6.1 Affine and mixed geometry

Already in July 1925 Einstein had laid aside his doubts concerning “the deepening of the geometric foundations”. He modified Eddington’s approach to the extent that he now took both a non-symmetric connection and a non-symmetric metric, i.e., dealt with a mixed geometry (metric-affine theory):

“[...] 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.”115View original Quote ([78], p. 414)

As in general relativity, he started from the Lagrangian ℒ = ˆgikRik, but now with ˆgik and the connection Γ l kj 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:

− ∂gik-+ grkΓ ilr + girΓ lkr + gikϕl + gilϕk = 0, Rik = 0, (134 ) ∂xl
i.e., 64 + 16 equations for the same number of variables. ϕk is an arbitrary covariant vector. The asymmetric g ik is related to ˆglm by
jr rj j --gik- ˆgirˆg = ˆgriˆg = δi , ˆgik = √ −-g . (135 )
The three equations (134View Equation) and
∂ ˆgik ∂ˆgki ---k − --k- = 0, Rik = 0, (136 ) ∂x ∂x
were the result of the variation. In order to be able to interpret the symmetric part of gik as metrical tensor and its anti(skew)-symmetric part as the electromagnetic field tensor, Einstein put ϕk = 0, i.e., overdetermined his system of partial differential equations. However, he cautioned:

“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 ϕk equal to zero.”116View original Quote

In comparing Equation (134View Equation) with ϕk = 0 and Equation (47View Equation), 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.” ([346Jump To The Next Citation Point], p. 187)

J. M. Thomas wrote Einstein’s Equation (134View Equation) in the form

--2--- r gik∕l = gikϕl + gilϕk, ϕl = − n − 1Ω rl, (137 )
with Ω being the skew-symmetric part of the asymmetric connection H k= Γ k+ Ω k ij (ij) [ij], and gij being the symmetric part of the asymmetric metric h = g + ω ij (ij) [ij]. The two covariant derivatives introduced by J. M. Thomas are ∂gij r r gij,k = ∂xk-− grjΓ ik − girΓ jk and ∂hij r r hij∕k = ∂xk-− hrjH ik − hirH jk. J. M. Thomas then could reformulate Equation (137View Equation) in the form
gij∕l = g[ri]Ω jrl + g[ir]Ωlrj, (138 )
and derive the result
gij,l + gjl,i + gli,j = 0 (139 )
(see [346Jump To The Next Citation Point], p. 189).

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 Fik,l + Fli,k + Fkl,i = 0 he only obtained ∑ ∂∂xl(Fik,l + Fli,k + Fkl,i) = 0.

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 ϕij of the metric aij = gij + ϕij 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 &tidle;E → &tidle;B and &tidle;B → − &tidle;E 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 ±e and unequal mass ([246Jump To The Next Citation Point], 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 ik ˆg 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.”120View original Quote ([99Jump To The Next Citation Point], 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.”121View original Quote [79]

In a paper dealing with the field equations

Rik − R-gik = − κTik, (140 ) 4
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 ([263264265Jump To The Next Citation Point266]), Einstein indicated that he had lost hope in the extension of Eddington’s affine theory:

“That the equations (140View Equation) 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.”122View original Quote (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 [323].

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 WeylEddington. The equations

1 Rik − -R gik = − κTik electromagnetic 4
I take as the best we have nowadays. They are 9 equations for the 14 variables gik and γ ik. New calculations seem to show that these equations yield the motion of the electrons. But it appears doubtful whether there is room in them for the quanta.”123View original Quote ([99Jump To The Next Citation Point], p. 216)

According to the commenting note by Tonnelat, the 14 variables are given by the 10 components of the symmetric part g(ik)124 of the metric and the 4 components of the electromagnetic vector potential “the rotation of which are formed by the γ[ik]125View original Quote.

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,

lm 1- lm Rik = gikflmf − 2flfkmg
gives me little satisfaction. It does not allow for electrical masses free from singularities. Moreover, I cannot bring myself to gluing together two items (as the l.h.s. and the r.h.s. of an equation) which from a logical-mathematical point of view have nothing to do with each other.”127View original Quote ([99Jump To The Next Citation Point], p. 230)

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