravitational, quadratic in the electromagnetic field), the static spherically symmetric solution contains only two arbitrary constants $e,m$ besides $2 p$ which can be identified with elementary charge and mass; they are separately selectable [33 *]. However, in place of the charge appearing in the solutions of the Einstein–Maxwell theory, now for $e2$ the expression $e2p2$ , and for $e, mpe2-$ occurred in the same solution. The definition of mass seemed to be open, now. For vanishing electromagnetic field, the solution reduced to the solution for the gravitational field of general relativity.

In a discussion concerning the relation of matter and geometry, viz. matter as a “source” of geometry or as an intrinsic part of it, exemplified by the question of the validity of Mach’s principle, J. Callaway tried to mediate between the point of view of A. Einstein with his unified field theory already incorporating matter, geometrically, and the standpoint of J. A. Wheeler who hoped for additional relations between matter and space-time fixing the matter tensor as in the case of the Einstein–Maxwell theory ([70 *], p. 779). Callaway concluded that “if the approach of field theory is accepted, it is necessary to construct a theory in which space-time and matter enter as equals.” But he would not accept UFT as an alternative to quantum theory.

9.4 Einstein 1954/55

The last paper with B. Kaufman was submitted three months before, and appeared three months after Einstein’s death [174 *]. In it, the authors followed yet two other methods for deriving the field equations of UFT. Although the demand for transposition-invariance was to play a considerable role in the setting up of the theory, in the end invariance under $λ$ -transformations became the crucial factor. In the first approach (§1), instead of the previously used connection $Γ$ , Einstein and Kaufman introduced another one, $Γ ∗$ , containing four new variables (a 1-form) $Λk$ “supernumerary to the description of the field”, and defined by:

During variation of the Lagrangian, $Γ ∗$ and $Λk$ were treated as independent variables; after the variation $Λk$ could be fixed arbitrarily (“normed”). This trick allowed that four variational field equations could replace four equations put in by hand as had been the equation $Γ k = 0$ in earlier approaches. The following relation resulted:

such that the Lagrangian could be written as

After the variation with respect to $Γ ∗ l,ˆgik ik$ , and $Λk$ the choice $Λk = 2Γ k 3$ was made leading to $Γ ∗= 0 k$ . Although the wanted transposition-invariant field equations did come out, the authors were unhappy about the trick introduced. “The reason for our difficulties is that we require the field equations to be transposition-invariant, but we start out from a variational function which does not have that property. The question arises naturally whether we cannot find a form of the variational function which will itself be transposition-invariant, […].” ([174 *], p. 131.) In order to obtain such a Lagrangian, they replaced the connection by a quantity $U l ik$ called “pseudo-tensor”. It is transforming like a tensor only under linear coordinate transformations (§2 – §3):

From (307 *) we see that $Uilk$ does not transform like a connection. As a function of $Uilk$ the Ricci tensor:

is transposition-invariant, i.e., $K− (Γ (U&tidle;))ik = K− (Γ (U))ki$ . With regard to a $λ$ -transformation (52 *) the “pseudo-tensor” $U$ transforms as:

A short calculation shows that $K− (Γ (U ))ik$ is invariant under (52 *).

As a Lagrangian, now $ℋ = ˆgikK (Γ (U ))ik −$ was taken. Variation with respect to the variables $U l,gˆik ik$ , i.e.,

led to the field equations:

1 1 Nˆ ik,s ≡ − ˆgik,s − ˆgrk(Uris − -δis)U rtt− ˆgir(U skr − -δksU trt) = 0, (311 ) 3 3 K− (Γ (U ))ik = 0, (312 )

with $K− (Γ (U))ik$ given by (308 *). Although the authors do not say it, Eqs. (311 *) and (312 *) are equivalent to the “weak” field equations (287 *) – (290 *). By inserting infinitesimal coordinate- and $λ$ -transformations into (310 *), five identities called “Bianchi-identities” result. Modulo the field equations,

holds as well. With the help of special infinitesimal coordinate transformations and the Bianchi-identities a “conservation law for energy and momentum” is derived:

where $ˆTk := (ˆgrsU k) i rs ,i$ .

The results of this paper [174 *] were entered into the 5th Princeton edition of The Meaning of Relativity, Appendix II [158 *].¹⁵⁵ In “A note on the fifth edition” dated December 1954, Einstein wrote:

“For I have succeeded – in part in collaboration with my assistant B. Kaufman – in simplifying the derivations as well as the form of the field equations. The whole theory becomes thereby more transparent, without changing its content” ([158 *], page before p. 1).¹⁵⁶

From a letter to his friend Solovine in Paris of 27 February 1955, we note that Einstein was glad: “At least, yet another significant improvement of the general theory of the gravitational field (non-symmetric field theory) has been found. However, the thus simplified equations also cannot be examined by the facts because of mathematical difficulties”. ([160 *], p. 138)¹⁵⁷ In this edition of The Meaning of Relativity’, he made a “remark on the physical interpretation”. It amounted to assign $ˆg[is,s]$ to the (vanishing) magnetic current density and $12ηiklmg [ik],l$ to the electric current density.

The paper with Kaufman ended with “Considerations of compatibility and ‘strength’ of the system of equations”, a section reappearing as the beginning of Appendix II of the 5th Princeton edition. The 16 + 64 variables $U ilk, ˆgik$ must satisfy the 16 + 64 field equations (311 *), (312 *). The argument is put forward that due to $λ$ -invariance (identification of connections with different $λ$ ) the 64 $Γ$ -variables were reduced to 63 plus an additional identity.

“In a system with no $λ$ -invariance, there are 64 $Γ$ and no counterbalancing identity. This is the deeper reason for the relative weakness of systems which lack $λ$ -invariance. We hold to the principle that the stronger system has to be preferred to any weaker system, as long as there are no special reasons to the contrary.” ([174 *], p. 137.)

However, it is to be noted that in the 5th Princeton edition the $λ$ -transformation is reduced to $λk = ∂kλ$ (Eq. (5) on p. 148). In a footnote, Appendix II of the 4th Princeton edition of The Meaning of Relativity is given as a reference for the concept of “strength” of a system of differential equations (cf. Section 9.2.3). W. Pauli must have raised some critical questions with regard to the construction of the paper’s Lagrangian from irreducible quantities. In her answer of 28 February 1956, B. Kaufman defended the joint work with Einstein by discussing an expression $(α1gik + α2gki)Rik = γikRik$ : “Now the point is here that $gik$ was introduced in our paper merely as a multiplying function such as to make, together with $Rik$ , a scalar. Hence $ik g$ can just as well be this multiplier. The field equations we would get from this Lagrangian would be identical with the equations in our paper, except that they would be expressed in terms of $gik$ .” As to scalars quadratic in curvature she wrote: “ […] our paper does not claim that the system we give is 100% unique. In order to do that one would have to survey all possible additional tensors which could be used in the Lagrangian. We only considered the most ‘reasonable’ ones.” ([492 ], pp. 526–527.)

Until 1955, more than a dozen people had joined the research on UFT and had published papers. Nevertheless, apart from a mentioning of H. Weyl’s name (in connection with the derivation of the “Bianchi”-identities) no other author is referred to in the paper. B. Kaufman was well aware of this and would try to mend this lacuna in the same year, after Einstein had passed away.

At the “Jubilee Conference” in Bern in July 1955,¹⁵⁸ based on her recent work with A. Einstein [174 *], B. Kaufman gave an account “[…] of the logical steps through which one goes when trying to set up this generalization”, i.e., of general relativity to the “theory of the non-symmetric field” ([322 *], p. 227). After she presented essential parts of the joint paper with Einstein, Kaufman discussed its physical interpretation and some of the consequences of the theory. As in [148 *, 150 *], and [156 *], the electric current density is taken to be proportional to $g[ik],l + g[kl],i + g[li],k$ . From this identification, transposition invariance receives its physical meaning as showing that “all equations of the theory shall be invariant under a change of the sign of electric charge” ([322 ], p. 229). With (252 *), i.e., $ˆg[ik,]k = 0$ , holding again in the theory, $ˆgi4$ (with $i = 1,2,3)$ is identified with the components of the magnetic field. In the linear approximation, the field equations decompose into the linear approximation of the gravitational field equations of general relativity and into the weaker form of Maxwell’s equations already shown in (210 *), (211 *) of Section 7.3.

In the section “Results in the theory” of her paper, Kaufman tried to sum up what was known about the “theory of the non-symmetric field”. Both in terms of the number of papers published until the beginning of 1955, and of researchers in UFT worldwide, she did poorly. She mentioned Schrödinger, Hlavatý, Lichnerowicz and M.-A. Tonnelat as well as one or two of their collaborators, and some work done in Canada and India. The many publications coming from Italian groups were neglected by her as well as contributions from Japan, the United States and elsewhere which she could have cited. Nevertheless, in comparison with Einstein’s habit of non-citation, her references constituted a “wealth” of material. Of the few general results obtained, Lichnerowicz’ treatment of the Cauchy initial value problem for the weak field equations of UFT and his proof that a unique solution exists seems to be the most important [369 *]. Unfortunately, his proof, within general relativity, that static, regular solutions behaving asymptotically like a Schwarzschild point particle (with positive mass) are locally Euclidean, could not be carried over to UFT. This was due to the complications caused by the field equations (311 *), (312 *). While (311 *) could be solved, in principle, for the $U ikl$ as functions of $ˆgik,ˆgik ,l$ , its subsequent substitution into (312 *) led to equations too complicated to be solved – except in very special cases. In his summary of the conference, Pauli mocked Kaufman’s report:

“We have seen how Einstein and Mrs. Kaufman struggled heroically […], and how this fight has been led with the particular weapon of the $λ$ -transformation. Certainly, all this is formally very correct; however, I was unable to make sense of the $λ$ -transformations, either physically or geometrically.”[486 *]¹⁵⁹

The search for solutions of the weak field equations had begun already with exact spherically symmetric, static solution derived by a number of authors (cf. [475 *, 31 *, 32 *]; see Sections 8.3, and 9.6).

In a final section, B. Kaufman discussed two alternatives to what she now called “Einstein’s theory” for the first time. The first is Schrödinger’s purely affine version of the theory as presented in his book [557 *]. His field equations replacing (289 *), (290 *) were (cf. Section 8.1, Eq. (237 *):

Here, $λ$ plays the role of cosmological constant. At first, in the affine theory, $g ik$ is defined by the l.h.s. of (315 *), but then this equations is read like an Einstein equation for the metric.

In a note added for the reprint in 1954 of his book, Schrödinger warned the readers of his chapter on UFT that he did not regard his unification of gravitation and electromagnetism:

“[…] as anything like a well-established theory. It must be confessed that we have as yet no glimpse of how to represent electrodynamic interaction, say Coulomb’s law. This is a serious desideratum. On the other hand we ought not to be disheartened by proofs, offered recently by L. Infeld, M. Ikeda and others, to the effect, that this theory cannot possibly account for the known facts about electrodynamic interaction. Some of these attempts are ingenious, but none of them is really conclusive.” ([557 *], reprint 1954, p. 119.)

9.5 Reactions to Einstein–Kaufman

Schrödinger found the paper by Einstein and Kaufman in the Festschrift for L. de Broglie [172 ] “very important” and set out to draw some consequences. In particular, by using the approximation-scheme of Einstein and Kaufman, he showed that “the electric current-four-vector is in general different from zero throughout the field” ([559 *], p. 13). In the strong field equations, $K [ik] = 0 −$ led to the vanishing of the electric current density. Dropping the so-called cosmological term for convenience, Schrödinger now wrote Eq. (316 *) of his “weak” field equations in the form:

with a free vector-variable $Xi$ .¹⁶⁰ Besides obtaining, in first approximation, Einstein’s vacuum field equations of general relativity and one set of Maxwell’s equations, he gave as the second set:

From this he concluded that “the curl of the current is essentially the dual of the curl of $Γ i$ ” (his notation for $X i$ ). Here, $g = η + g + g + ... ik ik 1ik 2ik$ .

In 2nd order, the charge-current tensor was defined by $sijk = g[ij],k + g [jk],i + g[ki],j 2 2 2 2$ , and the wave equation then $ηrs∂r∂sg [ik] = sijk + ηrm ηsng[rs]B nmik 2 2 1 1$ . $Bnmik 1$ is a linear combination of the 2nd derivatives of $g1ik$ . However, Schrödinger rejected this equation: “it is not invariant” ([559 *], p. 19). Since 1952, Cornelius Lanczos had come to Dublin, first as a visiting, then as a senior professor, and, ultimately, as director at the Dublin Institute for Advanced Studies. In his paper, Schrödinger acknowledged “discussions with my friend professor Cornel Lanczos” ([559 ], p. 20).

In the Festschrift on the occasion of de Broglie’s 60th anniversary, published only in 1953, C. F. von Weizsäcker expressed his opinion clearly that:

“[…] in the future, no reason exists for connecting the metric more closely to the electromagnetic field, and perhaps also to the meson field.” ([680 ], p. 141.)¹⁶¹

One year later, consistent with this, and with Einstein’s death “in April 1955, Schrödinger became quite depressed, for he was now convinced that his unified field theory was no longer tenable” ([446 *], p. 326). In any case, there is no further published research on UFT by him.

M. S. Mishra also studied Einstein’s last publication written together with B. Kaufman [174 ] and solved (311 *) for the connection. He obtained M.-A. Tonnelat’s result (364 *) of Section 10.2.3 [433 ]. Instead of beginning with (311 *) and (312 *) as Einstein and Kaufman had done, he then introduced “another set of field equations” by taking

with the contracted curvature tensor which is transposition-symmetric $S := U s − U tU s+ 1U tU r ik ik,s is tk 3 it rk$ . The solution to (319 *), (320 *) is given as:

where $Slmn$ is the torsion tensor, and $Sm (Γ )$ the torsion vector. Mishra then linearized the metric and showed the result to be equivalent to the linearized Einstein–Straus equations (cf. Section 7.3) In the same paper, Mishra suggested another set of field equations by starting from the transposed Ricci tensor and making it transposition invariant in the same manner as Einstein and Kaufman did in their case.

In a joint paper with M. L. Abrol, also directed to the Einstein–Kaufman version of Einstein’s theory, Mishra claimed: “It is shown […] that Infeld’s method [cf. [304 ]] of approximation, to find the equations of motion of charged particles from the system of field equations, fails in this particular theory” [437 ]. This was due to some unknown terms in the 2nd and 3rd order of the approximation. After a modification of the field equations according to the method of Bonnor [cf. [34 *]], the Coulomb force appeared in 4th order.

9.6 More exact solutions

9.6.1 Spherically symmetric solutions

A hope for overcoming the difficulty of relating mathematical objects from UFT to physical observables was put into the extraction of exact solutions. In simple cases, these might allow a physical interpretation by which the relevant physical quantities then could be singled out. One most simple case with high symmetry is the static spherically symmetric (sss) field. Papapetrou’s solution of Section 8.3 soon was generalized to the case $v ⁄= 0,w ⁄= 0$ by Wyman [709 *] which means that both, electric and magnetic fields, are now present. Wyman’s three different solutions of the weak field equations contain one arbitrary function of the radial coordinate $r$ , each. Wyman questioned the physical interpretation of $g(ij) = hij$ and $g[ij] = kij$ as standing for the gravitational field and the electromagnetic field, respectively. He built another expression:

with $u = √--hi--- i hsthsht$ and $h = h ksaS b i ab si$ . If $a ij$ is chosen as a metric, then the unique solution of general relativity in this sss case, e.i., the Schwarzschild solution, results. Although Wyman’s construction was very artificial, it clearly exemplified the unsurmountable impediment to UFT: the “embarras de richesses” in mathematical objects.

Wyman also questioned the boundary condition used at spacelike infinity: $limesr → ∞ gab = ηab$ , where $ηab$ is the Minkowski metric. By looking at his (or Papapetrou’s) sss solutions, he showed that different boundary condition could be set up leading to different solutions: $vr2 → 0$ for $r → ∞$ or $v → 0$ for $r → ∞$ .

Apparently, this left no great impression; the search for sss solutions continued. As two sets of field equations were competing against each other, Einstein’s (and Schrödinger’s) weak and strong equations (with or without cosmological constant $λ$ ), we must distinguish the solutions suggested. In the case of sss fields, only one additional field equation, e.g., $R [23] = 0$ , separates the strong from the weak equations. For the strong equations with $λ = 0$ , Bonnor [31 ] obtained the general exact solution in the cases for which either a magnetic or an electric field is present. He also generalized Papapetrou’s solution for the weak equations to the case where the function $v$ is real or imaginary. The solutions describe spread out charges while the masses are banned into singularities. All solutions display an infinite set of “singular” surfaces between the radial coordinate $r = 2m$ and $r = ∞$ .¹⁶² For some time, after a note by Bandyopadhyay [8 ], who claimed that for the strong equations $m e = 0$ where $m, e$ are the parameters for mass and charge,¹⁶³ a discussion took place whether isolated massless magnetic monopoles could exist. Since 1948, Papapetrou and Schrödinger had changed the assignment of components of $kij$ to the electric and magnetic fields; now $g ,g ,g [23] [31] [12]$ stood for the electric field [479 ]. Ikeda, in a paper of 1955, claimed to have shown “that a single magnetic pole cannot exist in the Einstein new theory, as in the Maxwell theory” ([298 *], p. 272).¹⁶⁴ This result depended on Ikeda’s identification of the electromagnetic field with

where $h = det(− hab)$ . In 1960, Bandyopadhyay came back to the question and claimed that “the ‘stronger’ equations will not allow isolated magnetic poles with mass whereas the ‘weaker’ equations will allow the existence of such entities” ([10 ], p. 427). Bonnor’s second paper of 1952 dealt with the strong equations in the case $vw ⁄= 0$ . Again, the exact solutions described spread out charges of both signs with an infinity of singular surfaces. They were unphysical because they contained no parameter for the mass of the sources [32 *].

In her book, M.-A. Tonnelat discussed these solutions; her new contribution consisted in the calculation of the components of her connection $Δ$ – Schrödinger’s star connection (27 *) - and the Ricci $W (Δ)$ -tensor formed from it for the more general case of time-dependent spherically symmetric fields [629 ], ([632 *], p. 71, 73). By help of this calculation, her collaborator Stamatia Mavridès could present a general result: for $g[23] = 0,g[10] ⁄= 0$ (as the non-vanishing components of $kij$ ) only static exact spherically symmetric solutions do exist [402 ]. Later in Italy, F. De Simoni published another generalization of Wyman’s and Bonnor’s solutions for the weak field equations; he used the Ricci tensor of Einstein and Straus (73 *) made Hermitian, i.e., $Pij + P&tidle;ij$ . His paper is not referred to in Tonnelat’s book [114 ]. J. R. Vanstone mistakenly believed he had found time-dependent spherically symmetric solutions, but the time dependency can be easily removed by a coordinate transformation [668 ]. Also B. R. Rao had calculated some, but not all components of the connection for the case of a time-dependent spherically symmetric field but had failed to find a time-dependent solution [502 ].

Unfortunately, all this work did not bring further insight into the physical nature of the sss solutions. The only physically “usable” solution remained Papapetrou’s. He also proved the following theorem: “Spherically-symmetric solutions periodical in time of the “weak” field equations satisfying the boundary conditions $ˆgik → ηik$ for $r → ∞$ are, in 1st approximation, identical to solutions of the “strong”field equations” [478 ].

9.6.2 Other solutions

Still worse, in 1958 a sobering contribution from the Canadian mathematician Max Wyman and his German colleague Hans Zassenhaus cast doubt on any hope for a better understanding of the physical contents of UFT by a study of exact solutions. They investigated solutions of Einstein’s non-symmetrical UFT with vanishing curvature tensor: in this case weak and strong field equations coincide. Unlike for flat space-time in general relativity, a large class of solutions resulted; a situation which, according to the authors, “[…] merely adds to the confusion and indicates that the form of the theory is far from complete”. They went on: “However, as far as a satisfactory physical interpretation of such a theory goes, almost complete chaos seems to result.” ([710 *], p. 228.) Strong words, indeed, but not without reason:

“[…] for a theory based on a non-symmetric tensor an infinity of tensors of all orders exist. The only hope to extract from this maze the proper mathematical expressions to use for physical quantities would thus have to be physical in nature. So far no such physical assumptions have been put forward” ([710 ], p. 229).

In the paper, seven distinct solutions were displayed. For one special case, i.e.,

the standard interpretation ( $Fij = kij$ electromagnetic field, $hij$ gravitational potential) “would attribute the electromagnetic field to a distribution of charge along the infinite planes $y = ±x$ or along any of the equipotential $x2 − y2 = constant$ .” For a weak gravitational field, the Newtonian potential would be $V = 1(x2 + y2) 8$ “corresponding to a distribution of mass of density $1 2$ inside an infinite cylinder”. This is due to the approximated Newtonian equations of motion following from the geodesic equation for $hij$ . Hence, for this solution, mass and charge are unrelated.

This result casts into doubt much of the work on exact solutions independently of any specific assignment of mathematical objects to physical variables. It vindicated Schrödinger’s opinion that exact solutions were of useless for a better understanding of the particle-aspect of the theory; cf. the quotation at the end of Section 8.2. Nevertheless, the work of assembling a treasure of exact solutions continued. In 1954, it had still been supported by Kilmister & Stephenson in this way: ‘The true test of this theory [i.e., Einstein’s weak field equations] as an adequate description of the physical world must await exact solutions of the field equations” [331 ].

Einstein did not live to see the results of Wyman & Zassenhaus; now some of the non-singular exact solutions of the field equations of UFT he so much wished to have had, were at hand:

“The big difficulty [of UFT] lies in the lack of a method for deriving singularity-free exact solutions which alone are physically interesting. Yet the bit we have been able to calculate has strengthened my trust in this theory.” (Einstein to Pauli, April 1, 1948 quoted from [489 *])¹⁶⁵

How would he have dealt with the fact, unearthed in 1958, that such non-singular solutions not always offered a convincing physical interpretation, or even were unphysical?

The symmetry of so-called “1-dimensional” gravitational fields of general relativity, i.e., those for which the metric components depend on only a single coordinate, is high enough to try and solve for them field equations of UFT. In fact, already in 1951, Bandyopadhyay had found such a solution of the weak equations with $g ⁄= 0, g = 0,g = g [10] [23] 22 33$ and had taken it as describing an infinite charged plate [7 *]. In 1953, E. Clauser presented another such 1-dimensional field as a solution of the weak equations with $g22 ⁄= g33$ and saw it as representing a magnetostatic field [79 ]. B. R. Rao in 1959 generalized Bandyopadhyay’s solution to the case $g[10] × g[23] ⁄= 0$ without attempting to provide a physical interpretation [503 ].

Plane wave solutions of the weak and strong field equations of the form

with $σ, ρ$ functions of $x1,x2,x3 − x0$ have been given by Takeno [601 ].

9.7 Interpretative problems

Already up to here, diverse assignments of geometrical objects to physical quantities (observables) were encountered. We now assemble the most common selections.

a) Gravitational Field

From the fact that the exact, statical, spherically symmetric solution of the weak field equations derived by A. Papapetrou did not coincide asymptotically, or far from the assumed location of the point source at $r = 0$ , i.e., for $r → ∞$ , with the corresponding solution of the Einstein–Maxwell equations (Reissner–Nordström) [475 *], a discussion of the relation between geometrical objects and physical observables arose. Perhaps the metric chosen to describe the gravitational potential ought not to be identified with $hik$ ! Let the inverse of $gik$ be given by

From a study of the initial value problem, A. Lichnerowicz suggested the use of the inverse $lik = l(ik) ⁄= hik$ of $ik l$ as the genuine metric ([371 *], p. 288). Schrödinger had already worked with it. A related suggestion made by several doctoral students of M.-A. Tonnelat (J. Hély, Pham Tan Hoang, M. Lenoir) was to use $a ij$ with $a aks = δk is i$ and $∘ -- aij = h hij g$ as metric [250 , 271 *, 272 *], ([359 *], p. 92). In Section 9.6.1 we have seen that by another, if only very contrived definition of the metric, complete separation of the gravitational and electromagnetic fields could be achieved: the Schwarzschild solution could be made part of an exact solution of unified field theory [709 ]. The torsion tensor appeared in the definition of this metric.

b) Electromagnetic Field and Charge Currents

The same ambiguity arose for the description of the electromagnetic field: Einstein’s specification that it be connected to $kij = g[ij]$ was taken over by the majority ([147 *], p. 583). For dimensional reasons, this interpretation implies that a constant of dimension “length” will appear in the theory. In her discussion of two possibilities, St. Mavridès suggested $∘ -- aij = ghhij$ for the metric and $∘ -- sij = gkij h$ for the electromagnetic field [403 , 400 *, 401 *, 404 ]. The first choice was supported by Pham Tan Hoang [271 *, 272 *]. Although making the electrical field regular at $r = 0$ :

this choice did not fuse gravitational and electromagnetic fields any better. Mavridès’ choice was guided by a comparison with the Born–Infeld non-linear electrodynamics¹⁶⁶ [42 ], cf. also Section 5. The fundamental invariants of the electromagnetic field now are given by $pr qs hpqhrsm m$ and $√−-l √m--$ with $ij l = det(lij), m = det(m )$ ([641 *], p. 345); cf. Section 5. Mme. Tonnelat, in her books, also discussed in detail how to relate observables as the gravitational and electromagnetic fields, the electric current density, or the energy-momentum tensor of matter to the geometric objects available in the theory ([632 *], Chapter VI; [636 *, 382 *]; cf. also Section 10.2.1). For the electromagnetic field tensor, four possibilities were claimed by her to be preferable:¹⁶⁷ $mik; mik;K [ik];𝜖ijklmkl$ . Tonnelat opted for $mik$ , and also for the electric current density vector $J i = √1--𝜖ijkl(m + m + m ) 6 −h [ij],k [ki],j [jk],i$ . The field induction is defined via: $ˆP ik = -∂ℒ- ∂mik$ . Schrödinger had identified the electromagnetic field with the anti-symmetrical part $K [ik]$ of the Ricci tensor [545 *, 549 *]; this suggestion was also made in [138 *, 141 , 5 *, 93 *]. On the other hand, Eq. (235 *) can be satisfied by $ˆg[ij] = 12𝜖ijkl(∂kAl − ∂lAk)$ such that $Fij := 𝜖ijklˆg[kl] = ∂iAj − ∂jAi$ would naturally constitute the relationship to the electromagnetic field tensor $F ij$ [203 *]. In this context, the electromagnetic induction would be $ij 1 ijkl H ∼ 2𝜖 kkl$ , and alternatively, $ijkl ∼ 𝜖 K− kl$ ([650 *], p. 370). We learned above in (323 *) of Section 9.6.1 that M. Ikeda used yet another definition of the electromagnetic field tensor.

If electrical currents are to be included, the following choices for the current density were considered by Einstein, by Straus, (cf. Sections 9.2.2 and 9.3), and by others [34 *, 637 *]:

The second choice would either violate the weak field equations or forbid any non-zero current density. These alternatives are bound to the choice for the induction. Two possibilities were discussed by Mavridès:

jijk := k{[ij],k}, (331 ) 1 √ ---- 4 πji := √----∂l( − hhirhlskrs). (332 ) − h

In the 2nd case, Papapetrou’s spherically symmetric static solutions would not make sense, physically [407 ].¹⁶⁸ Finzi proposed yet another expression: $0 jk = 12𝜖kpqr∇pKqr$ ([473 *], p. 288). Late in his life, Einstein gave $[is] ˆg ,s$ the interpretation of magnetic current density [158 *].

An ambiguity always present is the assignment of the electric and magnetic fields to the components $ki0$ or $kab,a,b = 1,2,3$ , or vice versa in order to arrive at the correct Maxwell’s equations.

Another object lending itself to identification with the electromagnetic field would be homothetic curvature encountered in Section 2.3.1, i.e., $j j j V+kl = K+ jkl = ∂kL lj − ∂lLkj$ : $j Llj$ could then play the role of the vector potential. This choice has been made by Sciama, but with a complex curvature tensor $s Kjjkl$ [565 *]. In this case $sj j j K jkl = ∂kL [lj] − ∂lL[kj] = 2S[j,k]$ . The vector potential thus is identified with the torsion vector.

In a paper falling outside of the period of this review, H.-J. Treder suggested to also geometrize spinorial degrees of freedom by including them in the asymmetric metric; it took the form [651 , 67 ]:

where $A hi$ are tetrad components, $ψ α$ a 2-component Weyl-spinor, $C$ a constant (with dimension) and $αβ β γij = σiαμ˙σj ˙νγμ˙ν˙$ . $σiα˙μ$ are Pauli-matrices, $γ ˙μ˙ν$ corresponds to the antisymmetric $2 × 2$ symbol $𝜖αβ$ . It is obvious that the assumed mapping of geometrical objects to physical variables had to remain highly ambiguous because the only arguments available were the consistency of the interpretation within unified field theory and the limit to the previous theories (Einstein–Maxwell theory, general relativity), thought to be necessarily encased in UFT. As we have seen, the hope of an eventual help from exact solutions had to be abandoned.

c) Matter tensor

In Einstein’s understanding of UFT, the matter tensor for a continuous matter distribution should also become part of geometry. Again a precise attribution to geometrical objects could not be found. One way of approaching the problem was to reshape part of the field equations into the form of the old or a newly defined “Einstein tensor”, and terms left over. These then were declared to constitute the “matter tensor”. In her book, M.-A. Tonnelat discussed this problem in detail ([632 *], Chapter VII, A, pp. 109–117); cf. also Section 10.3.1. As late as 1963, Schrödinger could write:

“Thus it is as yet undecided what interpretation of the various tensors and densities is most likely to let the theory meet observed facts” ([557 ], reprinted 1963, p. 115).

9.8 The role of additional symmetries

The symmetries Einstein had introduced, i.e., transposition invariance and $λ$ -transformations, played a major role in versions of UFT, but not in physics, in general. There are only a few papers with these symmetries as their topic. J. Winogradzki investigated the relationship of the results in Einstein’s final approach to UFT (made together with B. Kaufman) to the theorems by Emmy Noether [704 *, 703 *]. She called invariance with regard to the group $U$ composed of coordinate- and $λ$ -transformations which had been named the “extended” group by Einstein, $U$ - or $J$ - invariance:

with $Ω k ij$ being independent of $gij$ . As a necessary condition for the field equations following from a variational principle to be J-invariant, she derived $Ω k= δkλ ij i j$ . Thus, with the help of four axioms postulated by her, she arrived at the $λ$ -transformations (52 *).

According to the 2nd theorem by E. Noether, $λ$ -invariance leads to four identities which were written out by Winogradzki for a Lagrangian density $Aˆ$ (her “Hamiltonian”):

Equation (334 *) relates $[is] ˆg ,s = 0$ and the 16 field equations which are not invariant under $λ$ -transformations.

P. G. Bergmann also discussed Einstein’s $λ$ -transformations, but just in the special form used in the 5th edition of The Meaning of Relativity, i.e., with $λk = ∂k λ$ . No wonder that he then concluded: “[…] the $λ$ transformation appears to be closely related in its conception to Weyl’s original gauge transformation” ([23 ], p. 780).

	Abstract
1	Introduction
2	Mathematical Preliminaries
	2.1	Metrical structure
	2.2	Symmetries
	2.3	Affine geometry
	2.4	Differential forms
	2.5	Classification of geometries
	2.6	Number fields
3	Interlude: Meanderings – UFT in the late 1930s and the 1940s
	3.1	Projective and conformal relativity theory
	3.2	Continued studies of Kaluza–Klein theory in Princeton, and elsewhere
	3.3	Non-local fields
4	Unified Field Theory and Quantum Mechanics
	4.1	The impact of Schrödinger’s and Dirac’s equations
	4.2	Other approaches
	4.3	Wave geometry
5	Born–Infeld Theory
6	Affine Geometry: Schrödinger as an Ardent Player
	6.1	A unitary theory of physical fields
	6.2	Semi-symmetric connection
7	Mixed Geometry: Einstein’s New Attempt
	7.1	Formal and physical motivation
	7.2	Einstein 1945
	7.3	Einstein–Straus 1946 and the weak field equations
8	Schrödinger II: Arbitrary Affine Connection
	8.1	Schrödinger’s debacle
	8.2	Recovery
	8.3	First exact solutions
9	Einstein II: From 1948 on
	9.1	A period of undecidedness (1949/50)
	9.2	Einstein 1950
	9.3	Einstein 1953
	9.4	Einstein 1954/55
	9.5	Reactions to Einstein–Kaufman
	9.6	More exact solutions
	9.7	Interpretative problems
	9.8	The role of additional symmetries
10	Einstein–Schrödinger Theory in Paris
	10.1	Marie-Antoinette Tonnelat and Einstein’s Unified Field Theory
	10.2	Tonnelat’s research on UFT in 1946 – 1952
	10.3	Some further developments
	10.4	Further work on unified field theory around M.-A. Tonnelat
	10.5	Research by and around André Lichnerowicz
11	Higher-Dimensional Theories Generalizing Kaluza’s
	11.1	5-dimensional theories: Jordan–Thiry theory
	11.2	6- and 8-dimensional theories
12	Further Contributions from the United States
	12.1	Eisenhart in Princeton
	12.2	Hlavatý at Indiana University
	12.3	Other contributions
13	Research in other English Speaking Countries
	13.1	England and elsewhere
	13.2	Australia
	13.3	India
14	Additional Contributions from Japan
15	Research in Italy
	15.1	Introduction
	15.2	Approximative study of field equations
	15.3	Equations of motion for point particles
16	The Move Away from Einstein–Schrödinger Theory and UFT
	16.1	Theories of gravitation and electricity in Minkowski space
	16.2	Linear theory and quantization
	16.3	Linear theory and spin-1/2-particles
	16.4	Quantization of Einstein–Schrödinger theory?
17	Alternative Geometries
	17.1	Lyra geometry
	17.2	Finsler geometry and unified field theory
18	Mutual Influence and Interaction of Research Groups
	18.1	Sociology of science
	18.2	After 1945: an international research effort
19	On the Conceptual and Methodic Structure of Unified Field Theory
	19.1	General issues
	19.2	Observations on psychological and philosophical positions
20	Concluding Comment
	Acknowledgements
	References
	Footnotes
	Biographies