Distant parallelism

The geometric concept of “fields of parallel vectors” had been introduced on the level of advanced textbooks by Eisenhart as early as 1925–1927 [119

, 121

] without use of the concept of a metric. In particular, the vanishing of the (affine) curvature tensor was given as a necessary and sufficient condition for the existence of $D$ linearly independent fields of parallel vectors in a $D$ -dimensional affine space ([121 ], p. 19).

6.4.1 Cartan and Einstein

As concerns the geometry of “Fernparallelism”, it is a special case of a space with Euclidean connection introduced by Cartan in 1922/23 [31 , 30 , 32 ]. Pais let Einstein “invent” and “discover” distant parallelism, and he states that Einstein “did not know that Cartan was already aware of this geometry” ([241

], pp. 344–345). However, when Einstein published his contributions in June 1928 [84

, 83

], Cartan had to remind him that a paper of his introducing the concept of torsion had

“appeared at the moment at which you gave your talks at the Collège de France. I even remember having tried, at Hadamard’s place, to give you the most simple example of a Riemannian space with Fernparallelismus by taking a sphere and by treating as parallels two vectors forming the same angle with the meridians going through their two origins: the corresponding geodesics are the rhumb lines.”¹⁵⁴ (letter of Cartan to Einstein on 8 May 1929; cf. [50 ], p. 4)

This remark refers to Einstein’s visit in Paris in March/April 1922. Einstein had believed to have found the idea of distant parallelism by himself. In this regard, Pais may be correct. Every researcher knows how an idea, heard or read someplace, can subconsciously work for years and then surface all of a sudden as his or her own new idea without the slightest remembrance as to where it came from. It seems that this happened also to Einstein. It is quite understandable that he did not remember what had happened six years earlier; perhaps, he had not even fully followed then what Cartan wanted to explain to him. In any case, Einstein’s motivation came from the wish to generalise Riemannian geometry such that the electromagnetic field could be geometrized:

“Therefore, the endeavour of the theoreticians is directed toward finding natural generalisations of, or supplements to, Riemannian geometry in the hope of reaching a logical building in which all physical field concepts are unified by one single viewpoint.”¹⁵⁵ ([84 ], p. 217)

In an investigation concerning spaces with simply transitive continuous groups, Eisenhart already in 1925 had found the connection for a manifold with distant parallelism given 3 years later by Einstein [118

]. He also had taken up Cartan’s idea and, in 1926, produced a joint paper with Cartan on “Riemannian geometries admitting absolute parallelism” [40 ], and Cartan also had written about absolute parallelism in Riemannian spaces [33 ]. Einstein, of course, could not have been expected to react to these and other purely mathematical papers by Cartan and Schouten, focussed on group manifolds as spaces with torsion and vanishing curvature ([41 , 34 ], pp. 50–54). No physical application had been envisaged by these two mathematicians.

Nevertheless, this story of distant parallelism raises the question of whether Einstein kept up on mathematical developments himself, or whether, at the least, he demanded of his assistants to read the mathematical literature. Against his familiarity with mathematical papers speaks the fact that he did not use the name “torsion” in his publications to be described in the following section. In the area of unified field theory including spinor theory, Einstein just loved to do the mathematics himself, irrespective of whether others had done it before – and done so even better (cf. Section 7.3).

Anyhow, in his response (Einstein to Cartan on 10 May 1929, [50

], p. 10), Einstein admitted Cartan’s priority and referred also to Eisenhart’s book of 1927 and to Weitzenböck’s paper [393 ]. He excused himself by Weitzenböck’s likewise omittance of Cartan’s papers among his 14 references. In his answer, Cartan found it curious that Weitzenböck was silent because

“[...] he indicates in his bibliography a note by Bortolotti in which he several times refers to my papers.”¹⁵⁶ (Cartan to Einstein on 15 May 1929; [50 ], p. 14)

The embarrassing situation was solved by Einstein’s suggestion that he had submitted a comprehensive paper on the subject to Zeitschrift für Physik, and he invited Cartan to add his description of the historical record in another paper (Einstein to Cartan on 10 May 1929). After Cartan had sent his historical review to Einstein on 24 May 1929, the latter answered three months later:

“I am now writing up the work for the Mathematische Annalen and should like to add yours [...]. The publication should appear in the Mathematische Annalen because, at present, only the mathematical implications are explored and not their applications to physics.”¹⁵⁷ (letter of Einstein to Cartan on 25 August 1929 [50 , 35 , 89 ])

In his article, Cartan made it very clear that it was not Weitzenböck who had introduced the concept of distant parallelism, as valuable as his results were after the concept had become known. Also, he took Einstein’s treatment of Fernparallelism as a special case of his more general considerations. Interestingly, he permitted himself to interpet the physical meaning of geometrical structures¹⁵⁸ :

“Let us say simply that mechanical phenomena are of a purely affine nature whereas electromagnetic phenomena are essentially metric; therefore it is rather natural to try to represent the electromagnetic potential by a not purely affine vector.”¹⁵⁹ ([35 ], p. 703)

“In particular, I learned from Mr. Weitzenböck and Mr. Cartan that the treatment of continua of the species which is of import here, is not really new.[...] In any case, what is most important in the paper, and new in any case, is the discovery of the simplest field laws that can be imposed on a Riemannian manifold with Fernparallelismus.”¹⁶⁰ ([89 ], p. 685)

“For me, the great attraction of the theory presented here lies in its unity and in the allowed highly overdetermined field variables. I also could show that the field equations, in first approximation, lead to equations that correspond to the Newton–Poisson theory of gravitation and to Maxwell’s theory. Nevertheless, I still am far from being able to claim that the derived equations have a physical meaning. The reason is that I could not derive the equations of motion for the corpuscles.”¹⁶¹ ([89 ], p. 697)

The split, in first approximation, of the tetrad field $hab$ according to $¯ hab = ηab + hab$ lead to homogeneous wave equations and divergence relations for both the symmetric and the antisymmetric part identified as metric and electromagnetic field tensors, respectively.

6.4.2 How the word spread

Einstein in 1929 really seemed to have believed that he was on a good track because, in this and the following year, he published at least 9 articles on distant parallelism and unified field theory before switching off his interest. The press did its best to spread the word: On 2 February 1929, in its column News and Views, the respected British science journal Nature reported:

“For some time it has been rumoured that Prof. Einstein has been about to publish the results of a protracted investigation into the possibility of generalising the theory of relativity so as to include the phenomena of electromagnetism. It is now announced that he has submitted to the Prussian Academy of Sciences a short paper in which the laws of gravitation and of electromagnetism are expressed in a single statement.”

Nature then went on to quote from an interview of Einstein of 26 January 1929 in a newspaper, the Daily Chronicle. According to the newspaper, among other statements Einstein made, in his wonderful language, was the following:

“Now, but only now, we know that the force which moves electrons in their ellipses about the nuclei of atoms is the same force which moves our earth in its annual course about the sun, and it is the same force which brings to us the rays of light and heat which make life possible upon this planet.” [2 ]

Whether Einstein used this as a metaphorical language or, whether he at this time still believed that the system “nucleus and electrons” is dominated by the electromagnetic force, remains open.

The paper announced by Nature is Einstein’s “Zur einheitlichen Feldtheorie”, published by the Prussian Academy on 30 January 1929 [88

]. A thousand copies of this paper had been sold within 3 days, so the presiding secretary of the Academy ordered the printing of a second thousand. Normally, only a hundred copies were printed ([183

], Dokument Nr. 49, p. 136). On 4 February 1929, The Times (of London) published the translation of an article by Einstein, “written as an explanation of his thesis for readers who do not possess an expert knowledge of mathematics”. This article then became reprinted in March by the British astronomy journal The Observatory [86

]. In it, Einstein first gave a historical sketch leading up to the introduction of relativity theory, and then described the method that guided him to the new theory of distant parallelism. In fact, the only formulas appearing are the line elements for two-dimensional Riemannian and Euclidean space. At the end, by one figure, Einstein tried to convey to the reader what consequence a Euclidean geometry with torsion would have – without using that name. His closing sentences are¹⁶² :

“Which are the simplest and most natural conditions to which a continuum of this kind can be subjected? The answer to this question which I have attempted to give in a new paper yields unitary field laws for gravitation and electromagnetism.” ([86 ], p. 118)

A few months later in that year, again in Nature, the mathematician H. T. H. Piaggio gave an exposition for the general reader of “Einstein’s and other Unitary Field Theories”. He was a bit more explicit than Einstein in his article for the educated general reader. However, he was careful to end it with a warning:

“Of course the ultimate test of the theory must be by experiment. It may succeed in predicting some interaction between gravitation and electromagnetism which can be confirmed by observation. On the other hand, it may be only a ‘graph’ and so outside the ken of the ordinary physicist.” ([258 ], p. 879)

The use of the concept “graph” had its origin in Eddington’s interpretation of his and other peoples’ unified field theories to be only graphs of the world; the true geometry remained the Riemannian geometry underlying Einstein’s general relativity.

Even the French-Belgian writer and poet Maurice Maeterlinck had heard of Einstein’s latest achievement in the area of unified field theory. In his poetic presentation of the universe “La grande féerie”¹⁶³ we find his remark:

“Einstein, in his last publications comments to which are still to appear, again brings us mathematical formulae which are applicable to both gravitation and electricity, as if these two forces seemingly governing the universe were identical and subject to the same law. If this were true it would be impossible to calculate the consequences.”¹⁶⁴ ([214 ], p. 68)

6.4.3 Einstein’s research papers

We are dealing here with Einstein’s, and Einstein and Mayer’s joint papers on distant parallelism in the reports of the Berlin Academy and Mathematische Annalen, which were taken as the starting point by other researchers following suit with further calculations. Indeed, there was a lot of work to do, only in part because Einstein, from one paper to the next, had changed his field equations¹⁶⁵ .

In his first note [84

], dynamics was absent; Einstein made geometrical considerations his main theme: Introduction of a local “ $n$ -bein-field” $hkˆı$ at every point of a differentiable manifold and the related object $hkˆı$ defined as the collection of the “normed subdeterminants of the $hkˆı$ ”¹⁶⁶ such that $h ˆhk= δk. il ˆl i$ As we have seen before, the components of the metric tensor are defined by

“Fernparallelism” now means that if the components referred to the local $n$ -bein of a vector $ˆk ˆk l A = hlA$ at a point $p$ , and of a vector $ˆk B$ at a different point $q$ are the same, i.e., $ˆ ˆ Ak = B k$ , then the vectors are to be considered as “parallel”. There is an underlying symmetry, called “rotational invariance” by Einstein: joint rotations of each $n$ -bein by the same angle. All relations with a physical meaning must be “rotationally invariant”. Of course, in space-time with a Lorentz metric, the 4-bein-transformations do form the proper Lorentz group.

If parallel transport of a tangent vector $A$ is defined as usual by $dAk = − Δ lmkAldxm$ , then the connection components turn out to be

The (Riemannian) curvature tensor calculated from Equation (161

) turns out to vanish. As Einstein noted, by $gij$ from Equation (160

) also the usual Riemannian connection $k Γlm (g )$ may be formed. Moreover, $k k k Ylm := Γlm (g ) − Δ lm$ is a tensor that could be used for building invariants. In principle, distant parallelism is a particular bi-connection theory. The connection $Γlmk(g)$ does not play a role in the following (cf., however, de Donder ’s paper [48

]). From Equation (161

), obviously the torsion tensor $S k = 1(Δ k− Δ k) ⁄= 0 lm 2 lm ml$ follows (cf. Equation (21

)). Einstein denoted it by $k Λ lm$ and, in comparison with the curvature tensor, considered it as the “formally simplest” tensor of the theory for building invariants by help of the linear form $j l Λlj dx$ and of the scalars $gijΛimlΛ jml$ and $gij glrgmsΛ lmi Λrsj.$ . He indicated how a Lagrangian could be built and the 16 field equations for the field variables $h lj$ obtained.

At the end of the note Einstein compared his new approach to Weyl’s and Riemann’s:

In his second note [83

], Einstein departed from the Lagrangian $ij l m ℒ = h g Λ im Λjl$ , i.e., a scalar density corresponding to the first scalar invariant of his previous note¹⁶⁸ . He introduced $ϕk := Λkll$ , and took the case $ϕ = 0$ to describe a “purely gravitational field”. However, as he added in a footnote, pure gravitation could have been characterised by $∂ϕik − ∂ϕki ∂x ∂x$ as well. In his first paper on distant parallelism, Einstein did not use the names “electrical potential” or “electrical field”. He then showed that in a first-order approximation starting from $hiˆj = δiˆj + kiˆj + ...$ , both the Einstein vacuum field equations and Maxwell’s equations are surfacing. To do so he replaced $hiˆj$ by $gij = δij + 2k (ij)$ and introduced $( ) ϕk := 1 ∂kkjj− ∂kjjk- 2 ∂x ∂x$ . Einstein concluded that

“The separation of the gravitational and the electromagnetic field appears artificial in this theory. [...] Furthermore, it is remarkable that, according to this theory, the electromagnetic field does not enter the field equations quadratically.”¹⁶⁹ ([83 ], p. 6)

In a postscript, Einstein noted that he could have obtained similar results by using the second scalar invariant of his previous note, and that there was a certain indeterminacy as to the choice of the Lagrangian.

This shows clearly that the ambiguity in the choice of a Lagrangian had bothered Einstein. Thus, in his third note, he looked for a more reassuring way of deriving field equations [88

]. He left aside the Hamiltonian principle and started from identities for the torsion tensor, following from the vanishing of the curvature tensor¹⁷⁰ . He thus arrived at the identity given by Equation (29

), i.e., (Einstein’s equation (3), p. 5; his convention is $ˆı ˆı Λ ˆkˆl := 2S ˆkˆl$ )

The second identity used by Einstein follows with the help of Equation (27

) for vanishing curvature (Einstein’s equation (5), p. 5):

By replacing $ˆjk A$ by $ˆ ˆkˆlj V$ and using Equation (165

), the final form of the second identity now is

Einstein seems to have sensed that the average reader might be able to follow his path to the postulated field equations only with difficulty. Therefore, in a postscript, he tried to clear up his motivation:

“The field equations suggested in this paper may be characterised with regard to other such possible ones in the following way. By staying close to the identity (167 ), it has been accomplished that not only 16, but 20 independent equations can be imposed on the 16 quantities $ˆ hki$ . By ‘independent’ we understand that none of these equations can be derived from the remaining ones, even if there exist 8 identical (differential) relations among them.”¹⁷² ([88 ], p. 8)

“A deeper investigation of the consequences of the field equations (170 ) will have to show whether the Riemannian metric, together with distant parallelism, really gives an adequate representation of the physical qualities of space.”¹⁷³

In his second paper of 1929, the fourth in the series in the Berlin Academy, Einstein returned to the Hamiltonian principle because his collaborators Lanczos and Müntz had doubted the validity of the field equations of his previous publication [88

] on grounds of their unproven compatibility. In the meantime, however, he had found a Lagrangian such that the compatibility-problem disappeared. He restricted the many constructive possibilities for $ℒ = ℒ(h ˆk,∂h ˆk) i l i$ by asking for a Lagrangian containing torsion at most quadratically. His Lagrangian is a particular linear combination of the three possible scalar densities, as follows:

with $ˆj1 := h Sklm Skm l$ , $ˆj2 := h SklmSkl m$ , and $ˆj3 := h SjSj$ . If $𝜖1,𝜖2$ are small parameters, then his final Lagrangian is $ℒ = ˆH + 𝜖1H ˆ∗ + 𝜖2Hˆ∗∗$ . In order to prove that Maxwell’s equations follow from his Lagrangian, Einstein had to perform the limit $𝜖1 σ := 𝜖2 → 0$ in an expression termed $Gˆ∗ ik$ , which he assumed to depend homogeneously and quadratically on a linear combination of torsion¹⁷⁴ .

In a Festschrift for his former teacher and colleague in Zürich, A. Stodola, Einstein summed up what he had reached¹⁷⁵ . He exchanged the definition of the invariants named $ˆj2$ , and $ˆj3$ , and stated that a choice of $A = − B$ , $C = 0$ in the Lagrangian $ˆj = Aˆj1 + Bˆj2 + Cˆj3$ would give field equations

“[...] that coincide in first approximation with the known laws for the gravitational and electromagnetic field [...]”¹⁷⁶

with the proviso that the specialisation of the constants $A$ , $B$ , $C$ must be made only after the variation of the Lagrangian, not before. Also, together with Müntz, he had shown that for an uncharged mass point the Schwarzschild solution again obtained [87

Einstein’s next publication was the note preceding Cartan’s paper in Mathematische Annalen [89

]. He presented it as an introduction suited for anyone who knew general relativity. It is here that he first mentioned Equations (162

) and (163

). Most importantly, he gave a new set of field equations not derived from a variational principle; they are¹⁷⁷ .

“In fact, in the new theory of Mr. Einstein, it is natural to call a universe homogeneous if the torsion vectors that are associated to two parallel surface elements are parallel themselves; this means that parallel transport conserves torsion.”¹⁷⁸ ([35 ], p. 703)

From Equation (173

) with the help of Equation (171

), (172

), Einstein wrote down two more identities. One of them he had obtained from Cartan:

“But I am very grateful to you for the identity

ik k lm G ∥i − S lm G = 0,

which, astonishingly, had escaped me. [...] In a new presentation in the Sitzungsberichten, I used this identity while taking the liberty of pointing to you as its source.”¹⁷⁹ (letter of Einstein to Cartan from 18 December 1929, Document X of [50

], p. 72)

In order to show that his field equations were compatible he counted the number of equations, identities, and field quantities (in $n$ -dimensional space) to find, in the end, $n2 + n$ equations for the same number of variables. To do so, he had to introduce an additional variable $ψ$ via $F = ϕ − ∂log-ψ k k ∂xk$ . Here, $F k$ is introduced by $F = ∂ F − ∂ F = ∂ ϕ − ∂ ϕ ik k i i k k i i k$ . Einstein then showed that $∂lFik + ∂kFli + ∂iFkl = 0$ .

The changes in his approach Einstein continuously made, must have been hard on those who tried to follow him in their scientific work. One of them, Zaycoff , tried to make the best out of them:

“Recently, A. Einstein ([89 ]), following investigations by E. Cartan ([35 ]), has considerably modified his teleparallelism theory such that former shortcomings (connected only to the physical identifications) vanish by themselves.”¹⁸⁰ ([433 ], p. 410)

“It is not for the first time that such spaces are envisaged. From a purely mathematical point of view they were studied previously. M. Cartan was so amiable as to write a note for the Mathematische Annalen exposing the various phases in the formal development of these concepts.”¹⁸¹ ([92 ], p. 4)

Note that Einstein does not say that it was Cartan who first “envisaged” these spaces before. Later in the paper, he comes closer to the point:

“This type of space had been envisaged before me by mathematicians, notably by WEITZENBÖCK, EISENHART et CARTAN [...].”¹⁸² [92 ]

Some of the material in the paper overlaps with results from other publications [85

, 90

, 93

]. The counting of independent variables, field equations, and identities is repeated from Einstein’s paper in Mathematische Annalen [89

]. For $n = 4$ , there were 20 field equations $ik (G = 0$ , $l F = 0)$ for $16 + 1$ variables $hkˆı$ and $ψ$ , four of which were arbitrary (coordinate choice). Hence 7 identities should exist, four of which Einstein had found previously. He now presented a derivation of the remaining three identities by a calculation of two pages’ length. The field equations are the same as in [89

]; the proof of their compatibility takes up, in a slightly modified form, the one communicated by Einstein to Cartan in a letter of 18 December 1929 ([92

], p. 20). It is reproduced also in [90

Interestingly, right after Einstein’s article in the institute’s journal, a paper of C. G. Darwin, “On the wave theory of matter”, is printed, and, in the same first volume, a report of Max Born on “Some problems in Quantum Mechanics.” Thus, French readers were kept up-to-date on progress made by both parties – whether they worked on classical field theory or quantum theory [45 , 21 ].

A. Proca, who had attended Einstein’s lectures, gave an exposition of them in a journal of his native Romania. He was quite enthusiastic about Einstein’s new theory:

“A great step forward has been made in the pursuit of this total synthesis of phenomena which is, right or wrong, the ideal of physicists. [...] the splendid effort brought about by Einstein permits us to hope that the last theoretical difficulties will be vanquished, and that we soon will compare the consequences of the theory with [our] experience, the great stepping stone of all creations of the mind.”¹⁸³ [260 , 261 ]

Einstein’s next paper in the Berlin Academy, in which he reverts to his original notation $hˆkl$ , consisted of a brief critical summary of the formalism used in his previous papers, and the announcement of a serious mistake in his first note in 1930, which made invalid the derivation of the field equations for the electromagnetic field ([90

], p. 18). The mistake was the assumption on the kind of dependence on torsion of the quantity $Gˆ∗ ik$ , which was mentioned above. Also, Einstein now found it better “to keep the concept of divergence, defined by contraction of the extension of a tensor” and not use the covariant derivative $∇ ∗ l$ introduced by him in his third paper in the Berlin Academy [88

Then Einstein presented the same field equations as in his paper in Annalen der Mathematik, which he demanded to be

While these demands had been sufficient to uniquely lead to the gravitational field equations (with cosmological constant) of general relativity, in the teleparallelism theory a great deal of ambiguity remained. Sixteen field equations were needed which, due to covariance, induced four identities.

“Therefore equations must be postulated among which identical relations are holding. The higher the number of equations (and consequently also the number of identities among them), the more precise and stronger than mere determinism is the content; accordingly, the theory is the more valuable, if it is also consistent with the empirical facts.”¹⁸⁴ ([90 ], p. 21)

“The proof of the compatibility, as given in my paper in the Mathematische Annalen, has been somewhat simplified due to a communication which I owe to a letter of Mr. CARTAN (cf. §3, [16]).”¹⁸⁵

In linear approximation, i.e., for $¯ hik = δik + hik$ , Einstein obtained d’Alembert’s equation for both the symmetric and the antisymmetric part of $¯ hik$ , identified with the gravitational and the electromagnetic field, respectively.

Einstein’s next note of one and a half pages contained a mathematical result within teleparallelism theory: From any tensor with an antisymmetric pair of indices a vector with vanishing divergence can be derived [93

In order to test the field equations by exhibiting an exact solution, a simple case would be to take a spherically symmetric, asymptotically (Minkowskian) 4-bein. This is what Einstein and Mayer did, except with the additional assumption of space-reflection symmetry [106

]. Then the 4-bein contains three arbitrary functions of one parameter $s$ :

Possibly, W. F. G. Swan of the Bartol Research Foundation in Swarthmore had this paper in mind when he, in April 1930, in a brief description of Einstein’s latest publications, told the readers of Science:

“It now appears that Einstein has succeeded in working out the consequences of his general law of gravity and electromagnetism for two special cases just as Newton succeeded in working out the consequences of his law for several special cases. [...] It is hoped that the present solutions obtained by Einstein, or if not these, then others which may later evolve, will suggest some experiments by which the theory may be tested.” ([339 ], p. 391)

Two days before the paper by Einstein and Mayer became published by the Berlin Academy, Einstein wrote to his friend Solovine:

“My field theory is progressing well. Cartan has already worked with it. I myself work with a mathematician (S. Mayer¹⁸⁶ from Vienna), a marvelous chap [...].”¹⁸⁷ ([98 ], p. 56)

The mentioning of Cartan resulted from the intensive correspondence of both scientists between December 1929 and February 1930: About a dozen letters were exchanged which, sometimes, contained long calculations [50

] (cf. Section 6.4.6). In an address given at the University of Nottingham, England, on 6 June 1930, Einstein also must have commented on the exact solutions found and on his program concerning the elementary particles. A report of this address stated about Einstein’s program:

“The problem is nearly solved; and to the first approximations he gets laws of gravitation and electro-magnetics. He does not, however, regard this as sufficient, though those laws may come out. He still wants to have the motions of ordinary particles to come out quite naturally. [The program] has been solved for what he calls the ‘quasi-statical motions’, but he also wants to derive elements of matter (electrons and protons) out of the metric structure of space.” ([91 ], p. 610)

With his “assistant” Walther Mayer, Einstein then embarked on a very technical, systematic study of compatible field equations for distant parallelism [108 ]. In addition to the assumptions (1), (2), (3) for allowable field equations given above, further restrictions were made:

“Two of these are (non-trivial) generalisations of the original gravitational field equations, one of them being known already as a consequence of the Hamiltonian principle. The remaining two types are denoted in the paper by [...].”¹⁸⁸

With no further restraining principles at hand, this ambiguity in the choice of field equations must have convinced Einstein that the theory of distant parallelism could no longer be upheld as a good candidate for the unified field theory he was looking for, irrespective of the possible physical content¹⁸⁹ . Once again, he dropped the subject and moved on to the next. While aboard a ship back to Europe from the United States, Einstein, on 21 March 1932, wrote to Cartan:

“[...] In any case, I have now completely given up the method of distant parallelism. It seems that this structure has nothing to do with the true character of space [...].” ([50 ], p. 209)

What Cartan might have felt, after investing the forty odd pages of his calculations printed in Debever’s book, is unknown. However, the correspondence on the subject came to an end in May 1930 with a last letter by Cartan.¹⁹⁰ .

6.4.4 Reactions I: Mostly critical

About half a year after Einstein’s two papers on distant parallelism of 1928 had appeared, Reichenbach , who always tended to defend Einstein against criticism, classified the new theory [268 ] according to the lines set out in his book [267

] as “having already its precisely fixed logical position in the edifice of Weyl–Eddington geometry” ([267

], p. 683). He mentioned as a possible generalization an idea of Einstein’s, in which the operation of parallel transport might be taken as integrable not with regard to length but with regard to direction: “a generalisation which already has been conceived by Einstein as I learned from him” ([267

], p. 687)¹⁹¹ . As concerns parallelism at a distance, Reichenbach was not enthusiastic about Einstein’s new approach:

“[...] it is the aim of Einstein’s new theory to find such an entanglement between gravitation and electricity that it splits into the separate equations of the existing theory only in first approximation; in higher approximation, however, a mutual influence of both fields is brought in, which, possibly, leads to an understanding of questions unanswered up to now as [is the case] for the quantum riddle. But this aim seems to be in reach only if a direct physical interpretation of the operation of transport, even of the immediate field quantities, is given up. From the geometrical point of view, such a path [of approach] must seem very unsatisfactory; its justifications will only be reached if the mentioned link does encompass more physical facts than have been brought into it for building it up.”¹⁹² ([267 ], p. 689)

A first reaction from a competing colleague came from Eddington, who, on 23 February 1929, gave a cautious but distinct review of Einstein’s first three publications on distant parallelism [84

, 83

, 88

] in Nature. After having explained the theory and having pointed out the differences to his own affine unified field theory of 1921, he confessed:

“For my own part I cannot readily give up the affine picture, where gravitational and electric quantities supplement one another as belonging respectively to the symmetrical and antisymmetrical features of world measurement; it is difficult to imagine a neater kind of dovetailing. Perhaps one who believes that Weyl’s theory and its affine generalisation afford considerable enlightenment, may be excused for doubting whether the new theory offers sufficient inducement to make an exchange.” [62 ]

Weyl was the next unhappy colleague; in connection with the redefinition of his gauge idea he remarked (in April/May 1929):

“[...] my approach is radically different, because I reject distant parallelism and keep to Einstein’s general relativity. [...] Various reasons hold me back from believing in parallelism at a distance. First, my mathematical intuition a priori resists to accept such an artificial geometry; I have difficulties to understand the might who has frozen into rigid togetherness the local frames in different events in their twisted positions. Two weighty physical arguments join in [...] only by this loosening [of the relationship between the local frames] the existing gauge-invariance becomes intelligible. Second, the possibility to rotate the frames independently, in the different events, [...] is equivalent to the symmetry of the energy-momentum tensor, or to the validity of the conservation law for angular momentum.”¹⁹³ ([407 ], pp. 330–332.)

As usual, Pauli was less than enthusiastic; he expressed his discontent in a letter to Hermann Weyl of 26 August 1929:

“First let me emphasize that side of the matter about which I fully agree with you: Your approach for incorporating gravitation into Dirac’s theory of the spinning electron [...] I am as adverse with regard to Fernparallelismus as you are [...] (And here I must do justice to your work in physics. When you made your theory with $′ gik = λgik$ this was pure mathematics and unphysical; Einstein rightly criticised and scolded you. Now the hour of revenge has come for you, now Einstein has made the blunder of distant parallelism which is nothing but mathematics unrelated to physics, now you may scold [him].)”¹⁹⁴ ([251 ], pp. 518–519)

“By the way, I now no longer believe in one syllable of teleparallelism; Einstein seems to have been abandoned by the dear Lord.”¹⁹⁵ (Pauli to Ehrenfest 29 September 1929; [251 ], p. 524)

Pauli’s remark shows the importance of ideology in this field: As long as no empirical basis exists, beliefs, hopes, expectations, and rationally guided guesses abound. Pauli’s letter to Weyl from 1 July 1929 used non-standard language (in terms of science):

“I share completely your skeptical position with regard to Einstein’s 4-bein geometry. During the Easter holidays I have visited Einstein in Berlin and found his opinion on modern quantum theory reactionary.”¹⁹⁶ ([251 ], p. 506)

While the wealth of empirical data supporting Heisenberg’s and Schrödinger’s quantum theory would have justified the use of a word like “uninformed” or even “not up to date” for the description of Einstein’s position, use of “reactionary” meant a definite devaluation.

Einstein had sent a further exposition of his new theory to the Mathematische Annalen in August 1928. When he received its proof sheets from Einstein, Pauli had no reservations to criticise him directly and bluntly:

“I thank you so much for letting be sent to me your new paper from the Mathematische Annalen [89 ], which gives such a comfortable and beautiful review of the mathematical properties of a continuum with Riemannian metric and distant parallelism [...]. Unlike what I told you in spring, from the point of view of quantum theory, now an argument in favour of distant parallelism can no longer be put forward [...]. It just remains [...] to congratulate you (or should I rather say condole you?) that you have passed over to the mathematicians. Also, I am not so naive as to believe that you would change your opinion because of whatever criticism. But I would bet with you that, at the latest after one year, you will have given up the entire distant parallelism in the same way as you have given up the affine theory earlier. And, I do not wish to provoke you to contradict me by continuing this letter, because I do not want to delay the approach of this natural end of the theory of distant parallelism.”¹⁹⁷ (letter to Einstein of 19 December 1929; [251 ], 526–527)

“Your letter is quite amusing, but your statement seems rather superficial to me. Only someone who is certain of seeing through the unity of natural forces in the right way ought to write in this way. Before the mathematical consequences have not been thought through properly, is not at all justified to make a negative judgement. [...] That the system of equations established by myself forms a consequential relationship with the space structure taken, you would probably accept by a deeper study – more so because, in the meantime, the proof of the compatibility of the equations could be simplified.”¹⁹⁸ ([251 ], p. 582)

Before he had written to Einstein, Pauli, with lesser reservations, complained vis-a-vis Jordan:

“Einstein is said to have poured out, at the Berlin colloquium, horrible nonsense about new parallelism at a distance. The mere fact that his equations are not in the least similar to Maxwell’s theory is employed by him as an argument that they are somehow related to quantum theory. With such rubbish he may impress only American journalists, not even American physicists, not to speak of European physicists.”¹⁹⁹ (letter of 30 November 1929, [251 ], p. 525)

Of course, Pauli’s spells of rudeness are well known; in this particular case they might have been induced by Einstein’s unfounded hopes for eventually replacing the Schrödinger–Heisenberg–Dirac quantum mechanics by one of his unified field theories.

The question of the compatibility of the field equations played a very important role because Einstein hoped to gain, eventually, the quantum laws from the extra equations (cf. his extended correspondence on the subject with Cartan ([50

] and Section 6.4.6).

That Pauli had been right (except for the time span envisaged by him) was expressly admitted by Einstein when he had given up his unified field theory based on distant parallelism in 1931 (see letter of Einstein to Pauli on 22 January 1932; cf. [241 ], p. 347).

Born’s voice was the lonely approving one (Born to Einstein on 23 September 1929)²⁰⁰ :

“Your report on progress in the theory of Fernparallelism did interest me very much, particularly because the new field equations are of unique simplicity. Until now, I had been uncomfortable with the fact that, aside from the tremendously simple and transparent geometry, the field theory did look so very involved”²⁰¹ ([154 ], p. 307)

Born, however, was not yet a player in unified field theory, and it turned out that Einstein’s theory of distant parallelism became as involved as the previous ones.

Einstein’s collaborator Lanczos even wrote a review article about distant parallelism with the title “The new field theory of Einstein” [201

]. In it, Lanczos cautiously offers some criticism after having made enough bows before Einstein:

“To be critical with regard to the creation of a man who has long since obtained a place in eternity does not suit us and is far from us. Not as a criticism but only as an impression do we point out why the new field theory does not house the same degree of conviction, nor the amount of inner consistency and suggestive necessity in which the former theory excelled.[...] The metric is a sufficient basis for the construction of geometry, and perhaps the idea of complementing RIEMANNian geometry by distant parallelism would not occur if there were the wish to implant something new into RIEMANNian geometry in order to geometrically interpret electromagnetism.”²⁰² ([201 ], p. 126)

“It is indeed a courageous deed of the editors to accept an essay on a new field theory of Einstein for the ‘Results in the Exact Sciences’ [literal translation of the journal’s title]. His never-ending gift for invention, his persistent energy in the pursuit of a fixed aim in recent years surprise us with, on the average, one such theory per year. Psychologically interesting is that the author normally considers his actual theory for a while as the ‘definite solution’. Hence, [...] one could cry out: ’Einstein’s new field theory is dead. Long live Einstein’s new field theory!’ ”²⁰³ ([248 ], p. 186)

For the remainder, Pauli engaged in a discussion with the philosophical background of Lanczos and criticised his support for Mie’s theory of matter of 1913 according to which

“the atomism of electricity and matter, fully separated from the existence of the quantum of action, is to be reduced to the properties of (singularity-free) eigen-solutions of still-to-be-found nonlinear differential equations for the field variables.”²⁰⁴

Thus, Pauli lightly pushed aside as untenable one of Einstein’s repeated motivations and hoped-for tests for his unified field theories.

Lanczos, being dissatisfied with Einstein’s distant parallelism, then tried to explain “electromagnetism as a natural property of Riemannian geometry” by starting from the Lagrangian quadratic in the components of the Ricci tensor: $ℒ$ = $RikRlmgilgkm + C (Rikgik)2$ with an arbitrary constant $C$ . He varied $gik$ and $Rik$ independently [202

]. (For Lanczos see J. Stachel’s essay “Lanczos’ early contributions to relativity and his relation to Einstein” in [330 ], pp. 499–518.)

6.4.5 Reactions II: Further research on distant parallelism

The first reactions to Einstein’s papers came quickly. On 29 October 1928, de Donder suggested a generalisation by using two metric tensors, a space-time metric $gik$ , and a bein-metric $g⋆ik$ , connected to the 4-bein components $h lˆj$ by

Another application of Einstein’s new theory came from Eugen Wigner in Berlin whose paper showing that the tetrads in distant-parallelism-theory permitted a generally covariant formulation of “Diracs equation for the spinning electron”, was received by Zeitschrift für Physik on 29 December 1928 [419

]. He did point out that “up to now, grave difficulties stood in the way of a general relativistic generalisation of Dirac’s theory” and referred to a paper of Tetrode [344

]. Tetrode, about a week after Einstein’s first paper on distant parallelism had appeared on 14 June 1928, had given just such a generally relativistic formulation of Dirac’s equation through coordinate dependent Gamma-matrices²⁰⁵ ; he also wrote down a (symmetric) energy-momentum tensor for the Dirac field and the conservation laws. However, he had kept the metric $gik$ introduced into the formalism by

Zaycoff of the Physics Institute of the University in Sofia also followed Einstein’s work closely. Half a year after Einstein’s first two notes on distant parallelism had appeared [84

, 83

], i.e., shortly before Christmas 1928, Zaycoff sent off his first paper on the subject, whose arrival in Berlin was acknowledged only after the holidays on 13 January 1929 [429 ]. In it he described the mathematical formalism of distant parallelism theory, gave the identity (42

), and calculated the new curvature scalar in terms of the Ricci scalar and of torsion. He then took a more general Lagrangian than Einstein and obtained the variational derivatives in linear and, in a simple example, also in second approximation. In his presentation, he used both the teleparallel and the Levi-Civita connections. His second and third papers came quickly after Einstein’s third note of January 1929 [88

], and thus had to take into account that Einstein had dropped derivation of the field equations from a variational principle. In his second paper, Zaycoff followed Einstein’s method and gave a somewhat simpler derivation of the field equations. An exact, complicated wave equation followed:

“We may say that A. Einstein built a plane world which is no longer waste like the Euclidean space-time-world of H. Minkowski, but, on the contrary, contains in it all that we usually call physical reality.”²⁰⁶ ([428 ], p. 724)

A conference on theoretical physics at the Ukrainian Physical-Technical Institute in Charkow in May 1929, brought together many German and Russian physicists. Unified field theory, quantum mechanics, and the new quantum field theory were all discussed. Einstein’s former calculational assistant Grommer, now on his own in Minsk, in a brief contribution stressed Einstein’s path for getting an overdetermined system of differential equations: Vary with regard to the 16 bein-quantities but consider only the 10 metrical components as relevant. He claimed that Einstein had used only the antisymmetric part of the tensor $k k k P lm := Γ (g )lm − Δ lm$ , where both $k Γ (g)lm$ and $k Δ lm$ were mentioned above (in Einstein’s first note) although Einstein never used $Γ (g)lmk$ . According to Grommer the anti-symmetry of $P$ is needed, because its contraction leads to the electromagnetic 4-potential and because the symmetric part can be expressed by the antisymmetric part and the metrical tensor. He also played the true voice of his (former) master by repeating Einstein’s program of deriving the equations of motion from the overdetermined system:

“If the law of motion of elementary particles could be derived from the overdetermined field equation, one could imagine that this law of motion permit only discrete orbits, in the sense of quantum theory.”²⁰⁷ ([153 ], p. 646)

Levi-Civita also had sent a paper on distant parallelism to Einstein, who had it appear in the reports of the Berlin Academy [207

]. Levi-Civita introduced a set of four congruences of curves that intersect each other at right angles, called their tangents $k λˆı$ and used Equation (160

) in the form:

“Einstein’s new gravitational theory is intimately linked to the known theory of the orthogonal congruences of curves due to Ricci. In order to ease a comparison between both theories, we may bring together here the notations of Ricci and Levi-Civita [...] with those of Einstein.”²⁰⁸

A little after the publication of Levi-Civita’s papers, Heinrich Mandel embarked on an application of Kaluza’s five-dimensional approach to Einstein’s theory of distant parallelism [218 ]. Einstein had sent him the corrected proof sheets of his fourth paper [85

]. The basic idea was to consider the points of $M4$ as equivalent to the ensemble of congruences with tangent vector $Xi5$ in $M5$ (with cylindricity condition)²⁰⁹ . The space-time interval is defined as the distance of two lines of the congruence on $M5$ : $2 l l i k dτ = (γil − X5i X5l )(δk − X 5 X5k )dx dx .$ Mandel did not identify the torsion vector with the electromagnetic 4-potential, but introduced the covariant derivative $i ∂Ai i j e i l △kA := ∂xk + {kj}A + cX5k ℳ lA ,$ where the tensor $ℳ$ is skew-symmetric. We may look at this paper also as a forerunner of some sort to the Einstein–Mayer 5-vector formalism (cf. Section 6.3.2).

Before Einstein dropped the subject of distant parallelism, many more papers were written by a baker’s dozen of physicists. Some were more interested in the geometrical foundations, in exact solutions to the field equations, or in the variational principle.

One of those hunting for exact solutions was G. C. McVittie who referred to Einstein’s paper [88

“[...] we test whether the new equations proposed by Einstein are satisfied. It is shown that the new equations are satisfied to the first order but not exactly.”

He then goes on to find a rigourous solution and obtains the metric $2 ax1 2 −2ax1 2 −ax1 2 −ax1 2 ds = e dx4 − e dx1 − e dx2 − e dx3$ and the 4-potential $1 ϕ4 = 21√π-e2ax1$ [225 ]. He also wrote a paper on exact axially symmetric solutions of Einstein’s teleparallelism theory [226 ].

Tamm and Leontowich treated the field equations given in Einstein’s fourth paper on distant parallelism [85

]. They found that these field equations did not have a spherically symmetric solution corresponding to a charged point particle at rest²¹⁰ . The corresponding solution for the uncharged particle was the same as in general relativity, i.e., Schwarzschild’s solution. Tamm and Leontowitch therefore guessed that a charged point particle at rest would lead to an axially-symmetric solution and pointed to the spin for support of this hypothesis [342

, 342 ]. Wiener and Vallarta were after particular exact solutions of Einstein’s field equations in the teleparallelism theory. By referring to Einstein’s first two papers concerning distant parallelism, they set out to show that the

“[...] electromagnetic field is incompatible in the new Einstein theory with the assumption of static spherical symmetry and symmetry of the past and the future. [...] the new Einstein theory lacks at present all experimental confirmation.”

“Since writing this paper the authors have learned from Dr. H. Müntz that the new Einstein field equations of the 1929 paper do not yield the vanishing of the gravitational field in the case of spherical symmetry and time symmetry. In this case he has been able to obtain results checking the observed perihelion of mercury” ([416 ], p. 356)

In his paper “On unified field theory” of January 1929, Einstein acknowledges work of a Mr. Müntz:

“I am pleased to dutifully thank Mr. Dr. H. Müntz for the laborious exact calculation of the centrally-symmetric problem based on the Hamiltonian principle; by the results of this investigation I was led to the discovery of the road following here.”²¹¹

Again, two months later in his next paper, “Unified field theory and Hamiltonian principle”, Einstein remarks:

“Mr. Lanczos and Müntz have raised doubt about the compatibility of the field equations obtained in the previous paper [...].”

and, by deriving field equations from a Lagrangian shows that the objection can be overcome. In his paper in July 1929, the physicist Zaycoff had some details:

“Solutions of the field equations on the basis of the original formulation of unified field theory to first approximation for the spherically symmetric case were already obtained by Müntz.”

In the same paper, he states: “I did not see the papers of Lanczos and Müntz.” Even before this, in the same year, in a footnote to the paper of Wiener and Vallarta, we read:

The latter remark refers to a constant query Pauli had about what would happen, within unified field theory, to the gravitational effects in the planetary system, described so well by general relativity²¹² .

Unfortunately, as noted by Meyer Salkover of the Mathematics Department in Cincinatti, the calculations by Wiener and Vallarta were erroneous; if corrected, one finds the Schwarzschild metric is indeed a solution of Einstein’s field equations. In the second of his two brief notes, Salkover succeeded in gaining the most general, spherically symmetric solution [288 , 287 ]. This is admitted by the authors in their second paper, in which they present a new calculation.

“In a previous paper the authors of the present note have treated the case of a spherically symmetrical statical field, and stated the conclusions: first, that under Einstein’s definition of the electromagnetic potential an electromagnetic field is incompatible with the assumption of static spherical symmetry and symmetry of the past and future; second, that if one uses the Hamiltonian suggested in Einstein’s second 1928 paper, the electromagnetic potential vanishes and the gravitational field also vanishes.”

“None of the conclusions of the previous paper are vitiated by this investigation, although some of the final formulas are supplemented by an additional term.” ([417 ], p. 802)

“In recent papers Wiener and the author have determined the tensors $shλ$ of Einstein’s unified theory of electricity and gravitation under the assumption of static spherical symmetry and of symmetry of past and future. It was there shown that the field equations suggested in Einstein’s second 1928 paper [83 ] lead in this case to a vanishing gravitational field. The purpose of this paper is to investigate, for the same case, the nature of the gravitational field obtained from the field equations suggested by Einstein in his first 1929 paper [88 ].”

“that Wiener has shown in a paper to be published elsewhere soon that the Schwarzschild solution satisfies exactly the field equations suggested by Einstein in his second 1929 paper ([85 ]).”

Finally, Rosen and Vallarta [283 ] got together for a systematic investigation of the spherically symmetric, static field in Einstein’s unified field theory of the electromagnetic and gravitational fields [93 ].

Further papers on Einstein’s teleparallelism theory were written in Italy by Bortolotti in Cagliari, Italy [22 , 23 , 25 , 24 ], and by Palatini [242 ].

In Princeton, people did not sleep either. In 1930 and 1931, T. Y. Thomas wrote a series of six papers on distant parallelism and unified field theory. He followed Einstein’s example by also changing his field equations from the first to the second publication. After that, he concentrated on more mathematical problems , such as proving an existence theorem for the Cauchy–Kowlewsky type of equations in unified field theory, by studying the characteristics and bi-characteristic, the characteristic Cauchy problem, and Huygen’s principle. T. Y. Thomas described the contents of his first paper as follows:

“In a number of notes in the Berlin Sitzungsberichte followed by a revised account in the Mathematische Annalen, Einstein has attempted to develop a unified theory of the gravitational and electromagnetic field by introducing into the scheme of Riemannian geometry the possibility of distant parallelism. [...] we are led to the construction of a system of wave equations as the equations of the combined gravitational and electromagnetic field. This system is composed of 16 equations for the determination of the 16 quantities $hˆk i$ and is closely analogous to the system of 10 equations for the determination of the 10 components $gik$ in the original theory of gravitation. It is an interesting fact that the covariant components $ˆ h ki$ of the fundamental vectors, when considered as electromagnetic potential vectors, satisfy in the local coordinate system the universally recognised laws of Maxwell for the electromagnetic field in free space, as a consequence of the field equations.” [350 ]

This looks as if he had introduced four vector potentials for the electromagnetic field, and this, in fact, T. Y. Thomas does: “the components $ˆk hi$ will play the role of electromagnetic potentials in the present theory.” The field equations are just the four wave equations $∑ ˆı eˆkhˆj,ˆkˆk$ where the summation extends over $ˆk$ , with $ˆk = 1, ...,4$ , and the comma denotes an absolute derivative he has introduced. The gravitational potentials are still $gik$ . In his next note, T. Y. Thomas changed his field equations on the grounds that he wanted them to give a conservation law.

“This latter point of view is made the basis for the construction of a system of field equations in the present note – and the equations so obtained differ from those of note I only by the appearance of terms quadratic in the quantities $ˆı hj,k$ . It would thus appear that we can carry over the interpretation of the $h ˆk i$ as electromagnetic potentials; doing this, we can say that Maxwell’s equations hold approximately in the local coordinate system in the presence of weak electromagnetic fields.” [351 ]

The third paper contains a remark as to the content of the concept “unified field theory”:

“It is the objective of the present note to deduce the general existence theorem of the Cauchy–Kowalewsky type for the system of field equations of the unified field theory. [...] Einstein (Sitzber. 1930, 18–23) has pointed out that the vanishing of the invariant $ˆı hj,k$ is the condition for the four-dimensional world to be Euclidean, or more properly, pseudo-Euclidean. From the point of view of our previous notes this fact has its interpretation in the statement that the world will be pseudo-Euclidean only in the absence of electric and magnetic forces. This means that gravitational and electromagnetic phenomena must be intimately related since the existence of gravitation becomes dependent on the electromagnetic field. Thus we secure a real physical unification of gravitation and electricity in the sense that these concepts become but different manifestations of the same fundamental entity – provided, of course, that the theory shows itself to be tenable as a theory in agreement with experience.” [352 ].

In his three further installments, T. Y. Thomas moved away from unified field theory to the discussion of mathematical details of the theory he had advanced [353 , 354 , 355 ].

Unhindered by constraints from physical experience, mathematicians try to play with possibilities. Thus, it was only consequential that Valentin Bargmann in Berlin, after Riemann and Weyl, now engaged in looking at a geometry allowing a comparison “at a distance” of directions but not of lengths, i.e., only of the quotient of vector components, $Ai∕Ak$ [5 ]. In the framework of a purely affine theory he obtained a necessary and sufficient condition for this geometry,

Schouten and van Dantzig also used a geometry built on complex numbers, and on Hermitian forms:

“[...] we were able to show that the metric geometry used by Einstein in his most recent approach to relativity theory [84 , 83 ] coincides with the geometry of a Hermitian tensor of highest rank, which is real on the real axis and satisfies certain differential equations.” ([313 ], p. 319)

The Hermitian tensor referred to leads to a linear integrable connection that, in the special case that it “is real in the real”, coincides with Einstein’s teleparallel connection.

Distant parallelism was revived four decades later within the framework of Poincaré gauge theory; the corresponding theories will be treated in the second part of this review.

6.4.6 Overdetermination and compatibility of systems of differential equations

In the course of Einstein’s thinking about distant parallelism, his ideas about overdetermined systems of differential equations gradually changed. At first, the possibility of gaining hold on the paths of elementary particles – described as singular worldlines of point particles – was central. He combined this with the idea of quantisation, although Planck’s constant $h$ could not possibly surface by such an approach. But somehow, for Einstein, discretisation and quantisation must have been too close to bother about a fundamental constant.

Then, after the richer constructive possibilities (e.g., for a Lagrangian) became obvious, a principle for finding the correct field equations was needed. As such, “overdetermination” was brought into the game by Einstein:

“The demand for the existence of an ‘overdetermined’ system of equations does provide us with the means for the discovery of the field equations”²¹³ ([90 ], p. 21)

It seems that Einstein, during his visit to Paris in November 1929, had talked to Cartan about his problem of finding the right field equations and proving their compatibility. Starting in December of 1929 and extending over the next year, an intensive correspondence on this subject was carried on by both men [50

]. On 3 December 1929, Cartan sent Einstein a letter of five pages with a mathematical note of 12 pages appended. In it he referred to his theory of partial differential equations, deterministic and “in involution,” which covered the type of field equations Einstein was using and put forward a further field equation. He clarified the mathematical point of view but used concepts such as “degree of generality” and “generality index” not familiar to Einstein²¹⁴ . Cartan admitted²¹⁵ :

“I was not able to completely solve the problem of determining if there are systems of 22 equations other than yours and the one I just indicated [...] and it still astonishes me that you managed to find your 22 equations! There are other possibilities giving rise to richer geometrical schemes while remaining deterministic. First, one can take a system of 15 equations [...]. Finally, maybe there are also solutions with 16 equations; but the study of this case leads to calculations as complicated as in the case of 22 equations, and I was not fortunate enough to come across a possible system [...].” ([50 ], pp. 25–26)

Einstein’s rapid answer of 9 December 1929 referred to the letter only; he had not been able to study Cartan’s note. As the further correspondence shows, he had difficulties in following Cartan:

“For you have exactly that which I lack: an enviable facility in mathematics. Your explanation of the indice de généralité I have not yet fully understood, at least not the proof. I beg you to send me those of your papers from which I can properly study the theory.” ([50 ], p. 73)

It would be a task of its own to closely study this correspondence; in our context, it suffices to note that Cartan wrote a special note²¹⁶

“[...] edited such that I took the point of view of systems of partial differential equations and not, as in my papers, the point of view of systems of equations for total differentials [...]”²¹⁷

which was better suited to physicists. Through this note, Einstein came to understand Cartan’s theory of systems in involution:

“I have read your manuscript, and this enthusiastically. Now, everything is clear to me. Previously, my assistant Prof. Müntz and I had sought something similar – but we were unsuccessful.”²¹⁸ ([50 ], pp. 87, 94)

In the correspondence, Einstein made it very clear that he considered Maxwell’s equations only as an approximation for weak fields, because they did not allow for non-singular exact solutions approaching zero at spacelike infinity.

“It now is my conviction that for rigourous field theories to be taken seriously everywhere a complete absence of singularities of the field must be demanded. This probably will restrict the free choice of solutions in a region in a far-reaching way – more strongly than the restrictions corresponding to your degrees of determination.”²¹⁹ ([50 ], p. 92)

Although Einstein was grateful for Cartan’s help, he abandoned the geometry with distant parallelism.

6.4 Distant parallelism

6.4.1 Cartan and Einstein

6.4.2 How the word spread

6.4.3 Einstein’s research papers

6.4.4 Reactions I: Mostly critical

6.4.5 Reactions II: Further research on distant parallelism

6.4.6 Overdetermination and compatibility of systems of differential equations