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:
![Jijk = γ{[ij],k}](article2494x.gif)







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
![Γ s [δk+ ∗skrars] = HL j ij s ik](article2514x.gif)









For the field variables , in empty space the field equations following from (474*) are













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.)






By an approximation around Euclidean space which neglected the squares of
, the cubes of
, and interaction terms between
, Kursunŏglu obtained the following
results:
![Jij = 2∂ [iJj]](article2557x.gif)













![|E | = e2[1 − e−κr − κre− κr] − r](article2572x.gif)




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:
![Brs = 2∂[rBs]](article2580x.gif)
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 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
![∂[ωk μλ] = 0.](article2591x.gif)
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
![R [μλ] = 2 ∂[μX λ]](article2593x.gif)
coupled with
![α R [μλ] = − D αSμλ](article2594x.gif)
(where denotes the covariant derivative with respect to
). Hence
![S νμλ = 2X [μδνλ] + T νμλ,](article2597x.gif)
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:
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
field261
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:

“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]:

















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:
















![√ - √ - 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](article2684x.gif)

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

![∘ --- − 1 2 2L −1 C1 H = M [C1 exp(μx ) + C2exp (− μx)]3 × exp[− -√-------tan ---exp(μx ) ] μ C1C2 C2](article2689x.gif)


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 ],](article2692x.gif)





