“to the consideration of the small oscillations of the proton and single electron forming the hydrogen atom.” [18, 17].
Another one was the Russian physicist
G. Rumer, whose somewhat exotic suggestion, within a framework
of Kantian philosophy, was to remain in Riemannian geometry and
keep to Einstein’s vacuum field equations
but raise the number of the dimension of the underlying manifold.
He named such a manifold an and considered a
three-dimensional Riemannian subspace
embedded, locally
and isometrically, into
. Then, by use of the
Gauss-Codazzi and Ricci-Codazzi equations, he decomposed the Ricci
tensor of
into its part in
and the rest. His classification went as follows:
“We have seen that the itself is either an
(in this case it is
empty), or a subspace of an
(thus it contains a
gravitational field), or a subspace of
(then also an
electrical field is present). In the case in which ‘something’
exists which is neither a gravitational nor an electrical field,
the
must be a subspace of
(
). However, geometry shows that every
is a subspace of a particular
, i.e., the Euclidean or
pseudo-Euclidean space. This shows us that transition to
is the final step.” 274
([284],
p. 277)
The unidentified parts in the decomposition of
the Ricci tensor into its piece in and the rest Rumer
ascribed to “matter.” He acknowledged Born’s
“[...] stimulation and his interest extended
toward the completion of this paper.” 275
Born himself was mildly skeptical276 :
“[...] a young Russian surfaced here who
brought with him a 6-dimensional relativity theory. As I already
felt frightened by the various 5-dimensional theories, and had
little confidence that something beautiful would result in this
way, I was very skeptical.” 277
After Lanczos had (mildly) criticised Einstein’s parallelism at a distance [200], he seemed to have lost confidence in Einstein’s program for unification and became a “renegade”. He developed a theory by which
“[...] the basic properties of the
electromagnetic field may be derived effortlessly from the general
properties of Riemannian geometry by use of a variational principle
characterised by a very natural demand.” 278
([201],
p. 168)
For his Lagrangian, he took , with
being a constant. He
first varied with respect to the metric
and the Ricci
tensor
as independent variables, and then
expressed the variation
with
. The resulting variation is then set equal to zero.
In the process “spontaneously” a
“free vector appears for which, later, a restraining equation of the type of the equation for the [electromagnetic] potential results - as a consequence of the conservation laws for energy and momentum.”
Also Rainich’s approach [264] mentioned in Section 6.1, which, in the case of Maxwell’s equations without sources, and for non-null electromagnetic fields, did substitute a set of algebraic conditions on the Einstein tensor for Maxwell’s equations, might be seen as an alternative for the unification of gravity and electromagnetism. According to L. Witten:
“The only criterion for a unified field theory that these equations do not satisfy is that they are not derived from a variational principle by means of a Lagrange’s function involving geometric quantities alone.” ([422], p. 397)
Finally, van Dantzig’s program after 1934, which we briefly met in Section 1, might be considered. It aimed at showing, eventually, that the
“metric should turn out finally to be a system of some statistical mean values of certain physical quantities.” ([363], p. 522)
This meant turning upside down Einstein’s geometrization program for matter.