five conservation laws which are shown to be the differential laws for conservation
of energy, momentum, and charge.
b) A number of authors, in particular Jordan [Jl], have studied an extension
of this formalism to the case where the right-hand side of Eq. 17.16 is replaced
by a scalar field. Bergmann informed me that he and Einstein also had worked
on this generalization [B3].
c) In the 1980s, particle physicists have taken up the study of field equations in
(4 + 7V)-dimensional manifolds, where '4' refers to space-time and where the
extra N variables span a compact space-like TV-dimensional domain which is sup-
posed to be so small as not to influence the usual physics in inadmissible ways.
Various values of N are being considered for the purpose of including non-Abe-
lian gauge fields. Some authors advocate dropping constraints of the type 17.2 and
17.3, hoping that the compactness in the additional dimensions will result from
'spontaneous compactification,' a type of spontaneous symmetry breaking. The
future will tell what will come of these efforts. It seems fitting to close this section
by noting that, in 1981, a paper appeared with the title 'Search for a Realistic
Kaluza-Klein Theory' [Wl].*
- Two Options. Einstein spent much less energy on five-dimensional theories
than on a second category of unification attempts in which the four-dimensional
manifold is retained but endowed with a geometry more general than Riemann's.
At this point the reader is offered two options.
Option 1. Take my word for it that' these attempts have led nowhere thus far,
skip the next section, skim the two sections thereafter, and turn to the quantum
theory.
Option 2. If he is interested in what not only Einstein but also men like Edding-
ton and Schroedinger tried to do with these geometries, turn to the next section.
17d. Relativity and Post-Riemannian Differential GeometryIn his address on general relativity and differential geometry to the Einstein Cen-
tennial Symposium in Princeton [C3], the eminent mathematician Shiing-Shen
*In that paper, one will find references to other recent work in this direction.be the variational principle, where R is the 5-curvature scalar. All that is given
about X is that it is a scalar function of field variables and their covariant deriv-
atives. In addition, one must admit an explicit dependence of ^~on the coordinates
X". By extending the Noether methods to this more general situation, one can
derive an explicit expression for the source tensor T** in terms of jC and deriv-
atives of X with respect to the fields and to X". This tensor satisfies(17.18)
336 THE LATER JOURNEY