## 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 to Einstein’s Lagrangian of UFT [34]. They
are:^{254}

The assignment of to the gravitational potentials and of to the electromagnetic field was upheld while the electric current became defined as .

A linearization of Bonnor’s field equations up to the first order in led to:

In first approximation the electric current is given by such that the previous equation looked like . For a spherically symmetric particle at rest with radial coordinate , Bonnor obtained where is the charge density. For and for , the charge density will be restricted^{255}to . 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:
with real , imaginary , and . Correspondingly, the symmetrical linear
connection split into a real connection part and an imaginary valued tensor
.^{256}
His approach to UFT then differed considerably from Einstein’s. In place of (16*) of Section 2.1.1 now

^{257}Now, and . As a real Lagrangian, Moffat chose: where presumably is defined by the decomposition of the inverse of with , i.e., , 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 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 , in empty space the field equations following from (474*) are

Matter is introduced through the variation with “the complex-symmetric source term” , the Newtonian gravitational constant. According to Moffat: “The real tensor represents the energy-momentum of matter, while is the charge-current distribution.” A weak-field-approximation with real and imaginary is then carried through. In 1st approximation, the wave equation resulted where and . 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*]:

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.)

^{259}I assume that his Ricci tensor is the same as the one used by Einstein in [149], i.e., .

By an approximation around Euclidean space which neglected the squares of , the cubes of , and interaction terms between , Kursunŏglu obtained the following results:

where , and . Both and are identified with the electromagnetic field: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 (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 . In the note, a Lagrangian differing from (478*) appeared:

where 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
Relativity^{260}
by replacing the constraint on vector torsion by with constant , and by introducing
a vector-potential for the electromagnetic field tensor [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 . 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 [MR0068357]:

“The Einstein paper contains three separate sections. In the first section the author expresses the symmetric part of the unified field connection by means of its skew symmetric part and vice versa. Then he identifies the electromagnetic field with and imposes on it the first set of Maxwell conditions

The second set of Maxwell conditions is equivalent to the Einstein condition . 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

coupled with

(where denotes the covariant derivative with respect to ). Hence

where is a solution of

Therefore is equivalent to

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

#### 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 . 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 or, analogously, curvature and the classical spin tensor . 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 , and space-time with metric where , an attribution which we have seen before in the work of Lichnerowicz. This implied that spinless particles moved on geodesics of the metric , 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:

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*]: 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: . However,
here In order to reach this result he had introduced a complex tetrad
field^{261}
and defined a complex curvature tensor 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 are
geodesics of the Riemannian space with metric . These streamlines are not geodesics of the
metric , 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.)^{263}

Sciama’s main new idea was that the holonomy group plays an important rôle with its subgroup, Weyl’s , 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 like a (classical) Dirac spinor, he defined the spin-flux as where is a fitting representation of the Lorentz group. The indices are tetrad-indices (real tetrad ), introduced by . 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:

with . 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 group^{264}[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., . Then, in linear approximation, from (211*), the unacceptable restriction 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]:

with , 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 in the manner above must be abandoned.” ([65*], p. 1145.) With the asymmetric metric having gauge weight the determinant is of gauge weight for dimension of the manifold ). Buchdahl then set out to build a gauge-invariant unified field theory by starting from Weyl space with symmetric metric and linear connection . The gauge transformation is given by . Tensor densities now have both a coordinate weight [cf. (21*) of Section 2.1.1], and a gauge weight defined via the covariant derivative by: ([65], p. 90). As a gauge-invariant curvature tensor and its contractions were used, the curvature scalar then is of gauge weight . Consequently, a gauge-invariant Lagrangian density must contain terms quadratic in curvature like . Buchdahl used the gauge-invariant Hermitian Ricci tensor in Eq. (73*) of Section 2.3.2, and the field equations [66*]: 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 corresponding to of (55*) he used another one turning out to be equal to: . The two Lagrangians used were . The resulting field equations were:

A solution is given by R. S. Mishra refined Hlavatý’s classification of the skew-symmetric part by allowing all signatures (“indices of inertia”) for the symmetric part of the asymmetric metric and by splitting Hlavatý’s third class into two [432*]:^{265}with and being functions of the single variable [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 and , and showed (coordinate?) singularities. As a physical interpretation, Sarkar offered the analogue to a Newtonian gravitating infinite plane. The limit 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):

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

where, again are functions of . The solutions found are static and with coordinate singularities. To give just one metrical component: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:

^{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].