“[...] Also, my opinion about my paper which
appeared in these reports [i.e., Sitzungsberichte of
the Prussian Academy, Nr. 17, p. 137, 1923], and
which was based on Eddington’s fundamental idea, is
such that it does not present the true solution of the problem.
After an uninterrupted search during the past two years I now
believe to have found the true solution.” 115
([78],
p. 414)
As in general relativity, he started from the
Lagrangian , but now with
and the connection
being varied
separately as independent variables. After some manipulations, the
variation with regard to the metric and to the connection led to
the following equations:
“However, for later investigations (e.g., the problem of the electron) it is to
be kept in mind that the HAMILTONian principle does not provide an
argument for putting equal to zero.” 116
In comparing Equation (134) with
and Equation (47
), we note that the
expression does not seem to correspond to a covariant derivative
due to the
sign where a
sign is required. But
this must be due to either a calculational error, or to a printer’s
typo because in the paper of J. M. Thomas following
Einstein’s by six months and
showing that Einstein’s
“new equations can be obtained by direct
generalisation of the equations of the gravitational field
previously given by him. The process of generalisation consists in
abandoning assumptions of symmetry and in adopting a definition of
covariant differentiation which is not the usual one, but which
reduces to the usual one in case the connection is symmetric.”
([346], p. 187)
J. M. Thomas wrote Einstein’s Equation (134) in the form
After having shown that his new theory contains
the vacuum field equations of general relativity for vanishing
electromagnetic field, Einstein then proved that, in a
first-order approximation, Maxwell’s field equations result cum grano salis: Instead of he only obtained
.
This was commented on in a paper by Eisenhart who showed “more
particularly what kind of linear connection Einstein has employed” and who
obtained “in tensor form the equations which in this theory should
replace Maxwell’s equations.” He then pointed to some difficulty in
Einstein’s theory: When
identification of the components of the antisymmetric part of the metric
with
the electromagnetic field is made in first order,
“they are not the components of the curl of a vector as in the classical theory, unless an additional condition is added.” ([120], p. 129)
Toward the end of the paper Einstein discussed time-reversal;
according to him, by it the sign of the magnetic field is changed,
while the sign of the electric field vector is left unchanged117
. As he wanted to
obtain charge-symmetric solutions from his equations, Einstein now proposed to change
the roles of the magnetic fields and the electric fields in the
electromagnetic field tensor. In fact, the substitutions and
leave invariant Maxwell’s
vacuum field equations (duality transformations)118. Already Pauli had pointed to
time-reflection symmetry in relation with the problem of having
elementary particles with charge
and unequal mass
([242
], p. 774).
At first, Einstein seems to have been proud about his new version of unified field theory; he wrote to Besso on 28 July 1925 that he would have liked to present him “orally, the egg laid recently, but now I do it in writing”, and then explained the independence of metric and connection in his mixed geometry. He went on to say:
“If the assumption of symmetry119
is dropped, the laws
of gravitation and Maxwell’s field laws for empty space are
obtained in first approximation; the antisymmetric part of is the electromagnetic field. This is surely a
magnificent possibility which likely corresponds to reality. The
question now is whether this field theory is consistent with the
existence of quanta and atoms. In the macroscopic realm, I do not
doubt its correctness.” 120
([327
], p. 209)
We have noted before that a similar suggestion within a theory with a geometry built from an asymmetric metric had been made, in 1917, by Bach alias Förster.
Yet, in the end, also this novel approach did not convince Einstein. Soon after the publication discussed, he found his argument concerning charge symmetric solutions not to be helpful. The link between the occurrence of solutions with both signs of the charge with time-symmetry of the field equations induced him to doubt, if only for a moment, whether the endeavour of unifying electricity and gravitation made sense at all:
“To me, the insight seems to be important that
an explanation of the dissimilarity of the two electricities is
possible only if time is given a preferred direction, and if this
is taken into account in the definition of the decisive physical
quantities. In this, electrodynamics is basically different from
gravitation; therefore, the endeavour to melt electrodynamics with
the law of gravitation into one unity, to me no longer seems to be
justified.” 121
[79]
In a paper dealing with the field equations
which had been discussed earlier by Einstein [70], and to which he came back now after Rainich’s insightful paper into the algebraic properties of both the curvature tensor and the electromagnetic field tensor ([262, 263, 264“That the equations (140) have received only
little attention is due to two circumstances. First, the attempts of all of us were
directed to arrive, along the path taken by Weyl and Eddington or a similar one, at a
theory melting into a formal unity the gravitational and
electromagnetic fields; but by lasting
failure I now have laboured to convince myself that truth cannot be approached along this path.”
122
(Einstein’s italics; [80],
p. 100)
The new field equation was picked up by R. N. Sen of Kalkutta who calculated “the energy of an electric particle” according to it [322].
In the same spirit as the one of his paper, Einstein said good bye to his theory in a letter to Besso on Christmas 1925 in words similar to those in his letter in June:
“Regrettably, I had to throw away my work in the spirit of Eddington. Anyway, I now am convinced that, unfortunately, nothing can be made with the complex of ideas by Weyl-Eddington. The equations
According to the commenting note by Tonnelat, the
14 variables are given by the 10 components of the symmetric part
124
of the metric and the 4 components of the electromagnetic vector
potential “the rotation of which are formed by the
” 125
.
But even “the best we have nowadays” did not satisfy Einstein; half a year later, he expressed his opinion in a letter to Besso:
“Also, the equation put forward by myself126 ,