"On the History of Unified Field Theories. Part II. (ca. 1930 – ca. 1965)"
Hubert F. M. Goenner 
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

13 Research in other English Speaking Countries

13.1 England and elsewhere

We have met the work of W. B. Bonnor of the University of Liverpool on UFT already before. After having investigated exact solutions of the “weak” and “strong” field equations, he set up his own by adding the term p2mikK ik to Einstein’s Lagrangian of UFT [34]. They are:254

gi+k−||l = 0, (464 ) S = 0, (465 ) i HKer + p2U = 0, (466 ) − (ik) (ij) Her Her Her K [ik],l + K [kl],i + K [li],k + p2(U [ik],l + U [kl],i + U [li],k) = 0, (467 ) − − −
[rs] 1 [rs] Uij = g[ij] − g girgjs + -g grsgij. (468 ) 2

The assignment of g(ij) to the gravitational potentials and of g[ij] to the electromagnetic field was upheld while the electric current became defined as Jijk = g{[ij],k}.

A linearization gij = ηij + γij of Bonnor’s field equations up to the first order in γ led to:

γ[is],s = 0, γ(is),sk + γ(ks),si − γss,ki = 0, (469 ) γ = − 4p2γ . (470 ) {[ik],l},ss {[ik],l}
In first approximation the electric current is given by Jijk = γ{[ij],k} such that the previous equation looked like ∂s∂sJijk = − 4p2Jijk. For a spherically symmetric particle at rest with radial coordinate r, Bonnor obtained
1 d d ρ -2 --(r2---) = − 4p2ρ, (471 ) r dr dr
where ρ is the charge density. For 2 i p = a2 and ρ → 0 for r → ∞, the charge density will be restricted255 to ρ = const.1e−a2r r. As M.-A. Tonnelat remarked, Bonnor’s strategy was simply to add a term leading to Maxwell’s energy-momentum-stress tensor ([634], p. 919). Abrol & Mishra later re-wrote Bonnor’s field equations with help of the connections defined in (51*) and (52*) of Section 2.2.3 [2].

In Trinity College, Cambridge, UK, in the mid-1950s research on UFT was carried out by John Moffat as part of his doctoral thesis. It was based on a complex metric in (real) space-time: gij := sij + aij with real sij, imaginary aij, and ∗srisrk = δik. Correspondingly, the symmetrical linear connection Γ k = ∗Γ k + ˆΓ k ij ij ij split into a real connection part ∗ Γ k ij and an imaginary valued tensor ˆ k Γ ij.256 His approach to UFT then differed considerably from Einstein’s. In place of (16*) of Section 2.1.1 now

i Γ i ∂X-- i j ˆ i j i ˆ i j ∇kX := ∂xk + ∗Γ jk X + Γjk X = X ;k + Γjk X , (472 )
where the semicolon indicates covariant derivation with regard to the real connection part. In place of (30*) the compatibility condition
r r gij,k − grjΓ ik − girΓ jk = 0 (473 )
was introduced. From this equation it turned out that the connection is formally akin to Hattori’s (93*), i.e., Γ s [δk+ ∗skrars] = HL j ij s ik except that the imaginary aij is entering on both sides; cf. Eqs. (8) – (10), p. 624 in [439*]. It seems that Moffat did know neither Einstein’s papers concerning UFT with a complex metric [147, 148] (see Section 7.2) nor Hattori’s connection. This is unsurprising in view of his thesis advisors F. Hoyle and A. Salam which were not known as specialists in UFT. With these complex valued mathematical objects, Moffat now built a “generalization of gravitation theory” [440*] with the explicit purpose to find a theory yielding the correct equations of motion for charged particles (Lorentz force).257 Now, gij := ∗gij + gˆij and ∗sir ∗ grk = δki. As a real Lagrangian, Moffat chose:
√ --- rs rs ˆ ℒ= − g[∗g ∗ Rrs + ˆg Rrs], (474 )
where presumably ˆgrs is defined by the decomposition of the inverse ∗sij of gij = ∗grs + ˆgrs with g gks = δk is i, i.e., gij = ∗sij + ˆgij, although this relationship is not written down. His Ricci-tensors to be added to the list in Section 2.3.2 are the real and complex parts of K− jk :
∗Rij := − ∗ Γ ijr,r + ∗Γ irr,j + ∗Γrjs∗ Γisr− ∗ Γirj ∗ Γ rss + ˆΓrjsˆΓisr− ˆΓ ijrˆΓrss, (475 ) r r s r r s s r r s ˆRij := − ˆΓ ij,r + ˆΓir,j + ˆΓrj ∗ Γ is − ∗Γ ij ˆΓrs + ∗ Γrj ˆΓ is − ˆΓij ∗ Γrs . (476 )
W. Pauli’s objection in its strict sense still applies in spite of the Lagrangian being a sum of irreducible terms.

For the field variables √ --- √ --- − g ∗ gij, − gˆgij, in empty space the field equations following from (474*) are

∗Rij = 0, ˆRij = 0. (477 )
Matter is introduced through the variation -δℒ- 8π√ --- δgij = − c2 − g Tij with “the complex-symmetric source term” Tij = G ∗ Tij + ˆTij, G the Newtonian gravitational constant. According to Moffat: “The real tensor ∗T ij represents the energy-momentum of matter, while ˆT ij is the charge-current distribution.” A weak-field-approximation gij = ηij + hij + γij with real hij and imaginary γij is then carried through. In 1st approximation, the wave equation □ γ′ij = 16πTˆij resulted where γ′ij := γij − 1 δijγrr 2 and ηij ≃ − δij. For slowly moving point particles and weak fields, Maxwell–Lorentz electrodynamics was reached. After an application of the EIH-approximation scheme up to the 6th approximation omitting cross tems between charge and mass, Moffat concluded:258 “we have derived from the field equations the full Lorentz equations of motion with relativistic corrections for charged particles moving in weak and quasi-stationary electromagnetic fields.” In a note added in proof he claimed that his method of winning the equations of motion was valid also for “quickly varying fields and fast moving particle” ([440*], p. 487). In place of the Reissner–Nordström metric, he obtained as a static centrally symmetric metric [441*]:
( ) 2m-- ds2 = 1 − 2m-- dt2 − ----(1-−--r-)----− r2d Ω2, r (1 − 2mr-)2 + 4 𝜖2r2
where 𝜖 denotes the electric charge. Upon criticism by W. H. McCrea and W. B. Bonnor, Moffat included the “dipole procedure” of Einstein and Infeld in his derivation of the equations of motion [442]. R. P. Kerr found that the field equations (477*), together with the boundary conditions at spacelike infinity, are not sufficient to determine the spherically symmetric solution. This holds even when four coordinate conditions are added [324].

In the 1950s, the difficulty with the infinities appearing in quantum field theory in calculations of higher order terms (perturbation theory) had been overcome by Feynman, Schwinger, Tomonoga and Dyson by renormalization schemes. Nevertheless, in 1952, Behram Kursunŏglu as a PhD student in Cambridge, UK, expressed the opinion,

“[…] that a correct and unified quantum theory of fields, without the need of the so-called renormalization of some physical constants, can be reached only through a complete classical field theory that does not exclude gravitational phenomena.” ([343*], p. 1396.)

So he looked for such a classical UFT and attempted to derive the “structure of the electron” from it. In Section 9.3.3 we already met Kursunŏglu’s field equations, cf. (300*) – (302*), following from the Lagrangian:
√ --- ℒ= ˆgrsRrs − 2p2[ − b − √ − g-], (478 )
where b = det(bkl) and bkl is the inverse to the symmetric part (ij) g of the asymmetric metric gij = aij + iq−1ϕij.259 I assume that his Ricci tensor Rij is the same as the one used by Einstein in [149], i.e., K jk −.

By an approximation around Euclidean space gij = − δij + hij + i ϕij which neglected the squares of h ij, the cubes of ϕ ij, and interaction terms between h , ϕ ij ij, Kursunŏglu obtained the following results:

bij = − δij + hij − T ′, T ′= 1δijΣr,s(ϕrsϕrs) − Σr(ϕirϕjr), (479 ) ij ij 4 1- ∂sfsi = Ji, Ilrs = Σt(𝜖lrstJt), fij = − 2 Σk,l(𝜖ijklϕkl) = ∂iAj − ∂jAi, (480 ) 1 1 ( 1 ) --□hij + --[δijΣr(JrJr) − JiJj] + Σr,s(∂rϕjs∂sϕir) + ∂i∂j -Σr,s(frsfrs) 2 2 4 1- 2 + 2 [Σr (firJjr) + Σr(fjrJir) − δijΣr,s(frsJrs)] = p Tij, (481 )
where Jij = 2∂ [iJj], and ′ Tij = − Tij. Both ϕij and fij are identified with the electromagnetic field:
(ϕ , ϕ , ϕ ; ϕ , ϕ , ϕ ) = (iE , iE , iE ; H , H , H ). 23 31 12 41 42 43 1 2 3 1 2 3
By comparison of the equation for hij with the Einstein field equations of general relativity, the relation p2q− 2 = 2G∕c4 with the gravitational constant G ensued. Kursunŏglu then put the focus on the equation for the electrical current density derived from (302*) after a lengthy calculation:
(□ − κ2)Jk = κ2hJk + wk (hrs,flm,Jp ), (482 )
where wk(...) is a function describing the interaction terms; it consists of 11 products of h with f or h or J (and up to their 2nd derivatives). The r.h.s. of (482*) then was summarily replaced by
∫ +∞ jk = ds e δ4(x − ξ )Vk (ξ), −∞
and “the electrostatic field due to an electron at rest” derived to be |E | = e2[1 − e−κr − κre− κr] − r. Kursunŏglu concluded that UFT “describes the charge density of an elementary particle as a short range field. It is not possible to measure the effects of an electron “radius” κ−1 by having two electrons collide with an energy of the order of mc2. Quantum theoretically the wavelength corresponding to this energy is -ℏ- mc, which is much larger than κ−1.” ([343] , p. 1375.)

As to the equations of motion, Kursunŏglu assumed that (30*) is satisfied and, after some approximations, claimed to have obtained the Lorentz equation, in lowest order with an inertial mass m = -1-κ e2 0 2c2 (cf. his equations (7.8) – (7.10)); thus according to him, inertial mass is of purely electromagnetic nature: no charge – no mass! Whether this result amounted to an advance, or to a regress toward the beginning of the 20th century is left open.

In a short note with G. Rickayzen, Kursunŏglu pointed out that the Born–Infeld non-linear electrodynamics followed from his “version of Einstein’s generalized theory of gravitation” in the limit p → ∞. In the note, a Lagrangian differing from (478*) appeared:

2 rs 2 2 √ --- √ --- 2p ℒ= ˆg (Rrs − ip Brs) + 2p [ − b − − g ], (483 )
where Brs = 2∂[rBs] is an auxiliary field variable [509].

G. Stephenson from the University College in London altered Einstein’s field equation as given in Appendix II of the 4th London edition of The Meaning of Relativity260 by replacing the constraint on vector torsion Sj = 0 by SrSr = a with constant a, and by introducing a vector-potential Aj for the electromagnetic field tensor kij [588]. His split of (30*) for the symmetric part coincided with Tonnelat’s (363*), but differed for the skew-symmetric part from her (362*) of Section 10.2.3. The missing term is involved in Stephenson’s derivation of his result: Dirac’s electrodynamics {kij} ∇s ˇkis = Ai. Hence the validity of this result is unclear.

A year later, Stephenson delved deeper into affine UFT [589]. We quote from the review written by one of the opinion leaders, V. Hlavatý, for the Mathematical Reviews [External LinkMR0068357]:

“The Einstein paper contains three separate sections. In the first section the author expresses the symmetric part ν ν Λλμ = Γ (λμ) of the unified field connection ν Γλμ by means of its skew symmetric part ν ν Sλμ = Γ[λμ] and vice versa. Then he identifies the electromagnetic field with k λμ = g[λμ] and imposes on it the first set of Maxwell conditions

∂[ωk μλ] = 0.

The second set of Maxwell conditions is equivalent to the Einstein condition Sα = 0 λα. However, according to the author, there appears to be no definite reason for imposing the additional condition (1). In the second part the author considers Einstein’s condition

R [μλ] = 2 ∂[μX λ]

coupled with

α R [μλ] = − D αSμλ

(where D α denotes the covariant derivative with respect to Λ ν λμ). Hence

S νμλ = 2X [μδνλ] + T νμλ,

where ν ν Tμλ = T [μλ] is a solution of

D αTαμλ = 0.

Therefore Sλαα = 0 is equivalent to

1- α X μ = − 3 Tμα

and the field equations reduce to 16 equations (4) and R(μλ) = 0. In the third part the author considers all possibilities of defining ν Γλμ by means of the derivatives of gλμ with all possible combinations of Einstein’s signs (++ ), (+ − ), (− − ). He concludes that in both cases (i.e. for real or complex gλμ) only the (+ − ) derivation leads to a connection Γ ν λμ without imposing severe restrictions on g λμ.”

13.1.1 Unified field theory and classical spin

Each of the three scientists described above introduced a new twist into UFT within the framework of – real or complex – mixed geometry in order to cure deficiencies of Einstein’s theory (weak field equations). Astonishingly, D. Sciama at first applied the full machinery of metric affine geometry in order to merely describe the gravitational field. His main motive was “the possibility that our material system has intrinsic angular momentum or spin”, and that to take this into account “can be done without using quantum theory” ([565*], p. 74). The latter remark referred to the concept of a classical spin (point) particle characterized by mass and an antisymmetric “spin”-tensor sαβ, α,β = 1,2,3. Much earlier, Mathisson (1897 – 1940) [392, 391, 393], Weyssenhoff (1889 – 1972) [695, 694] and Costa de Beauregard (1911 – 2007) [87], had investigated this concept. For a historical note cf. [584]. Sciama did not give a reference to C. de Beauregard, who fifteen years earlier had pointed out that both sides of Einstein’s field equations must become asymmetric if matter with spin degrees of freedom is generating the gravitational field. Thus an asymmetric Ricci tensor was needed. It also had been established that the deviation from geodesic motion of particles with charge or spin is determined by a direct coupling to curvature and the electromagnetic field i dxj kl Rjklds-F or, analogously, curvature and the classical spin tensor Rijkldxjskl ds. The energy-momentum tensor of a spin-fluid (Cosserat continuum), discussed in materials science, is skew-symmetric. What Sciama attempted was to geometrize the spin-tensor considered before just as another field in space-time. Because he insisted on physical space as being described by Riemannian geometry, he had to cope with two geometries, the one with the full asymmetric metric gij, and space-time with metric lij where (rk) k lrjg = δj, an attribution which we have seen before in the work of Lichnerowicz. This implied that spinless particles moved on geodesics of the metric lij, even if the gravitational field is generated by a massive spinning source, while spinning particles move on non-geodesic orbits determined by the non-symmetric connection. Sciama’s field equations were:

g = 0, (484 ) i+k−||l Si = 0, (485 ) K ik(L) − 1grsK rs(L) gik + L irr,k − L krr,i − 1-grs(L rpp,s − Lsqq,r) gik = Tik, (486 ) − 2 − 2
which, as he deemed, are “slightly different from the Einstein–Straus equations” ([565*], p. 77). Conceptually, they are very different, because the matter tensor did not and in principle cannot enter the Einstein–Straus equations. Not very modestly, Sciama concluded that “further studies are required before one can decide whether the symmetric or the non-symmetric theory describes nature better.” C. de Beauregard did not share Sciama’s opinion concerning the motion of spinless particles; according to him, in linear approximation around flat space-time [89*]:
dui dg[il] ----= ul----. (487 ) ds ds

Perhaps, Sciama had convinced himself that mixed geometry was too rich in geometrical objects for the description of just one, the gravitational interaction. In any case, in his next five papers in which he pursued the relation between (classical) spin and geometry, he went into UFT proper [565*]. He first dealt with the electromagnetic field which he identified with an expression looking like homothetic curvature: V = Ksj = ∂ Γ − ∂ Γ kl jkl k l l k. However, here r Γ k := L [kr] = Sk In order to reach this result he had introduced a complex tetrad field261 and defined a complex curvature tensor s Ki jkl skew-symmetric in one pair of its indices and “skew-Hermitian” in the other. In analogy to Weyl’s second attempt at gauge theory [692], he arrived at the trace of the tetrad-connection as his “gauge-potential” without naming it such. He also introduced a “principle of minimal coupling” as an equivalent to the “equivalence principle” of general relativity: matter must not directly couple to curvature in the Lagrangian of a theory. M.-A. Tonnelat and L. Bouche [646] then showed that Sciama’s non-symmetric theory of the pure gravitational field [565*] “implies that the streamlines of a perfect fluid μν μ ν (T = ρv v ) are geodesics of the Riemannian space with metric g(ij). These streamlines are not geodesics of the metric g (ij), but deviate from them by an amount which, in first approximation, agrees with a heuristic formula occurring in Costa de Beauregard’s theory of the gravitational effects of spin [89]”.262

In his next paper, Sciama described his endeavour of geometrizing classical spin within a general conceptual framework for unified field theory. His opening words made clear that he found it worthwhile to investigate UFT:

“The majority of physicists considers with some reserve unified field theory. In this article, my intention is to suggest that such a reserve is not justified. I will not explain or defend a particular theory but rather discuss the physical importance of non-Riemannian theories in general.” ([566], p. 1.)263View original Quote

Sciama’s main new idea was that the holonomy group plays an important rôle with its subgroup, Weyl’s U (1), leading to electrodynamics, and another subgroup, the Lorentz group, leading to the spin connection. Although he gave the paper of Yang & Mills [712] as a reference, he obviously did not know Utiyama’s use of the Lorentz group as a “gauge group” for the gravitational field [661]. C. de Beauregard‘s reaction to Sciama’s paper was immediate: he agreed with him as to the importance of embedding spin into geometry but did not like the two geometries introduced in [565]. He also suggested an experiment for measuring effects of (classical) spin in space-time [88].

In another paper of the same year, Sciama opted for a different identification of classical spin with geometrical structure: the skew-symmetric part of the connection no longer was solely connected with the electromagnetic field but with the spin angular moment of matter [567*]. By introducing a field ψ like a (classical) Dirac spinor, he defined the spin-flux as i -∂ℒ- SAB := ∂ψ,iσAB ψ where σAB is a fitting representation of the Lorentz group. The indices A, B = 0,1,2,3 are tetrad-indices (real tetrad eA i), introduced by gij = ηABeA eB i j. Seemingly, at that point Sciama had not known Cartan’s calculus with differential forms and reproduced the calculation of tetrad connection and curvature tensor in a somewhat clumsy notation. The result of interest is:

L k= S k+ δkS (488 ) [ij] ij [i j]
with S ijk = SkABeAi eBj. Use of a complex tetrad allowed him to define the electromagnetic field as before. At the time, he must have had an interaction with T. W. B. Kibble who’s paper on “Lorentz Invariance and the Gravitational Field” introduced the Poincaré group as a gauge group264 [325]. Sciama’s next paper did not introduce new ideas but presented his calculations and interpretation in further detail [568]. Two years later, when the ideas of Yang & Mills and Utiyama finally had been accepted by the community as important for field theory, Sciama for the first time named his way of introducing the skew-symmetric part of the connection “the now fashionable ‘gauge trick’ ” ([569*], p. 465, 466). His interpretation of UFT had changed entirely:

“We may note in passing that the result (7) [here Eq. (488*)] suggests that unified field theories based on a non-symmetric connection have nothing to do with electromagnetism.” ([569], p. 467)

C. de Beauregard had expressed this opinion three years earlier; moreover his doubts had been directed against the “unified theory of Einstein–Schrödinger-type” in total [90]. In the 1960s, the subject of classical spin and gravitation was taken up by F. W. Hehl [245] and developed into “Poincaré gauge theory” with his collaborators [246].

13.2 Australia

H. A. Buchdahl in Tasmania, Australia, added a further definition for the electrical current, i.e., ˆjk = ˆg[kl] ,l. Then, in linear approximation, from (211*), the unacceptable restriction l k ∂l∂ j = 0 followed. In order to remedy this defect, Buchdahl suggested another set of field equations which, with an appropriate Lagrangian, did not imply any restriction on the thus defined electric current [64]:

+ik− ˆk [kl] ˆg ∥l = 0, g[ij] = ∂[kAi], j = ˆg ,l, (489 ) Bˆ(ij) = 0, ˆB[il]= 0, (490 ) ,l
with Bˆij = δℒ∕δgij, Ak an arbitrary vector. Unfortunately, from a linear approximation in which only the antisymmetric part of the metric is considered to be weak, an unacceptable result followed: “Consequently, if one wishes to maintain an unrestricted current vector is would seem that the introduction of a vector potential Ai in the manner above must be abandoned.” ([65*], p. 1145.) With the asymmetric metric gij having gauge weight +1 the determinant g is of gauge weight +2 (+ d2 for dimension d of the manifold ). Buchdahl then set out to build a gauge-invariant unified field theory by starting from Weyl space with symmetric metric gij and linear connection k k k 1 k L ij = {ij} − δ(ikj) + 2 gijk. The gauge transformation is given by gij → λgij,ki → ki + ∂i(ln λ). Tensor densities now have both a coordinate weight z [cf. (21*) of Section 2.1.1], and a gauge weight v defined via the covariant derivative by:
+ ˆ i − i ∇ Xˆi = ∂-X- + L iXˆr − z L rXˆi − vk Xˆi, ∇ Xˆi = ∂X-- + L iXˆr − z L rˆXi − vk Xˆi. (491 ) k ∂xk kr kr k k ∂xk rk rk k
([65], p. 90). As a gauge-invariant curvature tensor and its contractions were used, the curvature scalar R then is of gauge weight − 1. Consequently, a gauge-invariant Lagrangian density must contain terms quadratic in curvature like √ --- 2 − gR. Buchdahl used the gauge-invariant Hermitian Ricci tensor Her − K− ik in Eq. (73*) of Section 2.3.2, and the field equations [66*]:
Her K (ik) = 0, Si = 0. (492 ) −
Under scrutiny and by use of approximation methods and boundary conditions at (spatial) infinity, it turned out, according to H. A. Buchdahl, that these equations very likely did not have acceptable physical solutions ([66], p. 264). In view of the non-acceptance of Weyl’s original gauge theory of the gravitational and electromagnetic fields, it is not surprising that Buchdahl’s gauge-invariant UFT did not lead to much further research. One sequel was Mishra’s paper [436] in which an exact solution in place of Buchdahl’s approximate one for weak fields is claimed; closer inspection shows that it is only implicitly given (cf. Eq. (3.1), p. 84).

13.3 India

In a short note, the Indian theoretician G. Bandyopadhyay considered an affine theory using two variational principles such as Schrödinger [553] had sugested in 1946 [9]. Besides his Ricci tensor Rjk corresponding to K− jk of (55*) he used another one &tidle;Rjk turning out to be equal to: + K− jk − 2∇jSk − Vkj. The two Lagrangians used were ∘ --------- ∘ --------- ℒ = det(Rjk)),ℒ&tidle;= det(R&tidle;jk )). The resulting field equations were:

K i+k−∥l = 0, Si = 0. (493 ) −
A solution is given by
K = λ g , g = 0. (494 ) − jk jk i+k−∥l
R. S. Mishra refined Hlavatý’s classification of the skew-symmetric part kij by allowing all signatures (“indices of inertia”) for the symmetric part hij of the asymmetric metric and by splitting Hlavatý’s third class into two [432*]:
ˇjs class 3: kisk ⁄= 0 class 4: kiskˇjs = 0, k = 0.
Here det(kij)- k := det(hlm) and 1 ˇrs K := 4krsk. He then studied the solutions of (30*) for all classes and signatures and showed that for a Riemannian metric only the first two classes exist while for signature zero all four classes are possible. He also set out to show that the solution of M.-A. Tonnelat (cf. Section 10.2.3) is valid only for the first class [434]. The conditions for Eqs. (30*), or (448*) to have a unique solution or to have at least one solution are derived and discussed in extenso in several further papers [431, 432, 346*]. Tonnelat’s conditions (364*) are made more precise: kh > 0,g (a2 + b2) ⁄= 0 : kh < 0, g ⁄= 0;k = 0,g(g − 2) ⁄= 0. Except for re-deriving Tonnelat’s result for class 1 (cf. [346], Eqs. (1.29)e, (1.30), p. 223), and polishing up Hlavatý’s results by the inclusion of some degenerate subcases, e.g., for (1 + K )(1 − 3K ) ⁄= 0 for 2 K = k, no new mathematical ideas were introduced. Physics was not mentioned at all. Also, Mishra contradicted Kichenassamy’s papers in which Tonnelat’s results had been upheld contrary to criticism by Hlavatý [326, 327]. Like in Wrede’s paper, Mishra considered the generalization of “the concepts of Einstein’s unified field theory to n-dimensional space” as well and derived “some recurrence relations for different classes of gλμ[435]. In another paper with S. K. Kaul, Mishra generalized Veblen’s identities (71*) of Section 2.3.1 to mixed geometry with asymmetric connection. The authors obtained 4 identities containing 8 terms each and with a mixture of ±-derivatives [323]. I have seen no further application within UFT. From my point of view as a historian of physics, R. S. Mishra’s papers are exemplary for estimable applied mathematics uncovering some of the structures of affine and/or mixed geometry without leading to further progress in the physical comprehension of unified field theory (cf. also [429, 296]). The generation of exact solutions to the Einstein–Schrödinger theory became a fashionable topic in India since the mid 1960s. Following a suggestion of G. Bandyopadhyay, R. Sarkar assumed the asymmetric metric to have the form:265
( ) ds2 = H (dx0 )2 + 2Idx [1dx0] + (dx1)2 + G (dx2)2 + (dx3 )2 (495 )
with H, G and I being functions of the single variable 1 x [526]. In Hlavatý’s classification, the metric was of second class. In terms of physics, static, one-dimensional gravitational and electromagnetic fields were described. The particular set of solutions obtained consisted of metric components with algebraic functions of √3λ-1 sinh( 2 x ) and √3λ-1 cosh( 2 x ), and showed (coordinate?) singularities. As a physical interpretation, Sarkar offered the analogue to a Newtonian gravitating infinite plane. The limit λli→m0 in the metric components led back to Bandyopadhyay’s solution [7] referred to in Section 9.6.2 (with some printing errors removed by Sarkar):
√ - √ - 2 G = (k + 3- bx1)43, H = 1∕b(k + 3- bx1)− 23[a − ------d√------], I = √1- ------d√------- 4 4 (k + 3 bx1 )83 b (k + 3 bx1)53 4 4
with constants a,b,d, k.

In a sequel [527], Sarkar used the asymmetric metric:

2 0 2 [2 3] 1 2 ( 2 2 32) ds = H (dx ) + 2Kdx dx + (dx ) + G (dx ) + (dx ) , (496 )
where, again G, H, K are functions of x1. The solutions found are static and with coordinate singularities. To give just one metrical component:
∘ --- − 1 2 2L −1 C1 H = M [C1 exp(μx ) + C2exp (− μx)]3 × exp[− -√-------tan ---exp(μx ) ] μ C1C2 C2
with x = x1, √ --- μ = 3λ,M, C1, C2 constants. No physical interpretation was given.

In the same year 1965, H. Prasad and K. B. Lal engaged in finding cylindrically symmetric wave-solutions of the weak field equations (277*), (278*) with:

i j 0 2 3 2 1 2 22 i j [1 [2 3] 0] hijdx dx = C[(dx ) − (dx ) ] − A (dx ) − B(dx ),kijdx dx = (ρdx + σdx )[dx − dx ],
where A,B, C are functions of x3 = z,x0 = t, and ρ,σ functions of x3 − x0 = z − t. The electromagnetic field is defined by F := 1√ − g-𝜖 grs ij 2 ijrs. All 64 components of the connection were calculated, exactly. However, in order to determine the components of the Ricci-tensor, second and higher powers of ρ,σ and their derivatives were omitted (“weak electromagnetic field”). Consequently, the solutions obtained, are only approximate. This holds also for solutions of the strong field equations (268*) likewise considered.266 Sometimes, exact solutions were announced but given only implicitly, pending the solution of nonlinear 1st order algebraical or/and differential equations; for wave solutions cf. [347].

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