Einstein–Schrodinger Theory in Paris

10 Einstein–Schrödinger Theory in Paris

Research on unified field theory in Paris centered around the mathematician A. Lichnerowicz, a student of Georges Darmois, and the theoretical physicist Marie-Antoinette Tonnelat. It followed two main lines: the affine or metric-affine approaches of Einstein and Schrödinger, and the 5-dimensional unification originating with G. Nordström and Th. Kaluza. The latter theme was first studied in Paris by Y. Thiry, a former student of A. Lichnerowicz (“Jordan–Thiry-theory”), and by students of M.-A. Tonnelat; the first topic, “Einstein–Schrödinger theory”, mainly by Tonnelat and her coworkers but no strict divide did exist. Between 1950 and the mid 1960s at least two dozen doctoral theses on topics in unitary field theory were advised by Lichnerowicz and/or by Tonnelat. Whereas the work of Tonnelat’s students could be classified as applied mathematics, Lichnerowicz’s interest, outside of pure mathematics, was directed to mathematical physics with its rigid proofs. This joined attack on unsolved questions and problematic features of classical unified field theory has made clear that (1) the theories under scrutiny were mathematically consistent, but (2) they could not be transformed into an acceptable part of physics.

10.1 Marie-Antoinette Tonnelat and Einstein’s Unified Field Theory

We first make contact with Marie-Antoinette Tonnelat and her research group in the Institut Henri Poincaré. She had studied with Louis de Broglie. During the German occupation of Paris she continued to work with de Broglie and on her own in the field of (relativistic) “spin-particles”, also under the influence of a gravitational field. She applied de Broglie’s “méthode de fusion” to massive spin 1 particles (called photons) in order to arrive at particles with maximal spin 2: spin 2 corresponded to the graviton. The theory contained the graviton, three photons and two scalar particles (spin 0) all with non-vanishing mass. Each relevant field component satisfied the Klein Gordon equation. From there, she arrived at Maxwell’s equations and the linearized version of the equation for Einstein spaces $Rij = λgij$ . She carefully looked at the theory for a particle with spin 2 as a “unitary theory” and preferred to call it “a unitary formalism” ([616 *], p. 163, 164):

“[…] the theory of maximal spin 2 allows to show how a unitary theory presents itself, approximately, but in the framework of wave mechanics.”¹⁶⁹

In this paper and in others in the early 1940s she also wrote down the standard commutation relations for the quantized spin-2 field [611 *, 616 *].

Many of her papers were published in the proceedings of the prestigious Academy of Sciences in Paris¹⁷⁰ [613 , 612 , 614 , 610 , 611 , 615 *, 616 *, 617 *]. The Academy’s sessions had been interrupted for a mere three weeks due to the German occupation. According to its president:

“Despite the ordeal which oppresses the country, the Comptes Rendus attest that scientific research has not bent, and that the Academy of Sciences remains a focus of ardent and fruitful work. […] Let us work.”¹⁷¹

Right after the war in 1946, like other young French scientists, Mme. Tonnelat apparently spent some time in Dublin with the group of E. Schrödinger. On the background of her previous work on a “unitary formalism” emerging from spin-2 particles, her interest in the unified field theories of Einstein and Schrödinger might have been brought forward during this stay with Schrödinger. Her scientific teacher, L. de Broglie, supported her research in UFT, although he himself stayed away from it. After briefly listing “innumerable attempts […] to complete the general theory of relativity […] and transform it into a ‘unified theory’ ”, he went on to say:

“Einstein’s efforts in this direction, ever characterized by the salient originality of his thought, will not be examined here. Despite their indisputable interest, they have not, to the best of our knowledge, attained any decisive success […]. Moreover, the nature of the electromagnetic field is so intimately bound to the existence of quantum phenomena that any non-quantum unified theory is necessarily incomplete. These are problems of redoubtable complexity whose solution is still ‘in the lap of the gods’ ” ([113 ], p. 121).

At the 8th Solvay Congress in 1948 in Brussels, Mme. Tonnelat presented a paper by L. de Broglie on the photon as composed of two neutrinos. Schrödinger asked a question afterwards about a tiny mass of the photon ([446 *], p. 444). During her work on unified field theory Tonnelat continued to study spin-particles, e.g., to regard a spin-1-particle as composed of two spin-1/2-particles [625 ].

10.2 Tonnelat’s research on UFT in 1946 – 1952

As we will see, M.-A. Tonnelat made several attempts until she reached the final version of her unified field theory. In the warm-up, i.e., in her first paper on the subject, Tonnelat referred only to Schrödinger’s theory [545 , 549 ] and to Eyraud’s thesis of 1926 (cf. Section 5 of Part I). By her theory, she intended to describe the gravitational, electromagnetic and mesonic fields [618 ]. The paper was reviewed by the American theoretical physicist A. H. Taub in Mathematical Reviews [ External Link 0017205]:

“The author states without proof some formal consequences of a variational principle in which the action function is an unspecified function of a symmetric second order tensor and three (of which two are independent) anti-symmetric second order tensors. These four tensors are defined in terms of a general affine connection in a four dimensional space, and its derivatives. The connection is not assumed to be symmetric. The paper does not explain how the invariant element of volume entering into the action principle is defined.”

By her, the torsion tensor $Λ k ij$ is defined through

where $k k Γij = Γ(ij)$ and $k k Λij = Λ[ij].$ Vector torsion is denoted by her as $k Λj = Λjk$ and corresponds to our $2Sk$ . She introduced four tensors of rank two $K μν,Sμν,Tμν,F μν$ corresponding in our notation (in the same order) to:

with the torsion vector $Si. Tμν$ corresponds to homothetic curvature $Vkl = K j − − jkl$ defined by (66 *) in Section 2.3.1. In the spirit of Schrödinger, a metric density is introduced by:

The Lagrangian is defined as a functional of these four tensors. By help of a “geometrically” defined matter tensor the resulting field equations are rewritten in the form of Einstein’s equations.

In three following papers, Tonnelat had become acquainted with some of the past literature on the subject, and included references to Einstein [619 , 620 , 621 ]. Her intention was to generalize this past work for an arbitrary affine connection in a more systematical way such as to “augment the interpretative possibilities of the theory.” The same tensors as before (one symmetric, three antisymmetric) were introduced with a bewildering change of her own notation: $G μν$ instead of $K ,F ∼ S ,S ∼ T μν μν μν μν μν$ , and $H ∼ F μν μν$ . Again, the Lagrangian is a functional of these four tensors and, possibly, the torsion vector. After dual (conjugate) field tensor densities were defined through:

and a new quantity $¯ ik R$ introduced by its symmetric and antisymmetric parts $¯ik ¯ ik ¯ik R := G − F$ , the field equations could be written in the form:

It is to be noted that this equation is purely formal: no definite expression for the Lagrangian had been chosen.

In the 2nd of the three notes, Tonnelat introduced a symmetric metric and split (340 *) into its symmetric and antisymmetric parts (S) and (A). According to her, the exact general solution to (A) was very difficult to produce; this is seen from the complicated expression in terms of quantities like the metric, its first derivative and torsion, known in principle. In the third note, the formalism is exemplified in second-order approximation and with additional restrictions, e.g., in de Sitter space. Proca’s equation for the mesonic field emerges. In her next paper, (340 *) is rewritten in the form of a metric compatibility condition:

where the covariant derivative $′ ∇$ is defined with help of a new connection $′ L$ . $ik r$ is related to $ik ¯R$ by $ik √ --ik R¯ = rr$ with $r := det(rik)$ and $rik$ the inverse of $rik$ .¹⁷² The decomposition

$¯ik ¯ ik ¯ ik R = G − F$ is also used. By a combination of symmetric and antisymmetric parts $rik = γik − ϕik$ two further tensors and their inverses are introduced, i.e., $ik γik = γ(ik),γ$ and $ϕik = ϕ[ik],ϕik$ . Now

Here, $f¯l = γlmϕmnf n$ , and $0 fn = √-1−g ¯F n = ∇lF¯nl$ with $¯Fnl$ having been defined in (339 *). Apart from ingenious manipulations of a very general formalism, no physical progress with regard to Schrödinger’s and Einstein’s theories had been reached in this paper [622 *]. As Tonnelat herself confessed: “The physical interpretation of the thus formed tensors is far from being immediately clear.”¹⁷³

Two summaries of her attempts with regard to affine field theory were presented by Tonnelat in 1951 and 1952 [627 *, 628 *]. It would be natural to discuss them together but M.-A. Tonnelat again complicated matters by an altered notation from one paper to the next. In 1951, for the first time, she introduced the transposed connection $k &tidle;Lij$ and thus worked with both $Pjk (L ) := − K− (L )jk$ and $&tidle; Qjk(L ) := − K+ (L)jk$ . Further basic tensors chosen were $s s S (L)jk := ∂kL sj − ∂jL sk$ and $T (&tidle;L)jk := ∂kL s− ∂jL s js ks$ , and a fifth quantity $Ijk := 4SjSk$ with $2Sk$ corresponding to her $s Λ ks$ . Having read the 2nd edition of Einstein’s The Meaning of Relativity – as she claimed,¹⁷⁴ she accepted Einstein’s suggestion that only Hermitian quantities should be used as fundamental quantities in a Lagrangian. The following objects were introduced by her:

1 (2)Rik = Pik(L) + Pki(&tidle;L) − -(S (L)ik + S (&tidle;L)ki) 4 = ∂lLikl− ∂kL (il)l+ L ik lL(lmm) − LimlL lkm , (343 ) &tidle; Hik = S(L )ik + S(L )ki = 2(∂iSk − ∂kSi), (344 ) Ijk := 4SjSk, (345 ) 1 Lik = Pik(L) − Pki(&tidle;L) − -(S (L)ik − S (&tidle;L)ki) = Likr Sr − ∂iSk + ∂kSi, (346 ) 2 K = − 1(S(L ) − S (L&tidle;) ) = ∂ L l− ∂L l. (347 ) ik 2 ik ki k (il) i (kl)

From these, Einstein had formed the linear combination:

which remains unchanged by a transformation of the connection:¹⁷⁵

By choosing $1 ϕk = − 3Sk$ , a new connection $∗ L$ with vanishing torsion vector is obtained. Furthermore, $∗ U (L )ik = U (L )ik$ . In place of the field densities (339 *), now the three quantities are used:

which are then manipulated similarly as in the previous paper.¹⁷⁶ Eventually, Eq. (341 *) is obtained again, where the connection $′ L$ now is derived by beginning with $∗ L$ . The new Eq. (341 *) remains unchanged if the connection $′ L ij k$ and $rik$ are replaced by the transposed objects. Up to here, a general formalism has been developed. In order to proceed, M.-A. Tonnelat then picked the same Lagrangian $ℒ$ as Schrödinger had in Eq. (229 *), i.e., $ℒ = 2∘ −-det(R---) λ rs$ , yet based on her vector-torsion-free connection $∗ L$ . The field equations derived by her decompose into the fundamental ones (variation with regard to $U (ik)$ ) and such of a “Maxwellian” type (variation with regard to $H ik$ ). With the decomposition $rik = γik − ϕik$ of the previous paper and the additional $ik ik ik r = g − f$ together with the inverses of these four tensors, eight basic fields float around in the theory waiting for physical interpretation. Although still in a purely affine theory, the particular interpretation relating $rik$ or $rik$ with a metric could make the transition to mixed geometry [cf. Section 2.1 and Eqs. (3 *), (4 *)]).¹⁷⁷ The field equations look very complicated such that an approximative handling seemed appropriate. For weak fields $ϕik$ , i.e., up to 2nd order, a Proca-type equation was obtained. For the approximated Ricci tensor, field equations of the form of Einstein’s equations with cosmological constant and including the “metric” $r = γ (ik) ik$ were reached. The matter tensor is a complicated functional of the connections $L$ and $′ L$ . Thus, formally, “matter” has been geometrized. Tonnelat did not give physical interpretations to the tensors appearing except for identifying $¯Il = 𝜖ijklIijk$ , $Iijk := ∂iϕjk + ∂jϕki + ∂kϕij$ , or its dual $J k = 16𝜖klrsIlrs$ with the electric current density.

In a separate note, M.-A. Tonnelat discussed a possible relation $5λ = μ2 6$ between the cosmological constant $λ$ appearing in the Proca equation obtained from her unified field theory in an approximation up to 2nd order in $kij$ :

and the Proca equation for spin-1 or spin-2 particles of rest mass $μ$ in a space of constant curvature derived by her [615 ]:

While she abstained from over-interpreting this relation in the sense of bringing both types of theories closer to each other, she still had some hope concerning the understanding of elementary particles:

“[…] the possibility remains of finding, thanks to the exact solution to the equation $0 = gik∥l +−$ , a solution valid even in the case of strong fields, an explanation of the nature of the elementary particles. However, as Schrödinger very strongly emphasized, the realization of this hope remains quite problematical despite all efforts.” ([626 ], p. 832) ¹⁷⁸

In her subsequent paper [628 *], Tonnelat pledged to adopt “Einstein’s notations”.¹⁷⁹ Two connections were used, both free of vector torsion: $(1) k k 2 k Δ ij := L ij + 3δ[iLj ]$ and $(2)Δ ijk:= L ijk + 23δki Lj (Lj = Sj)$ . $(1)Δ$ is the same as Schrödinger’s “star”-connection; cf. Eq. (232 *) in Section 8.1¹⁸⁰ She then discussed at length which Ricci-tensor to take as basic variable, and listed four possible expressions: $R (0)≃ − K (L) ,R(1)≃ K ((1)Δ ) ,R (2) ≃ K ((2)Δ ) μν − jk μν − jk μν − (jk)$ , and the Hermitian $R (3μ)ν ≃ 1(K ((2)Δ )(jk) + K ((2)(&tidle;Δ)(jk)) 2 − −$ . The formalism then is carried out with all $R(μAν),A = 0,1,2,3$ . For the first time, $rik = γik + ϕik$ is identified as the metric tensor.¹⁸¹ As before, it ensued from $ˆik -∂ℒ- √ -----ik R = ∂Rik = − rstr$ and its reciprocal. The field equations following from variation with regard to the $R (μAν)$ looked like:

where the connection $′ Lijk$ in the covariant derivative is formed with the help of $(1)Δ, rik,γik,ϕlm$ , and $∂lF ˆlk.$ The same Lagrangian as before is used such that the field equations simply are:

For the further evaluation, for each of the four tensors $Ri(Ak)$ the corresponding connection as a functional of the fields must be inserted into (354 *). This leads to very complicated manipulations such that, again, Tonnelat decided to take the antisymmetric part $kik$ of $rik$ to be small of the order $𝜖$ , expand in $𝜖$ and neglect all terms of the order $3 𝜖$ . The relation between the cosmological constant $λ$ expressed by the curvature radius of de Sitter space $ℛ$ , i.e., $λ = − 3ℛ-$ now is alternatively, $μ2 = − 2λ3$ , or $μ2 = − λ 2$ .

A comparison of M.-A. Tonnelat’s research with respect to Einstein’s and Schrödinger’s shows that, though first generalizing the class of possible Lagrangians enormously by including four tensor fields, in the end she went back to only one: she used Schrödinger’s Lagrangian corresponding to Einstein’s Lagrangian for general relativity. She avoided the additional equation within the field equations which demands that the torsion vector vanish by directly starting with a connection with zero torsion vector. Although this approach was new,¹⁸² most characteristic and important for her research seems to be that she directed her attention to “metric compatibility” in the sense of (200 *) of Section 7.2 and succeeded to “solve” it for the connection; cf. Section 10.2.3. She also showed that out of a purely affine theory, by proper definitions and interpretations, a theory within mixed geometry could be made. It was such a theory that she finally adopted.

10.2.1 Summaries by Tonnelat of her work

M.-A. Tonnelat summed up her research within Einstein–Schrödiner theory of almost a decade, published in many short notes in Comptes Rendus and in papers in other journals, in a monograph in 1955 [632 *]; it eventually became translated into English [642 *]. Unassumingly, she assessed the book as “[…] a collection of works with the sole aim of facilitating research on the subject” ([632 *], p. IX.)¹⁸³ In the Mathematical Reviews [ External Link MR0076499], A. H. Taub not only gave a detailed description of its contents but also put the book into a larger perspective:

“In this book the author summarizes and discusses a great body of material on the Einstein and Schrödinger unified field theories. […] The previous work of the author is collected and presented in a logical coherent fashion. The results obtained by other workers are also presented and compared. Thus, in this single volume containing an introduction and seven chapters one can obtain a well written complete and succinct account of the recent work in the field.”

Tonnelat’s associate J. Winogradzki, in her report on the book, gave a condensed list of the contents and found “that the major part of the work is devoted to the mathematical study of the field equations. The two last chapters deal with some physical content of the theory” [705 ]. Tonnelat clearly drew the line with regard to work by Lichnerowicz, e.g., the initial value problem. For her, the balance between the remaining problems of UFT and the results obtained was positive: “[…] Einstein’s theory binds together the realization of a satisfying synthesis, originating from a very general principle, and the possibility of new provisions.” ([632 *], p. 11) ¹⁸⁴ For her, UFT was a fruitful and important theory.

After another decade, in 1965, she published a second monograph on unified field theories reflecting now the development of research in all more prominent approaches to unified field theory. Her work and the results of her group concerning the Einstein–Schrödinger theory took up only one chapter [641 *]. A few of the doctoral theses she had advised were referred to. In its third part, the book aimed at presenting some connection between classical and quantum field theory. As a first approximation to an unknown nonlinear theory, Tonnelat’s alternative theory of gravitation (linear gravity) investigated by her in the 1960s is included (cf. Section 16.1). In comparison with her first book, she had become even more modest:

“Whatever the future of the unitary theories might be, this book will have reached its objective, if it has somehow shown that the ties between electromagnetism and gravitation form a history of rebouncings the outcome of which is far from being written.” ([641 *], p. IX)¹⁸⁵

In the following, some of the main aspects of her approach to UFT will be described.

10.2.2 Field equations

As she had moved from pure affine to mixed geometry during her research, M.-A. Tonnelat then started from the Lagrangian density $ˆℒ = ˆgijR ij$ where the reciprocal metric density $ˆgij$ is defined in (13 *), and $Rij$ is one out of the list of possible Ricci-tensors. $ij t t ℒˆ= ˆℒ(ˆg ,Lrs ,∂vL rs )$ . Variation with respect to $ˆgij$ and $L k ij$ led her to the field equations. With $Rij =!− K ij −$ she arrived at:

ik ˆg+− = − 2δkgˆisSs + ˆgikSl, ∂lmˆlk = 0, (355 ) ||l 3 l K ij = 0, (356 ) −

where $∘ ---------- ˆmik := − det(gst) mlk$ with $mik$ being the skew-symmetric part of the inverse metric, and $Si := L j [ij]$ .¹⁸⁶ . After going over to the connection previously named $(2) k Δ ij$ , i.e., Schrödinger’s “star”-connection, Einstein’s weak field equations followed from the variational principle in the form:

where $Dl$ stands for the covariant derivative with respect to the new connection; $K ij −$ is the Ricci tensor $(2) K− ( Δ )ij$ formed with $(2) Δ$ and named $Wij$ by Tonnelat (cf. [632 *], Eq. II, p. 31). If $Si = 0$ is added to Eqs. (355 *) and (356 *) Einstein’s strong field equations obtain; they no longer follow from the variational principle. M. Lenoir used the fiber bundle of affine reference frames and the transformation groups implied by it to arrive at Tonnelat’s field equations [358 ].

10.2.3 Removal of affine connection

A first objective was to use the equation $i+k− gˆ ||l = 0$ or, equivalently, (30 *) to express the affine connection $L ijk = L ijk(grs;∂kgrs)$ as a functional of the asymmetric metric $gij$ and its first derivatives in the same way as the Christoffel symbol had been expressed by the metric and its first derivatives. Now, the system comprises 64 linear equations for 64 variables $L k ij$ . As an already solved algebraic problem this might not create much interest for “pure” mathematicians: an inverse matrix must be found, if only a large one with functions as its elements. V. Hlavatý called for an “elementary algebraic device” to be invented. As a problem in applied mathematics, even in the computer age, it takes quite an effort to do this by computer algebra. The wish to obtain the solution in tensorial form aggravates matters. According to Hlavatý: “Finding such a device is by no means an easy task” ([269 *], p. 50).

In Riemann–Cartan theory, i.e., in a theory with symmetric metric and arbitrary linear connection, we obtain:

If $L [ilk] ⁄= 0$ , the usual method of solving (359 *) for the Christoffel symbol as a functional of the metric and its first derivatives still works,¹⁸⁷ but no longer for (30 *).

After E. Straus had not been able to present a manageable solution (cf. Section 7.3), M. A. Tonnelat invested a lot of work into the same methodical approach. In a series of steps involving many intermediate expressions which had to be replaced in the end, she obtained a solution. In her first attempt during the early 1950s [622 , 623 ], summed up in her monographs [632 *, 642 ], the solution is achieved by first splitting $gμν$ into its irreducible parts. If $u k ij$ is defined similarly as in (39 *) but with a different $u k ij$ by

the decomposition of (30 *) leads to

h u l= S&tidle; lk + S&tidle; lk , (361 ) jl ik ij lk kj li 1 {kij} hjlS&tidle;ikl = --fijk + ∇j kik − (uijl klk − ukjl kli), (362 ) 2

where¹⁸⁸ $&tidle; l S kj$ is the torsion tensor of the connection $&tidle;k Lij$ , and $fijk = ∂ikjl + ∂jkli + ∂lkij$ . We have met (361 *), (362 *) in principle already in Section 2.1.2. A. H. Taub had reviewed one of her papers on the subject from 1950 [ External Link

MR0037634]: “These systems are then solved explicitly. Straus [592 ] has also solved this problem by another method after remarking that the method used in this paper is feasible.” The main conclusion is that the symmetric part of the affine connection may be expressed by its antisymmetric part from (361 *), while (362 *) then determines the antisymmetric part, in terms of “the fields” $k {ij}h,hik,kik$ . In the next step thus $uijk$ is removed from (362 *). By a lengthy calculation, torsion is expressed by “the fields”. Thus, the connection

is fully known.¹⁸⁹ The procedure works if

where $∘ --- a =: 2 − g+ 6k,b =: 2 -k-[3 − g + k] h h −h h h$ and $g,h,k$ are the determinants of $gik,hik,kik$ , respectively. $a$ and $b$ turn up during the elimination process in which $g ⁄= 0$ and $h ⁄= 0$ is always assumed. $g = 2h, k = 0$ obviously leads to $a = b = 0$ and thus leaves the solution indeterminate; this is a result also obtained in an independent calculation by H. Takeno and coworkers [602 *]. They acknowledged Tonnelat’s solution in a note added in proof but found her condition $k ⁄= 0$ “too stringent”. M.-A. Tonnelat dealt with the case $k = 0$ in another paper and confirmed her general result, i.e., the condition $-g h ⁄= 2$ for a solution to exist [630 *]. Further careful investigations of possible subcases were made later by Hlavatý [269 *] and Mishra [434 *]; cf. Section 13.3.

A reproduction of Tonnelat’s calculations would not bring further insight, the more so as lots of auxiliary symbols were introduced by her including indices with one and two strokes. M.-A. Tonnelat has presented the method in detail not only in her books but also in an article [633 ], and in a talk given at the outstanding Relativity Jubilee Conference in 1955 in Bern ([631 *], p. 192 – 197). She was keen on securing priority, i.e., for having found the solution already in 1949 – 1950. This seemed imperative to her because in the meantime V. Hlavatý [257 , 259 ], and N. S. Bose [52 , 50 *] had also published solutions of $g−ik+∥l = 0$ by other methods (for Hlavatý cf. Section 12.2). In fact, Hlavatý had reviewed Tonnelat’s paper in Mathematical Reviews [ External Link MR0066128], in which she had shown that $det(kij) = 0$ did not affect her solution [630 ], and he added that “for the solution in the exceptional cases $g = 0,2 h$ ” one should consult a forthcoming paper of his [265 ]. While the limit $kij → 0$ leads back to the well known results in general relativity, the other limit $hij → ηij$ seemingly has not been discussed intensively by Tonnelat.

Indeed, the whole procedure is drastically shortened and becomes very transparent if $hij = ηij$ is assumed. In this context, apparently, no one did look at this particular case. N. N. Ghosh began with another simplified metric built like the general spherically symmetric metric, i.e., with only $h ,h ,h ,h ⁄= 0 00 11 22 33$ and $k ,k ⁄= 0 10 23$ , but with all components being functions of the four coordinates $0 3 x ,...,x$ However, he managed to solve (30 *) for the connection only by adding 4 conditions for the first derivatives of $hij$ and $kij$ in an ad hoc manner [222 *]. S. N. Bose rewrote (30 *) into an inhomogeneous linear equation for tensorial objects $Ti[jk],Ui[jk]$ , i.e., $lh(Ti[jk]) = Ui[jk]$ where $lh(T )$ is homogeneous and linear in $T$ [50 ]. Considered as matrix equation, its solution is $T = BU$ . The matrix $ˇ j js C i := h ksi$ , its eigenvalues and eigenvectors play an important role. Although the method is more transparent than Tonnelat’s, the solution is just as implicitly given as hers. Interestingly, at first Einstein seems to have had some doubts about her method of solution, because torsion expressed by the fields $hij,kij$ , and their first derivatives depended on the choice of the object used as the (symmetric) metric and raised a question of compatibility: “Mr. A. Einstein has directed my attention to this difficulty.” ([624 ], p. 2407).¹⁹⁰ In the paper, M.-A. Tonnelat could disperse Einstein’s reservations. In a letter to A. Einstein of 21 June 1951, L. de Broglie wrote:

“I am glad to learn that one of my former pupils, Mme. Tonnelat, who really is a remarkable person, has had contact with you with regard to her papers on the unitary theories, and that you have shown an interest in her results”.¹⁹¹

P. G. Bergmann’s report on the Jubilee Conference was noncommittal: “A. Tonnelat of the Sorbonne reported on some mathematical results she had obtained on this theory independently of Einstein and Kaufman.” And a little later: “The papers by Kaufman and Tonnelat are too technical to be reported here.”([22 *], p. 493.)

In a later approach by M.-A. Tonnelat [636 ], the affine connection is expressed by the metric as above but without a decomposition of $g μν$ – in a similar but very much more complicated way as in the case of the Levi-Civita connection (the Christoffel symbol). This second method does not work if $4h + 12k = 3g$ . An improvement of it was given by Dautcourt [110 ] who also showed that $4h + 12k ⁄= 3g$ does not guarantee a solution. V. Hlavatý used still another method to express the affine connection as a functional of the metric; cf. Section 12.2.

St. Mavridès applied Tonnelat’s method in the case of $lij$ and $mij$ , i.e., the inverses of $ij ij l ,m$ in $ij g$ being used as metric and electromagnetic field.¹⁹² As an existence-condition (364 *) appeared as well [406 ]. A plenitude of further work concerning this problem of how to express the affine connection by the asymmetric metric, its derivatives and torsion was done, with the uniqueness proof by Hlavatý & Saenz among them [270 ]. It amounted to a mathematical discussion of all logically possible cases and subcases without furthering UFT as a physical theory; cf. Sections 12.2 and 13.3.

As a functional of the metric, its first and second derivatives, the Ricci tensor becomes a rather complicated expression. To then find exact solutions of the remaining field equation in (340 *) is a difficult task. In a paper dealing with approximations of the field equations, M.-A. Tonnelat tried to show the superiority of her method by applying a scheme of approximations to her (weak) field equations [634 *]. However, the resulting equations of 4th order for weak electromagnetic fields $kij$ and of 1st order for weak gravitational fields $hij$ are still as complicated as to not allow a physical interpretation. In fact, the solution of the problem to remove the connection from the field equations neither helped the search for exact solutions nor contributed to a convincing physical interpretation of the theory. Nonetheless, it was of crucial importance for the proofs given by A. Lichnerowicz that the initial value problem could be well posed in UFT.

In a later development, Eq. (200 *) had been made inhomogeneous:

According to Tonnelat, $Aikl$ in (365 *) is a linear function of the first derivatives of the metric and an additional vector field with 2 free parameters; cf. Section 10.3.3, particularly Eq. (383 *). The simple expression J. Lévy [361 *] used, by starting from $i+k− ˆg ||l$ , was:

leads back to the first equation of (355 *). As was shown in Section 9.6, the most promising approach seemed to investigate special cases (spherical or axial symmetry), or approximate solutions. This also was done by Tonnelat at the end of one of her papers of 1955 [631 ], and tried again, later, in a dissertation advised by her [59 *]. There, the components of the connection were calculated and compared with the results of Bonnor [32 ]. In fact, for high symmetry, the whole work of solving (30 *) is unneeded: the field equations were solved at the same time for both the the metric $gij$ and the connection; cf. [475 ].

A special application refers to Schrödinger’s “star”-connection. In this case, for the skew-symmetric part:

where the bracket $⟨...⟩$ denotes cyclic permutation ([305 ], p. 745).

Remark:

It is interesting to confront Tonnelat’s position with Schrödinger’s. In his paper of 1951, he limited himself to an approximate solution of (30 *) by splitting the connection into its symmetrical and skew parts ([558 *], Eqs. (1,4), (1,5)):

L k= {k}h − hkr (kjsL s+ kisL s) = {k}h − hkr (kjsS s+ kisS s), (368 ) (ij) ij [ir] [jr] ij ir jr L k= S k= 1hkr (k + k + k ) + hkr (k L s− k L s ), (369 ) [ij] ij 2 ir,j rj,i ij,r js (ir) is (jr)

similarly as M.-A. Tonnelat had done in (361 *) and (362 *). Equations (368 *) and (369 *) are more convenient for the setting up of an approximation scheme. In Schrödinger’s words: “ […] from (369 *), if the $kik$ are small, the components of the tensor [torsion] are small of the same order. Hence, from (368 *) the symmetric affinity differs from the Christoffel-brackets only by quantities of the second order. Thus, using in (369 *) the Christoffel-brackets for $k L(ij)$ etc., one gets the tensor, with an error of the third order; and if this is used in (368 *), one gets $L (ikj)$ with an error of the fourth order.” From this, “a series of ascending powers of the $kik$ ” can be developed. Such an approximation up to a certain order, then can be inserted into the remaining field equations.¹⁹³

10.3 Some further developments

From the mathematical point of view, the results of Tonnelat and Hlavatý may be interpreted as having simplified the study of the weak field equations to some degree. For physics, no new insights were gained. In order to make progress, topics like exact solutions, equations of motion of test particles, or the problem how to express continuously distributed “matter” had to be investigated. It is here that the conflict between the “dualistic” approach to UFT separating the fields and their sources, and the “purely geometric” one showed up clearly. In the latter, the (total) field itself defines its own sources.

10.3.1 Identities, or matter and geometry

As a consequence of the contracted Bianchi-identities, in general relativity the divergence of the Einstein tensor $Gij := Rij − 1Rgij 2$ identically vanishes: $∇ Gij = 0 j$ . An implication of the field equations $ij ij G = − κT$ then is the vanishing of the divergence of the so-called “matter” tensor or energy-momentum-stress tensor of “external” matter, a quantity without geometric significance. In general relativity, $T ij$ is a functional of the metric, the matter variables, and eventually the connection through covariant derivatives according to the principle of minimal coupling. $∇jT ij = 0$ is used to derive equations of motion for point particles or field equations for matter fields. In UFT, according to Einstein, no external matter is allowed to occur; matter variables are to be defined from within the geometry.¹⁹⁴ Hence, a distinction between external and internal regions as exemplified by the corresponding Schwarzschild solutions in general relativity would be unnecessary. In the words of M.-A. Tonnelat:

“The immediate advantage of a unitary theory is that from the theory itself the form of the electromagnetic energy-momentum tensor and, perhaps, of the matter tensor can be extracted. The expression of this tensor then would be imposed by the very geometric principles, and not by conclusions from an alien theory as interesting as it might be.” ([635 *], p. 6)¹⁹⁵

An example for an electromagnetic energy-momentum tensor built from geometric quantities is given in (422 *) of Section 10.5.4.

As we have seen in Section 9.2, Einstein derived an identity corresponding to the contracted Bianchi-identity; cf. (260 *). It may be rewritten in various forms such as $∂ [K ˆglk + K ˆgkl − δlK ˆgmn] = 0 l − ik − ki i− mn$ [409 *], or

where the brackets {} now denote cyclic permutation [81 *]. Prior to this, a direct derivation by use of the Lie derivative and the invariance of the Lagrangian had been presented by Schrödinger and Lichnerowicz [556 , 368 *, 369 *]. Lichnerowicz gave the identity the form ([370 *], Eq. (6.2) or [371 *], p. 272–273):

with $Nˆ j = √ −-gN j:= √ − g-(Hj − 1δjHs ) k k k 2 k s$ and $2Hj = − K ksgjs − K skgsj k − −$ .

A reformulation led to

where the Ricci tensor $K− rs$ is derived from an arbitrary connection $k L ij$ . Expressed by the connection with vanishing vector torsion $(2)Δ k ij$ , i.e., after the interchange of $L k ij$ and $(2)Δ k ij$ , (372 *) remains formally unchanged for $(2) k K− ik = K− ik( Δ ij )$ . (372 *) can also be expressed as an ordinary divergence:

where $ˆ j 1 ˆ js ˆ sj Tk = 2(Tksg + Tskg )$ and $ˆ √ --- Tij = − g Tij$ with $1 K− ij − 2gijK− = κTij$ [555 , 409 ], ([632 *], p. 110–113), ([641 *], p. 305–308). Expressions for a formal “matter tensor” and an “energy pseudo-tensor” $j tk$ may be read off here (Cf. Eq. (7.22) on p. 113 in [632 *]). The procedure is ambiguous, though, as is known from general relativity. In Mme. Tonnelat’s group, (370 *) was named “identité de conservation” or “conditions de conservation”, and exhibited in several forms. For a Lagrangian of the form $Hrs(Lijk, ∂lL ikj ) ˆgrs + ℳˆ$ , where the scalar density $ℳˆ$ is independent of the connection $L k ij$ , the “identities of conservation” can be given the form ([359 *], p. 89):

For each solution of the corresponding field equations they are identically satisfied.

In the same spirit, (373 *) is often called “conservation law” although the quantity conserved need not correspond to a physical observable. An example of this is given by Kursunŏglu [343 *] who chose as his pseudo stress-energy-momentum tensor “beyond the shadow of doubt” to be:

with $k k k s ℬ ij = L ij − δjL (is)$ , and $ij s r s r ℬ = ˆg ℬij,ℬij = Lij L (sr) − Lir L(js)$ . $p$ is a constant defined in Section 9.3.3. Using the notations of Kursunŏglu, Clauser derived what he named contracted Bianchi-Einstein identity, i.e., (370 *): $∂j ˆTkj= 0$ with $ˆTkj = − 2(K (lk)ˆg(lj)) + K [lk]ˆg[lj]) + δ jk (ℬ+K lm ˆglm ) − ℬjlm ˆglm,k − − −$ . It is called a “strong” conservation law, because only Eqs. (250 *) and (251 *) of the weak field equation have been used, while for a“weak” conservation law all of the weak field equations would be needed [81 *]. $ˆTij$ is a tensor density only with regard to linear coordinate transformations.

Another approach for the introduction of the energy-momentum tensor of matter $Tik$ is the following. First a symmetric metric must be be chosen, e.g., $hmn$ . Then in the symmetrical part of the Ricci tensor $K(ij)(L )$ a term of the form of the Einstein tensor $Gik(hmn )$ is separated out. The field equations of UFT are then re-written as formal field equations of the type of Einstein’s equations in general relativity plus terms left over. This remainder is identified as $Tik ∼ Gik$ . The method is applicable because mixed geometry can always be re-interpreted as Riemannian geometry with many extra fields (geometric objects). Its ambiguity lies in the choice of the Riemannian metric. Taking $h ij$ as the metric, or the reciprocal of $∘ -- l lij g$ or $∘ -- g lij l$ like in [273 *, 390 *], or another of the many possible choices, makes a difference. By the formulation within a Riemannian geometry, the unifying strength of a more general geometry is given up, however. Also, according to a remark by M.-A. Tonnelat, the resulting equations $∇sT is = 0$ are satisfied identically if $K [ij](L) = 0$ holds. Thus, the information about the gravitational field contained in the symmetric part of the field equation $K (ij)(L) = 0$ does not influence the equations of motions of matter following from $∇sT is = 0$ [637 *]. Related with this is the fact that the matter tensor “seems to vanish together with the electromagnetic field $g[ij]$ , or at least with a field the properties of which remind of the electromagnetic field” ([635 *], p. 7).¹⁹⁶

In H.-J. Treder’s access to a “matter” tensor in “the asymmetric field theory of Einstein”, the subtraction was done not on the level of the Einstein tensor, but for the Lagrangian: from the Lagrangian density of UFT the Einstein Lagrangian was subtracted. An advantage is that the variational principle ensures the existence of an identity [649 *]. A disadvantage is that the Lagrangian density for the matter part depends not only on the metric but also on its derivatives of 1st and 2nd order. For the metric Treder took the symmetric part $hij$ of the asymmetric fundamental tensor $gij$ . His references went to Infeld and to Schrödinger’s work, none to Tonnelat’s. We conclude that Tonnelat’s hope presented in the the first quotation above remained unfulfilled.

10.3.2 Equations of motion

Unlike in general relativity, in UFT, the equations of motion, in general, are no longer a direct consequence of the field equations. Ambiguities are bound to arise.¹⁹⁷ A methodological concern was how to properly derive equations of motion for matter, in particular for point-particles, possibly charged and massive. Two problems arise: In principle, for a single non-interacting particle certain types of world lines could be defined as paths like the geodesics or auto-parallels in general relativity. But, what world lines should be investigated, the geodesics of $g(ij)$ , or of another object chosen as a representation of the gravitational potential (field)? The geodesics do not depend on $g[ij] = kij$ ; in the usual interpretation of $kij$ as representing the electromagnetic field, its influence on such paths would be naught:

Unlike this, in the equation for auto-parallels

the electromagnetic field would show up. Again, what kind of connection is to be used, with or without vector torsion (“star”-connection), axial torsion etc?

Secondly, in correspondence to the vacuum field equations of general relativity, the method of treating the motion of matter as motion of singular point particles as Einstein, Infeld & Hoffmann had done with their approximation scheme in general relativity (EIH-method) [171 , 170 ], would be in conceptual conflict with the spirit of UFT. Was it possible, here, to only consider the region outside of a world-tube around the moving body where the matter tensor $Tij = 0$ ? The alternative method of Fock with $Tij ⁄= 0$ included the interior of the moving bodies as well. In order to avoid infinitely many degrees of freedom for extended bodies, some limit procedure had to be introduced. In the EIH-method, $δ$ -functions, i.e., distributions are used as matter sources, although Einstein’s equations do not admit distributions as exact solutions. Nonetheless, many authors applied the EIH-method also in UFT. E. Clauser showed in great detail that the method is applicable there for charged particles [83 *]; cf. also Section 15. Pham Tan Hoang wrote his doctoral thesis by applying this “singularity-method” to unified field theory [273 ] (cf. Section 10.4.1). However, we have mentioned already in Section 9.3.3 that both, L. Infeld and J. Callaway were not even able to derive the results of Einstein–Maxwell theory.

With the singularity-method employed, it turned out to be non-trivial to reach the Lorentz-force, or even the Coulomb force in a “slow-motion-” and “weak-field-”approximation: $(0) (1) 2 (2) (1) 2 (2) hij = hij + 𝜖 hij + 𝜖 hij + ...;kij = 𝜖 kij + 𝜖 kij + ...$ , with only terms $v ∼ c$ being retained. In first approximation, only the motion of uncharged particles was described properly by the weak field equations if $hij$ is taken as the metric and $kij$ as the electromagnetic field [69 *]. This negative result remained valid up to 4th order in $𝜖$ for $lij (lij ⁄= hij)$ chosen as the metric and $mij (mij ⁄= kij)$ as the electromagnetic field [271 , 272 *]. Better results in which the Coulomb force could be made to appear were achieved by Treder and Clauser [650 , 81 *, 82 *], and later by N. P. Chau, in a slightly changed theory, [77 *]; cf. also Section 15.1.

The electric potential is introduced via $g(2) = 𝜖 ϕ(2) [mn] mns ,s$ , where $n,m, s = 1, 2$ , where $g(2) [mn ]$ is the metric coefficient in the lowest order of an expansion. Here, the equations of motion for two electrical point-charges $1e, 2e$ were following from 4th-order equations for the electrostatic potential $ϕ$ obeying

and leading to the solution $ϕ = Ar-+ B + Cr + Dr2$ instead of only the Coulomb potential. After the integration constants $B, D$ were argued away, the resulting equations of motion turned out to be, for the first point-charge:

2 1 2 12 1d x− m mx− 12 2 1 eex− 1 2x− m --2-= − GN ---3--+ (A C + A C )--3-− 2C C --, (378 ) dt r12 r12 r12

where $x− := x−1− x−2$ and $r12 := |x−1 − x−2|$ ; underlined letters denote 3-vectors. The constants $1 2 A, C,⋅⋅⋅ + ...$ . are referring to the electric potential of the two charges ([477 *], p. 197). However, in addition to the Coulomb force an unphysical force independent of distance also showed up. It could not be made to vanish without an accompanying loss of the Coulomb force. In order to remove this defect, the field equations had to be changed; cf. [77 *] and Section 10.4. In Section 9.3.3 we already encountered such an altered field equation.

In Section 13.1 we will present Bonnor’s field equations which are leading, in lowest order, to Newton’s equations for two charges $1 2 e,e$ :

Here, $q$ is the ratio of $kik$ to the electromagnetic field strength, and $p$ a constant in an additional term in the Lagrangian. Bonnor ascribed the success due to this added term, quadratic in the skew-symmetric part of the metric $gik$ , to the circumstance that “the corresponding terms [in the field equations] arising from the $kik$ refer not to potentials but to the electromagnetic field strengths. For this reason it is hardly to be expected that any minor modification of the tensor $R(ik)$ will lead to the Coulomb force” ([33 ], p. 377). This is also obvious when Bonnor’s term in the Lagrangian $p2 ˆmikkik$ is compared with the usual $FikFik$ in Maxwell’s theory, because Bonnor is interpreting $kik$ not as an electromagnetic potential but as the electromagnetric field cf. [35 ].

A. Papapetrou, in a review concerning the problem of motion, pointed out that if Schrödinger’s equations with a cosmological constant $λ$ are considered, i.e., Eqs. (236 * and (237 *), then from $R {[kl],i} − λg{[kl],i} = 0$ the equation for the electrostatic potential follows [477 *]:

For large distances, Coulomb’s law then remained valid for an approximate solution
$−2λrA- 2 ϕ ≃ e [r + B + Cr + Dr ]$ with vanishing $B, D$ . The constant $C$ represents electric charge. A characteristic length $1 λ$ showed up. As Papapetrou wished to apply UFT to elementary particles like the electron, the problem was that $1λ$ should be of the order of an electron radius and not be of cosmological significance. His general conclusion was:

“It appears impossible to come to a direct phenomenological use of this theory which would allow a satisfactory treatment of macroscopic problems. But this does not prove anything with regard to the applicability of the theory in the micro-physical domain.” ([477 *], p. 203.)¹⁹⁸

A review by M. Lenoir [357 ] of progress made in the papers by Clauser [81 *], Treder [649 ] and Papapetrou [477 ] with regard to the Coulomb force, was reviewed itself by W. B. Bonnor in Mathematical Reviews [ External Link MR0119977].

For alternative field equations following from M.-A. Tonnelat’s model cf. (383 *) – (385 *) in the next Section 10.3.3.

For continuous matter, a third approach within general relativity employed the vanishing of the covariant divergence of the matter tensor $T il;l = 0$ . This assumed a convincing answer to the question of how to properly define a matter tensor within UFT as described in the previous Section 10.3.1. M.-A. Tonnelat, in particular, started from the form given to an identity by E. Schrödinger:

which is rewritten and re-interpreted within Riemannian geometry ([382 *], p. 207 – 209). The equations of motion are derived from volume-integration over equations of the form of (373 *) as will be seen in the next section.

10.3.3 Tonnelat’s extension of unified field theory

After having spent years of research with Einstein’s Lagrangian $ˆℒ = ˆgijRij$ , M.-A. Tonnelat felt confident that this was not the way to proceed:

“Nevertheless, we are convinced that a modification of the generalization of the theory suggested by Einstein can lead, at least partially, to the goal Einstein himself had set. […] if one wants to cling to the original form of this theory which has caused many hopes and initiated a flood of papers, he would not know how to achieve the objectives which had been proposed at first, within the strict scope of the theories’ principles.” ([302 ], p. 117–118)¹⁹⁹

Hence she expanded her Lagrangian by including a normed 4-vector ([641 *], p. 353):

where $Lrst$ is a connection without vector torsion ( $L Si = 0$ ). $Γ i$ is the new 4-vector (1-form), $Aˆp, σ$ are Lagrangian multipliers, and $K$ is a constant not to be mixed up with the curvature scalar. For $K ij$ the expression is chosen:

$W ij$ is the Ricci tensor formed with the connection $L t rs$ . Note the four additional constant parameters $α, β,p,q$ ! In a previous paper of 1960, $α = β = 0$ had been assumed [637 *]. The ensuing field equations now were:²⁰⁰

−ij+ 2 2 α j 2β j ˆg ∥k = − --δik∂sˆg[js] −---(δik∂sˆg[js] + δk∂sgˆ[is]) − 2βˆgijΓ k +--(δikˆgsj + δkˆgis)Γ s, (383 ) 3 3 3 2 {q + 5-β(1 + 2α)} ∂ ˆg[is] = (4β--− p − σ )ˆg(is)Γ , (384 ) 3 s 3 s ωij := Kij + σ(Γ iΓ j − K2gij ) − 𝜃ij = 0. (385 )

The choice $α = β = σ = p = 0;q = − 1 (q ⁄= 0)$ leads back to Einstein’s strong field equations for a connection without torsion vector. Thus, M.-A. Tonnelat always assumed $4β2 − p − σ ⁄= 0 3$ . The subcase $α = β = 0$ first has been studied by doctoral students of Tonnelat in [361 *], and [53 *, 56 *]. For this case, (383 *) leaves $Γ i$ undetermined.

If the Lagrangian density $ℒˆ$ is augmented by a phenomenological matter-Lagrangian density $ˆℒmat$ , then through $Θij := − √1--δℒmaijt-= 𝜃ij − 1gijgrs𝜃rs −g δg 2$ a phenomenological matter-tensor can be described.

However, it is possible to relinquish the phenomenological matter tensor by using the method of reformulating mixed or metric-affine geometry as Riemannian geometry. In $Kij$ , a Riemannian part $Rie K ij(a)$ relative to an arbitrarily chosen (symmetric) metric $akj$ must be exhibited. Eq. (385 *) will then be replaced by

We need not go through all the works similar to the approach in the previous theory of M.-A. Tonnelat, but will only list the following items ([641 *], p. 363):

as a metric, the quantity $∘ -- ghlij$ was chosen; ²⁰¹
geometrically, $Γl i := lisΓ s$ corresponds to vector torsion $Si$ ; physically, according to (384 *), it is linked to the current $(ir) ˆg Γ r$ and proportional to the 4-velocity $k u$ of a particle;
from the equations of motion, in linear approximation, an acceleration term $s i ≃ Γ ∂sΓ$ and the Lorentz force showed up as well as further terms characterizing other forces of unknown significance.

The last result needs a comment:

1) As admitted by Tonnelat, it is obtained only after partial neglect of the variational principle: instead of $ω[ij] = 0$ which follows from (386 *) $ω [ij] ⁄= 0$ must be required in order to give a meaning to an equation like ([382 *], (108), p. 217):

(387 *) follows from (386 *) after a longer calculation omitting terms $∼ ∂k(σK2 )$ . The covariant derivative is formed with regard to the Levi-Civita connection for the metric $∘ -- glij h$ . “Hence, in the extended version of the asymmetric theory like in the initial version, the equations of motion can make sense only if at least one of the expressions $ω[ij]$ or $Θ [ij]$ does not vanish.”²⁰²

2) As usual, the equations of motion of charged particles were derived in linear approximation in which the electromagnetic field $kij$ is taken to be small of first order. It was expressed by a vector potential $ϕk$ and an axial potential vector through $--- χij = √ − l 𝜖ijlm ∂[lχm ]$ with $l = det(lij)$ such that:

In the end, a simplified version of her extended theory is considered with $α = β = p = 0,k = σ ⁄= 0 q$ . The action is

with

In the static case, only $χ123 =: χ$ and $ϕ0 =: ϕ$ remain²⁰³ and satisfy the equations $△ △ χ = 0$ and $△ ϕ = kΓ$ with $p+σ−4β2∕3 σ Γ = Γ 0,k = q+-5β(1+2α)-= q ⁄= 0 3$ .

10.3.4 Conclusions drawn by M.-A. Tonnelat

In M.-A. Tonnelat’s understanding, Einstein’s unified field theory shows a number of new perspectives:

the dynamics of both the electromagnetic and the gravitational fields are modified such that there appears to be also an influence of the gravitational field on the electromagnetic one;
As a nonlinear electrodynamics follows, new effects will appear – as, e.g., “a diffusion of light by light”.
The relation between field strengths and inductions is similar as in nonlinear Born–Infeld theory ([632 *], p. 10.).

She believed that her extension of Einstein’s field equations had alleviated the way toward the equations of motion for charged particles.

Altogether, she seems to have been optimistic about the importance of the theory although she was aware of the fact that its range of validity was unknown, and many conceptional questions still remained unanswered. At the time, Mme. Tonnelat’s opinion possibly was the same as the one ascribed by her to the celebrities: “One may find with Einstein and Schrödinger a mixture of a certain discourage and of great hopes for the subject.” [632 *], p. 11.)²⁰⁴ Four years later, after progress made had been criticized as irreconcilable with the axioms of UFT, her attitude still persisted:

“Nevertheless, the discouraging results obtained from different directions never have definitely compromised the theory; it is the ambiguity of the possible interpretations (choice of metric, interpretation of the skew-symmetric fields, etc) which have set straight issues for the inveterate and totalitarian unifiers.” [382 *], p. 200).²⁰⁵

The choice of words impregnated by ideology like “inveterate” and “totalitarian” speaks for itself.

As has been remarked before, W. Pauli had criticized unified field theory approached through metric-affine geometry: he demanded that the fundamental objects must be irreducible with regard to the permutation group and also referred to Weyl ([487 ], Anm. 23, p. 273). In this view, an admissible Lagrangian might be $ℒ = a ˆg(ik)K (ik) + b ˆg[ik]K [ik]$ rather than $ℒ = gˆikKik$ . By an even stricter application, Pauli’s principle would also rule out this Lagrangian, cf. Section 19.1.1.

In 1971, M.-A. Tonnelat veered away from the “unitary spirit” by giving more philosophical comments:

“It would be childish to think that, for Einstein, the existence of the unified fields would resolve into an ontological criss-cross of torsions and curvatures. It would be likewise improper to reduce such schemes to a pure formalism without any relation to a universe the objectivity of which they propose to present. […] The objective pursued by unitary physics presents itself not as a realization with well defined contours but as a possible direction. […]” ([645 ], p. 396)²⁰⁶

10.4 Further work on unified field theory around M.-A. Tonnelat

Much of the work initiated by M.-A. Tonnelat has been realized in doctoral theses, predominantly in the framework of Einstein–Schrödinger field theory. About a dozen will be discussed here. They are concerned with alternative formulations of the field equations, with the identities connected with them, with exact solutions, and with equations of motion for (charged) particles in different approximation schemes.

10.4.1 Research by associates and doctoral students of M.-A. Tonnelat

As to the associates of M.-A. Tonnelat, it is unknown to me how Stamatia Mavridès got into theoretical physics and the group around M.-A. Tonnelat. She had written her doctoral dissertation in 1953 outside of physics [405 ]. For 5 years, since 1954, she contributed, alone and with Mme. Tonnelat, to Einstein–Schrödinger unitary field theory in many different aspects. Some of her publications have already been encountered. In Section 9.7, her assignment of physical variables to geometrical objects was noted, in Section 10.2.3 her contribution to the removal of the connection. Moreover, her contribution to spherically symmetric exact solutions mentioned in Section 9.6.1 must be kept in mind. Mme. Mavridès also took part in the research on linear field theories; cf. Section 16.1. Since the 1970s, her research interests have turned to astrophysics and cosmology [417 ].

Judith Winogradzki was a student of L. de Broglie with a thesis on “the contribution to the theory of physical quantities attached to spin-1/2-particles” [701 ]. With the topic of her thesis, she easily could have come into contact also with M.-A. Tonnelat. Although she had shown an interest in affine spaces before writing her dissertation [700 ], the concept of spinors permeated her subsequent research even more than unitary field theory. As noted in Section 9.8, group theory and conservation laws in special relativistic field theories and general relativity were also dear to her [706 ]. As was mentioned, she determined the identities following from Noether’s theorems if applied to the group U ( $λ$ -and coordinate transformations) and asserted Einstein’s claim that invariance under the group U is able to determine the field equations²⁰⁷ . Also, Einstein’s $λ$ -transformations as an “extension of the relativistic group” were investigated by her [702 ]. In another paper, she set out to determine a “gauge”-group, named J-transformations, satisfying:

where $Λ m ik$ does not depend on the connection L; transposition symmetry is also excluded from (390 *). A variational principle with Lagrangian density $rs ˆg Rrs$ is considered where $Rik$ can equal $K− ik$ or $K+ ik$ . J. Winogradzki then proved that $Λ m = δlλk ik i$ , is necessary, e.g., Einstein’s $λ$ -transformations appear. The weak field equations of Einstein and Straus are the only ones admitted [703 ]. Since 1956, no papers on UFT by her seem to exist.
We now come to some of the doctoral theses. Jack Lévy’s Lagrangian for unified field theory is a slight generalization of Tonnelat’s:

where $Γ k$ is the torsion vector of the arbitrary (asymmetric) connection $k Γij$ , and $ki$ an arbitrary one-form which can be a function of the arbitrary vector field $Xk$ . The third term is introduced “in order to apply the normalization condition to an arbitrary vector $Xk$ and not necessarily to the torsion vector $Γ i$ . This is done to avoid any identification or anticipated interpretation of the fields” ([361 ], p. 249) ²⁰⁸ Note that in Lévy’s notation $Xi$ corresponds to $Si$ . The field equations follow from variation with regard to $ij t ˆg ,Γrs ,Xk$ , and the Lagrange multipliers $ˆp A , σ$ . Among the (weak) field equations, (235 *) is replaced by $[is] 3 gˆ ,s = 2Aˆi$ which allows the introduction of a non-vanishing electrical current if $ˆg[is]$ is related to the electromagnetic field (induction). Furthermore, in the field equations, Lévy put either $ki = 0$ , or $ki = kXi$ . In the first case the torsion vector drops out, while $Xk$ comes in. Also $[is] (is) (m ˆg ),s = σˆg Xs$ with $m ⁄= 0$ . In the second, the torsion vector stays and is normed: $κ2 √ --- ˆgijΓ iΓ j = 2k2 − g)$ ; furthermore $[is] σ∕k ˆg ,s = m+2k∕3ˆg(is)Γ s$ with $m ⁄= − 2k 3$ . Lévy then showed that both versions are mathematically and physically equivalent. The impression here and in the contributions below is that mathematical tricks were played to better the consistency of the approach without an improvement in the physical understanding of the theory.

Pham Tan Hoang’s dissertation dealt with equations of motion. First, an explication of the EIH-method for the derivation of such equations for point particles in general relativity, and an introduction into the basics of unified field theory were given [274 *]. A useful result by him is the following. A coordinate system defined by

for an affine connection $Lk ij$ had been named “isothermal” by M.-A. Tonnelat.²⁰⁹ In general relativity, in linear approximation, (392 *) reduces to the coordinate system introduced for obtaining the wave equation. According to Pham Tan Hoang one can set up a coordinate system in UFT such that it is also isothermal with regard to the particular Riemannian metric $aij = ∘ g-lij h$ if vector torsion is vanishing. This he ascribes to the relation ([272 ], p. 67):

Then, Pham Tan Hoang applied the EIH-method to the weak field equations. When $∘ --- aij = g∕l lij$ with $l = det(lrs)$ is taken as the metric and $ij ∘ --- ij q = l∕g m$ as electromagnetic field, he obtained the same negative result as J. Callaway [69 ]: in linear approximation only uncharged particles can be described properly ([274 ], p. 89). No cure for this failure was found. In the end, the author could only bemoan the ambiguity inherent in the basis of the theory – implying structural richness, on the other side. We shall see that higher approximations had to be calculated in order to get the Coulomb field and the Lorentz force; cf. Section 15.3. Although dependent on the identifications made, another difficulty pointed out by Pham Tan Hoang is the vanishing of the charge current with the vanishing of $g[ij]$ . Moreover, the identification of a geometric object corresponding to the energy-momentum tensor of matter could not be made unambiguously.

In her thesis, Liane Bouche-Valere hoped to find acceptable equations of motion for a charged particle by “a method analogous to the one which provides them in the interior electromagnetic case in general relativity.” ([56 *], p. 2–3.) By this, we must understand a method working with “conservation conditions”. Her Lagrangian was (up to notation):

where $Lk$ is the torsion-vector of the arbitrary (asymmetric) connection $k L ij$ , $Wij$ the Ricci tensor formed from this connection, and $ˆAp,σ$ are Lagrangian multipliers. $Γ i$ is a normed vector field [53 , 54 , 55 , 56 ]. With regard to Tonnelat’s Lagrangian in Section 10.3.3, the terms with constants $α$ and $β$ are omitted. L. Bouche demonstrated the existence of 4 characteristic cones, three of which are the same as in (421 *) in Section 10.5.4 below, the 4th was not determined explicitly by her; cf. however the result of Nguyen Phong-Chau in Section 10.4. Her application of approximations (up to the second one) to the equations of motion has shown that the Lorentz-force could no longer be obtained as soon as the antisymmetric part of the field equation was satisfied. Also, her study of spherically symmetric solutions of the new field equations, due to their complexity could not be carried to a successful end.

The aim of the thesis by Marcel Bray was to study exact spherically and axially symmetric solutions of the weak field equations (417 *) – (419 *) and to compare them with exact solutions in general relativity.²¹⁰ For a possible physical interpretation he had to make a choice among differing identifications between mathematical objects and physical observables. For the metric, interpreted as describing the inertial-gravitational potentials, he investigated two choices: the metric suggested by Maurer-Tison, cf. (412 *) of Section 9.7, and $∘ ---- gij = h∕g lij$ . Unfortunately, his hope that his research “perhaps could also provide some helpful guiding principles for the choice of the metric” (p. 1) did not materialize. No solutions of physical interest beyond those already known were displayed by him [59 ].

A further systematic study of the possible field equations of Einstein–Schrödinger UFT was done by Nguyen Phong-Chau in his thesis [77 , 456 *]. He started from a transposition invariant expression for the Ricci tensor with 7 parameters (cf. Section 2.3.2, Eqs. (381 *) and (382 *)) originally proposed by M.-A. Tonnelat:

where $Rij,Pij$ are the two contractions of the curvature tensor of the asymmetric, torsionless affine connection $Γ k ij$ , whereas $R&tidle; ,P&tidle; ij ij$ belong to the transposed connection $&tidle;Γ k = Γ k ij ji$ . $Γ i$ is an arbitrary vector field.²¹¹ He concluded that only two cases had to be considered: version A of Einstein–Schrödinger with $Kij = Wij + 2p∂[iΓ j]$ ; and version B with $Kij = Wij + rQij$ . Here, $Wij, Qij$ are the contractions of the curvature tensor belonging to a connection $L k ij$ with zero vector torsion while $p ⁄= 0,r$ are free constants. Thus, in version B only 76 (instead of 80) unknown components have to be determined and, for $1 r = 2$ , transposition invariance holds. Nguyen Phong-Chau succeeded in determining explicitly for arbitrary $r$ the fourth characteristic which L. Bouche had announced to exist for case B, but had not been able to specify in the case $r = 0$ (for the other three cf. (421 *) below in Section 10.5.4) :

It thus seems possible that further linear combinations of the 3 quantities used for the definition of the “lightcone”, i.e., $hij,lij,nij$ may occur.

In the framework of non-linear electromagnetism which should follow from UFT, he suggested an interpretation different from Maxwell’s theory: the electromagnetic potential ought to be described by a tensor potential identified with the antisymmetric part of the metric $kij$ , not just a 4-vector. In first approximation $□kij = 0$ . A consequence would be that elementary particles must be described differently; beyond mass and charge, an electron would obtain further characteristics incompatible with spherical symmetry ([456 ], p. 354).

Further dissertations dealing with the generalization of Kaluza–Klein theory and with linear theories of gravitation in Minkowski space are discussed in Sections 11.1.1 and 16.1, 16.2, respectively.

10.5 Research by and around André Lichnerowicz

Within the Institut Henri Poincaré, a lively interaction between theoretical physicists, mathematicians and natural philosophers took place which tried to grab some of the mysteries from “the lap of the gods”. One of the Paris mathematicians sharing Mme. Tonnelat’s interest in metric affine geometry was André Lichnerowicz . He looked mainly at problems of interest for a mathematician. In gravitation – both in general relativity and the “non-symmetric theory” – questions concerning the integration of the systems of partial differential equations representing the field equations were investigated, be it identities for curvature, the Cauchy problem arising from field equations in affine spaces [368 *, 370 *], existence and uniqueness of solutions and their global properties, or the compatibility of the field equations of both general relativity and UFT. In his own words: “[…] I could attack what interested me – the global problems of relativity, the keys to a real understanding of the theory.” ([379 ], p. 104.) This has also been subsumed in Lichnerowicz’s contribution to the Chapel Hill Conference of 1957 on the role of gravitation re-published in 2011 ([120 *], 65–75). For scalar-tensor theory with its 15th scalar variable $ϕ$ , it is to be noted that Lichnerowicz not only discussed $−1 ϕ ∼ κ$ , ( $κ$ the gravitational “constant”) as a possibility like Ludwig and Just [385 ] but accepted this relation right away ([371 *], p. 202).

10.5.1 Existence of regular solutions?

In Section 7.1, it was pointed out that Einstein thought it imperative to banish singular solutions from his theory of the total field. Therefore it was important to get some feeling for whether general relativity theory would allow non-singular solutions or not. Einstein and Pauli set out to prove theorems in this regard [177 *]. Their result was that the vacuum field equation $Rik = 0$ did not admit any non-singular static solution describing a field with non-vanishing mass. For distances tending to infinity, the asymptotic values of the Schwarzschild solution were assumed. The proof held for any dimension of space and thus included the theory by Kaluza and Klein. However, prior to Einstein and Pauli, Lichnerowicz had proven a theorem almost identical to theirs; he had shown the non-existence of non-trivial regular stationary, asymptotically Euclidean vacuum solutions with²¹² $g00 = 1 − 𝜖00,𝜖00 > 0$ [363 , 362 , 364 ]; cf. also Section 8.3. In a letter of 4 September 1945 ([489 *], p. 309), a double of which he had sent to Einstein, Lichnerowicz pointed this out to Pauli.²¹³ In his response of 21 September 1945, Pauli found the condition on $g00$ unphysical: why should $g00 > 1$ be impossible near infinity? After he had studied the paper of Einstein and Pauli in more detail, Lichnerowicz commented on it in a further letter to Pauli of 11 November 1945. There, he also confessed to be “a bit shocked” about the fact that Einstein and Pauli had only proven “non-existence” of regular solutions while he had shown that Euclidean space is the only regular solution ([489 *], p. 325–326). Pauli, in his answer of 15 November 1945, apparently had suggested a related problem. On 15 December 1945, Lichnerowicz wrote back that he had solved this problem, outlined the structure of the proof, suggested a joint publication in Comptes Rendus, and congratulated Pauli for receiving the Nobel prize ([489 *], p. 333–335). A co-authored paper did not appear but Lichnerowicz published a short note: “W. Pauli signaled me his interest in the possibility to avoid any auxiliary hypothesis: he thought that this could be reached by a synthesis of our respective methods. In fact, this has happened: an important problem in relativity theory has been solved” [367 ]. A further proof of the occurrence of singularities for static gravitational fields in general relativity was given by A. Lichnerowicz and Y. Fourès-Bruhat [380 ]. That Pauli was impressed by Lichnerowicz’s theorem is shown by his detailed discussion of it in his special lectures on relativity in 1953 as reported in ([194 *], p. 389–390).

As already mentioned in Section 8.3, A. Papapetrou, working at the time in Dublin with Schrödinger,

extended the theorem of Einstein and Pauli to a non-symmetric metric, i.e., to UFT with the strong field equations $Rik = 0, g ik∥l = 0, ˆg[is],s = 0 +−$ [474 ].

10.5.2 Initial value problem and discontinuities

The field equations for “Einstein’s unitary theory” were presented by A. Lichnerowicz in the form [370 *]:

r r g+ik−∥l := gik,l − grkLil − girLlk = 0, (397 ) [is] ˆg ,s = 0, (398 ) 4- Pik = 3 ∂[iSk], (399 )

where $Sk$ is a covariant vector to be identified with the torsion vector and $j Lik$ an “a priori arbitrary” affine connection corresponding to Schrödinger’s star-connection [cf. (291 *)]: $j L ik → ∗Γ ikl:= Γ ikl+ 23δil Γ k.Pik$ is the Ricci-tensor $− K jk −$ formed from $L j ik$ defined by (56 *). Note that from (399 *) follows $∂kP [ij] + ∂jP[ki] + ∂iP [jk] = 0$ , but the reasoning backward holds only locally (local existence of a potential) ([369 *], p. 500), ([371 *], 267). In Section 10.3.1 , the form given to the generalized Bianchi-identities by Lichnerowicz is shown.

The local initial value problem is the following: Let be given on a spacelike hypersurface $0 S (x = 0)$ , or $k f (x ) = 0$ , of the manifold (space-time) the components of the non-symmetric tensor $gij$ of class ( $C1,C3$ piecewise) and an affine connection of class ( $C0,C2$ piecewise).²¹⁴ With (397 *) having been solved, i.e., the connection expressed by the metric and its 1st derivatives, the task now is: Determine in a neighborhood of $S(x0 = 0)$ the tensor $gij$ and the torsion vector $Sk$ such that they satisfy (398 *), (399 *). In his main results up to 1954, Lichnerowicz had shown that the local initial value problem for real and analytical data on $S$ and with $g00 ⁄= 0$ has a unique solution [368 *, 370 *, 371 *], Part II, Chap. VI), [369 *]. The proof included a normalization-condition for the torsion vector: $∘ --- ∂s(gstSt |g|) = 0$ which was also used when in 1955, during the Jubilee-conference in Bern, A. Lichnerowicz reviewed global problems and theorems “of the relativistic equations”. He now considered part of the field equations given above:

with the Ricci tensor (previously denoted $Pik)$ $Rab = Rab (L)$ , where $Labc$ is the torsion-free connection. He could show that (400 *) “presents the same local mathematical coherence as the system of equations of general relativity”²¹⁵ ([372 *], p. 182). The first notice in Comptes Rendus [368 *] had been reviewed by V. Hlavatý, and then a paper by him containing the proofs announced appeared in the journal of Hlavatý’s home university [369 ]. For the Einstein–Maxwell field equation the requirement that the initial data be analytic had been weakened by Lichnerowicz’ student, Mme. Fourès-Bruhat, who also gave the first proof of existence and uniqueness of the local Cauchy problem for Einstein’s field equations [217 ]. Lichnerowicz conjectured that such a proof also could be achieved for unitary field theory. Between January and June 1954, a correspondence between Lichnerowicz and Einstein on this topic has taken place; on 11 May 1954 he sent Einstein his paper on the compatibility of the field equations of UFT [368 ].²¹⁶ W. Pauli was impressed by Lichnerowicz’ lecture at the conference in Bern:

“I believe, the most important else we have heard, was the report by Lichnerowicz on the Cauchy initial value problem in the nonlinear field equations of general relativity. I attach great importance to the study of such problems, because I suppose that it also will play an essential role with field quantization.”²¹⁷ [486 ].

Following Hadamard, Lichnerowicz had displayed the discontinuities of the Riemannian curvature tensor denoted by $[Rij,rs]$ on a characteristic $f = 0$ by beginning with²¹⁸ $i i [∂rLjs] = ujslr, li = ∂if$ which led to $i i i [R j,rs] = lrujs − lsujr$ . As a consequence,

and

Although A. Lichnerowicz had done the decisive steps in the formulation of the Cauchy initial value problem, a study of the case where it cannot be solved uniquely, i.e., $g00 = 0$ , was in order.

In several short notes in Comptes Rendus [291 *, 293 *, 292 , 295 *], S. I. Husain applied the methods of A. Lichnerowicz for an investigation of the discontinuities of the curvature and Ricci tensors in mixed geometry, first with Einstein’s “strong” and then the “weak” field equations²¹⁹ . Independently of the field equations, for the discontinuities the expressions of Lichnerowicz hold:

where $Ai ,bk js$ are his discontinuity-parameters and $Ri = K i (L r) j,rs − jkl pq$ . Thus, (401 *) reappeared for the more general curvature tensor. For the “strong” equations, Husain derived

using $k kl l := g ll$ [291 ]; the 2nd equation is non-trivial because there is no skew symmetry in the first two indices of the curvature tensor. For the “weak” system, he obtained instead [293 ]:

In the next paper, he switched to the definition $lk := g(kl)ll = lklll$ and arrived at

where $Ds$ is the covariant derivative with regard to the (Riemann) connection defined by $l ij$ , and $P m (L) jkl$ the curvature tensor belonging to the connection²²⁰ $L prq$ . The congruence-sign in (407 *) means “up to terms in $[Rij,rs]$ ”. In a further paper, Husain obtained even $lsDs [P mjil] ≃ 0$ and concluded that the wave front of radiation propagates “with the fundamental speed” [295 ].

10.5.3 Characteristic surfaces

The propagation of waves with their characteristic surfaces was of great interest also in Einstein–Schrödinger theory. A naive mathematical approach would take $gijdxidxj = 0$ as the defining relation for the characteristic surface. However, from the point of view of physics, what is the “lightcone” of the gravitational field? This obviously depends on the identification of the gravitational potential (field) with a geometric object of unitary field theory. As we have seen in Section 9.7, different identifications were made. The hypersurface $0 S (x = 0)$ represented by $0 1 2 3 f(x ,x ,x ,x ) = 0$ and tangent to the cone $grs∂rf∂sf = 0$ is a wave surface of the metric field $grs$ . We already noted that Lichnerowicz preferred the inverse of $lij = g(ij)$ as the “gravitational tensor” and thus defined the light cone through

([371 *], p. 288; [370 ]; [372 ]). Lichnerowicz was interested in the initial value problem and in wave surfaces, he also looked at gravitational shock waves, characterized by discontinuities in the connection. Here, the task is to rewrite the field equations in terms of tensor distributions. As an application, Lichnerowicz took relativistic hydrodynamics and magnetohydrodynamics.²²¹ He advised the thesis of Pham Mau Quan on relativistic hydrodynamics [501 ], in which the various characteristic surfaces were investigated across which discontinuities of mass density, pressure, fluid velocity and heat transport vector, or of their gradients will occur. Lichnerowicz then succeeded to again solve the initial value problem for magnetohydrodynamics [377 ]. In Jordan–Thiry theory interpreted as a UFT, Mme. F. Hennequin and R. Guy studied fluid dynamics in more detail; cf. Section 11.1.1.

In view of the many possibilities of identifying a geometric object with the gravitational field, it can be understood why V. Hlavatý [260 *] and E. Clauser [80 ] did not follow the approach by Lichnerowicz. They used the inverse of $hij = g(ij) ⁄= lij$ for the definition of the wave fronts:

As seen in Section 2.1, $hij ⁄= lij$ . Clauser dealt with a special case of Hlavatý’s classification with connection:

where $1 g 1 k pk := 6(log h),Ki := − 2(h),i$ , $s ! i ig+2k−h is Ki − psL i= 0,Lk := δk h − k.. ksk. g,h,k$ are the determinants of the corresponding tensors. Indices are moved with $hij$ ; the original spot of an index is noted by a dot. Clauser then was able to show that for the system:

the initial value problem is well defined and the characteristic surfaces are given by (409 *).

An important achievement was reached by Françoise Maurer-Tison who continued the investigation of Lichnerowicz. She pointed out that the characteristic cone of the metric (“light cone”), locally, can be decomposed into two cones described by the metrics $lij$ and $nij$ with

from which

follows. This left the use of the quantity $nij$ as a further possibility for the metric; cf. [408 ], ([398 *], p. 243–244), ([641 *], p. 339–344). In principle as many “light cones” as different interpretations made for the gravitational field can be found. However, in the literature studied, we will meet four different light cones being discussed; cf. also (421 *) and (396 *).

When two further approaches to the discontinuities of curvature tensors within the framework of UFT were published in 1961 in Comptes Rendus, the respective authors did not take notice of each other. In the first half of the year, L. Mas and A. Montserrat presented their three papers on “wave fronts” in unified field theory, while in the second half J. Vaillant published on “discontinuities” of the curvature tensor in Einstein–Schrödinger theory. Both continued the work of Lichnerowicz and Maurer-Tison.

In their first paper [390 ], Mas and Montserrat used the “compatibility equation” (33 *) in the form:

using the asymmetric connection $L kij$ and introduced two further connections $M ijk= L(kij)$ and the Riemannian connection obtained from $lij$ named $Γ k ij$ . They referred to Husain’s doctoral thesis [294 ] and like him called $m P jik$ the curvature tensor belonging to $k Lij$ . (405 *) and the second equation of (406 *) were reproduced, and the equation for the discontinuities of the Ricci tensor $[Pij] = 0$ added. For the curvature tensor $Rmjik$ belonging to $Γikj$ , the results of Lichnerowicz (401 *) and the first equation of (402 *) as well as $[Rij ] = 0$ were shown to hold also in the case of Einstein’s weak field equations. In the second paper, the light cone was defined by $hijlilj = 0$ with $hij = g (ij)$ and the corresponding discontinuities for curvature derived [444 ]. The third paper then brought an investigation including three different characteristic surfaces [445 ]. Let $1 2 3 gij := lij,gij := hij,gij := nij$ where $nij$ is defined by (412 *), (413 *). Then, on the characteristics $Σ(s),s = 1,2,3$ defined by $s gijlilj = 0$ or $g00 = 0,h00 = 0, n00 = 0$ , respectively, for the curvature tensors $s Rmjik$ belonging to the Riemannian connections calculated from $sg ij$ , the following discontinuities obtained:

In his three notes in Comptes Rendus, Jean Vaillant took up as well the investigation of the discontinuities of the curvature tensor for the weak field equations [663 *, 664 ].²²² In 1964 he finished his PhD thesis on this and related subjects.²²³ For the discontinuities of the curvature tensor on a characteristic, J. Vaillant noted more precisely $1 pqrs [Pijkl] = 2Kliljη lpkrs(lkhlq − llhkq)$ . This expression satisfies $i li[P j,rs] = 0$ and is consistent with (401 *) and (405 *). Vaillant also looked at the characteristics defined from (412 *) and by $rs h lrls = 0$ of F. Maurer-Tison. For both, the discontinuities

were shown to hold. If $k kr γ := n lr$ , then also

where the right hand sides contain linear combinations of $i [P j,rs]$ and $[∇iSj ]$ , respectively. In his third paper [665 ], Vaillant concluded that the only surfaces on which discontinuities of the Ricci tensor can arise, are the characteristics $hrslrls = 0$ The results of S. I. Husain (cf. Section 10.5.2) were not mentioned by him.

10.5.4 Some further work in UFT advised by A. Lichnerowicz

The doctoral theses inspired by A. Lichnerowicz are about equally directed to Einstein–Schrödinger and Jordan–Thiry (Kaluza) theory. As interesting as the study of the Cauchy problem initiated by Lichnerowicz was, it also could not remove the ambiguities in the choice for the metric.

In her thesis “Aspects mathématiques de la théorie du champ unifié d’Einstein–Schrödinger”, Françoise Maurer-Tison first wrote an introductory part on the geometrical background of unified field theory; she developed the concept of “coaffine connection”, i.e., an infinitesimal connection on the fiber bundle of affine reference frames. In Part 2, Maurer-Tison investigated in detail the Cauchy initial value problem. The last Part 3 of her thesis is devoted to the “physical interpretation”.

The “weak” field equations are written in the form:²²⁴

σ σ ∂ρgλμ − Lλρgσμ − Lρμgλσ = 0, (417 ) ∂ (g[ρβ]√ − g) = 0, (418 ) ρ Pαβ − 2-(∂αΣ β − ∂ βΣα) = 0, (419 ) 3

where $Γ σ ρμ$ is a linear connection with torsion-vector $Σβ$ and $Lσ ρμ$ a linear connection with vanishing torsion vector $Σ β(= S β) = 0.P αβ(L)$ is the Ricci tensor [398 *].

For the Cauchy initial value problem, (418 *), (419 *) are rewritten into time-evolution equations

and constraint equations on the initial surface $x0 = 0$ containing $∂ (g[ρk]√ − g-) = 0 ρ$ , and an equation for the Ricci tensor not written down here enclosing the metric, its first derivatives and the vector field $Σi$ ([398 *], p. 229). The existence of a solution is proven, and, in the case of analytic initial data, also its uniqueness. There exist three characteristic cones met before and defined by:

where $ij ij 2h ij ij n = γ = -g h − l$ [cf. (412 *) and (413 *)]. The first one (with $γ$ ) is declared to be the light cone, while the interpretation of the other two (time oriented) as wave fronts remains unclear (p. 241).

In Part 3, after a detailed calculation departing from a proper reformulation of the field equations and the “conservation equations”, Maurer-Tison arrived at what she named energy-momentum tensor of the electromagnetic field:

Electromagnetic field $H αβ$ and induction $K ′αβ$ are identified as follows: $H = 1 √−-g 𝜖 m γδ;K ′αβ = − -√1--𝜖αβγδΔ k . αβ 2 αβγδ 2 − g + − γδ$ Here, $Δ + −$ is a generalized Laplacian (cf. her Eq. (40.4) on p. 255). Mme. Maurer-Tison did not only comment in technical terms on her impressive work; she also described the whole field very much to the point with accurate words:

“The unified field theory of Einstein–Schrödinger is attractive by its apparent simplicity and repellent by the finicky calculations it requires: it is a young theory with moderate baggage as long as it is investigated with rigour, but an immense load when the efforts are taken into account which have been tried to explore its possibilities” ([398 ], p. 187).²²⁵

The following doctoral thesis by Marcel Lenoir constitutes a link with the next Section 11.1. In it, he gave as his aim the introduction of a geometrical structure which permits the incorporation of Bonnor’s supplementary term into the Lagrangian of UFT (cf. Section 13.1) resulting from the contraction of a suitable Ricci tensor ([359 *], p. 7). This is achieved by the introduction of space-time as a hypersurface of a 5-dimensional space $V5$ with metric tensor and asymmetric linear connection. The wanted supplementary terms follow from the curvature of $V 5$ . Lenoir’s approach to Bonnor’s field equations is an alternative to (and perhaps more convincing) than the earlier derivation, in space-time, by F. de Simoni (cf. Section 15.1).

For a background, in the first two chapters of the thesis, the geometry of fiber bundles and of hypersurfaces was summed up. Lenoir stated the “weak” field equations in the form:

i− j+ ij ij s is j sj i ij 2- i js ˆg ∥k(= ∂kˆg − ˆg Γks + ˆg Γks + ˆg Γsk ) = gˆ Γ k − 3 δkˆg Γ s, (423 ) [is] ∂sˆg = 0, (424 ) Kij = 0. (425 )

Following Kichenassamy, he distinguished UFT’s of “type Einstein–Schrödinger” with $[is] ∂sˆg = 0$ and of type “Einstein–Tonnelat” with $∂sˆg[is] = ˆFi ⁄= 0$ (cf. Section 10.3.3, Eq. (384 *)). From an effective Lagrangian conjured up from the curvature scalar in $V5$ , i.e.,

Lenoir was able to derive extended field equations in space-time from which, by specialization, all three types of field equations emerged: Einstein–Schrödinger’s, Einstein–Tonnelat’s (cf. Section 10.3.3) and Bonnor’s (pp. 61–71 of [359 ]). Lenoir also suggested alternatives for energy-momentum tensors in order to obtain the equations of motion through the “conservation equations”. He showed that identities will result as soon as all field equations are satisfied. In addition, a static, spherically symmetric solution of Bonnor’s field equation was given and an extension of Birhoff’s theorem obtained.

In the last chapter, Lenoir investigated whether Lichnerowicz’s theorem on the non-existence of regular solutions could also be proven for his extended unitary theory but did not arrive at a conclusive result.

The doctoral thesis of another student of Lichnerowicz, Albert Crumeyrolle, contained two different topics [93 ]. In the larger part, research on the equations of motion of charged particles and on the energy-momentum tensor (corresponding to the “matter” tensor in UFT) was resumed in the framework of Einstein–Schrödinger theory. As to the equations of motion for charged particles, Pham Tan Hoang had simplified calculations by a more complete use of the isothermal condition (392 *) and by further improvements as mentioned in Section 10.4. Yet the negative result remained the same as the one already obtained by E. Clauser and H.-J. Treder, cf. Section 10.3.2. The same applies to Crumeyrolle’s approximative calculation of the equations of motions in ([94 *], p. 390).

Because the energy-momentum tensor he constructed had to contain a metric field, Crumeyrolle investigated which of the three possibilities for the metric, i.e., $hij,lij$ , and $γij$ emerging from the Cauchy-problem (cf. (421 *) of Section 10.5.4) would be best for reaching the special relativistic energy-momentum tensor of the electromagnetic field. In fact, none was good enough. In first approximation, $hij$ fared best [94 *].

In the 2nd part of his thesis, an 8-dimensional auxiliary space was introduced in order to obtain more possibilities for field variables and (modified) field equations.²²⁶ A. Crumeyrolle provided this space $V8$ with coordinates $i ∗i x ,x ,(i = 1, 2,3,4)$ (“natural reference systems” in comparison with his adapted coordinates $i i ∗i z = x + 𝜖x$ ; cf. Section 2.6); in $V8$ , he embedded space-time by $∗i x = 0$ . By use of the covariant derivative $+ ∇$ in $V8$ , an affine connection with parts $j j j∗ j∗ πi,πi∗,πi ,πi∗$ followed. In the special coordinate system, named “natural diagonal reference system”, projection of the affine connection in $V8$ into a 4-dimensional space led to both a connection $Lkij = Lki∗∗j = Lkj∗i∗ = Lkj∗∗i∗ = Lk∗ij∗$ , and a tensor $Λkij = Lkij∗= Lki∗j = Lkji∗$ in space-time. Due to the increase in the number of field variables, he could derive two Ricci-tenors in this 4-dimensional space, identified with space-time:

and two tensorial objects:

With these tensors, modified field equations which contained the “weak” Einstein equations including additional terms in $𝒫jk$ then could be introduced ([96 *], p. 103–128). But a number of extra field equations had to be joined such as, still among others, $s Λ [sj] = 0,d𝒫¯[jk] = 0, ¯𝒫 (jk) = 0$ . cf. ([96 *], p. 126). Another approach by Crumeyrolle using a field of numbers different from the real numbers will be discussed in Section 11.2.2. In its Section XV ([97 *], pp. 126–130), it contains a new attempt at a unified field theory with a slightly changed formalism. As Crumeyrolle’s aim was to regain the old Einstein–Schrödinger theory from a theory with additional field variables and field equations, his approach could not bring progress for an eventual physical interpretation of UFT.

	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