Affine and mixed geometry

“[...] 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 $ℒ = ˆgikRik$ , 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:

“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.”¹¹⁶

In comparing Equation (134

) with $ϕk = 0$ 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)

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 ([246

], 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.”¹²⁰ ([99 ], 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 ]

“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)

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

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.”¹²³ ([99

], 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 $γ[ik]$ ”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 ,

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.”¹²⁷ ([99

], p. 230)

6.1 Affine and mixed geometry