“I show in the present paper that his [Einstein’s] new equations can be
obtained by a direct generalisation of the equations of the
gravitational field previously given by him []. [...] In the final section I show
that the adoption of the ordinary definition of covariant
differentiation leads to a geometry which includes as a special
case that proposed by Weyl as a basis for the electric
theory; further that the asymmetric connection for this special
case is of the type adopted by Schouten for the geometry at the
basis of his electric theory.” ([346],
p. 187)
We met J. M. Thomas’ paper before in section 6.1.
During the period considered here, a few
physicists followed the path of Eddington and Einstein. One who had absorbed Eddington’s and Einstein’s theories a bit later
was Infeld
of Warsaw128. In January 1928, he followed
Einstein by using an asymmetric metric the symmetric part of which stood for the gravitational potential, the
skew-symmetric part
for the electromagnetic field. However,
he set the non-metricity tensor (of the symmetric part
of the metric)
, and assumed
for the skew-symmetric part
,
Three months later, Infeld published a note in Comptes Rendus of the Parisian Academy in which he now presented the exact connection as
whereThe Japanese physicist Hattori embarked on a
metric-affine geometry derived purely from an asymmetric metrical
tensor . He defined an affine
connection
“The preceding equation shows that electrical
charge and electrical current are distributed wherever an
electromagnetic field exists.” 129
Thus, the same problem obtained as in Einstein’s theory: A field without electric current or charge density could not exist [155]130 .
Infeld quickly reacted to
Hattori’s paper by noting that Hattori’s voluminous calculations
could be simplified by use of Schouten’s Equation (39) of Section 2.1.2. As in Hattori’s
theory two connections are used, Infeld criticised that Hattori had
not explained what his fundamental geometry should be: Riemannian
or non-Riemannian? He then gave another example for a theory
allowing the identification of the electromagnetic field tensor
with the antisymmetric part of the Ricci tensor: He displayed again
the well-known connection with vector torsion used by Schouten [298
] without referring
to Schouten’s paper [164
]. He also claimed
that Hattori’s Equation (145
) is the same as the
one that had been deduced from Eddington’s theory by Einstein in the Appendix to the
German translation of Eddington’s book ([60
], p. 367). All
in all, Infeld’s critique tended to deny
that Hattori’s theory was more general than Einstein’s, and to point out
“that the problem of generalising the theory of
relativity cannot be solved along a purely formal way. At first,
one does not see how a choice can be made among the various
non-Riemannian geometries providing us with the gravitational and
Maxwell’s equations. The proper world geometry which ought to lead
to a unified theory of gravitation and electricity can only be
found by an investigation of its physical content.” 131
([164],
p. 811)
Infeld could as well have applied
this admonishment to his own unified field theory discussed above.
Perhaps, he became irritated by comparing his expression for the
connection (142) with
Hattori’s (145
).
In June 1931, von Laue submitted a paper of the Genuese mathematical physicist Paolo Straneo to the Berlin Academy [331]. In it Straneo took note of Einstein’s teleparallel geometry, but decided to take another route within mixed geometry; he started with a symmetric metric and the asymmetric connection
with both non-vanishing curvature tensorBy a remark of Straneo, that auto-parallels and
geodesics have to be distinguished in an affine geometry, the
Indian mathematician Kosambi
felt motivated to approach affine geometry from the system of
curves solving with an arbitrary parameter
. He then defined two covariant “vector-derivations”
along an arbitrary curve and arrived at an (asymmetric) affine
connection. By this, he claimed to have made superfluous the
five-vectors of Einstein and Mayer [107
]. This must be read
in the sense that he could obtain the Einstein-Mayer equations from his formalism
without introducing a connecting quantity leading from the space of
5-vectors to space-time [194]. Einstein, in his papers, did not
comment on the missing metric compatibility in his theory and its
physical meaning. Due to this complication - for example even a
condition of metric compatibility would not have the physical
meaning of the conservation of the norm of an angle between vectors
under parallel transport, and the further difficulty that much of
the formalism was very clumsy to manipulate; essential work along
this line was done only much later in the 10940s and 1950s (Einstein, Einstein and Strauss, Schrödinger,
Lichnerowicz, Hlavaty, Tonnelat, and many others). In this work a
generalisation of the equation for metric compatibility, i.e., Equation (47
), will play a central
role. The continuation of this research line will be presented in
Part II of this article.