First steps in the development of unified field theories

“Perhaps, there exists a covariant 6-vector by which the appearance of electricity is explained and which springs lightly from the $g μν$ , not forced into it as an alien element.”⁴³

“The aim of dealing with gravitation and electricity on the same footing by reducing both groups of phenomena to $gμν$ has already caused me many disappointments. Perhaps, you are luckier in the search. I am totally convinced that in the end all field quantities will look alike in essence. But it is easier to suspect something than to discover it.”⁴⁴ (16 November 1917 [321 ], Vol. 8A, Doc. 400, p. 557)

In his next letter, Förster gave results of his calculations with an asymmetric $gμν = sμν + aμν$ , introduced an asymmetric “three-index-symbol” and a possible generalisation of the Riemannian curvature tensor as well as tentative Maxwell’s equations and interpretations for the 4-potential $Aμ$ , and special solutions (28 December 1917) ([321

], Volume 8A, Document 420, pp. 581–587). Einstein’s next letter of 17 January 1918 is skeptical:

“Since long, I also was busy by starting from a non-symmetric $gμν$ ; however, I lost hope to get behind the secret of unity (gravitation, electromagnetism) in this way. Various reasons instilled in me strong reservations: [...] your other remarks are interesting in themselves and new to me.”⁴⁵ ([321 ], Volume 8B, Document 439, pp. 610–611)

Einstein’s remarks concerning his previous efforts must be seen under the aspect of some attempts at formulating a unified field theory of matter by G. Mie [229 , 230 , 231 ]⁴⁶ , J. Ishiwara, and G. Nordström, and in view of the unified field theory of gravitation and electromagnetism proposed by David Hilbert.

“According to a general mathematical theorem, the electromagnetic equations (generalized Maxwell equations) appear as a consequence of the gravitational equations, such that gravitation and electrodynamics are not really different.”⁴⁷ (letter of Hilbert to Einstein of 13 November 1915 [162 ])

Einstein’s answer to Hilbert on 15 November 1915 shows that he had also been busy along such lines:

“Your investigation is of great interest to me because I have often tortured my mind in order to bridge the gap between gravitation and electromagnetism. The hints dropped by you on your postcards bring me to expect the greatest.”⁴⁹ [101 ]

Even before Förster alias Bach corresponded with Einstein, a very early bird in the attempt at unifying gravitation and electromagnetism had published two papers in 1917, Reichenbächer [270

, 269

]. His paper amounts to a scalar theory of gravitation with field equation $R = 0$ instead of Einstein’s $Rab = 0$ outside the electrons. The electron is considered as an extended body in the sense of Lorentz–Poincaré, and described by a metric joined continuously to the outside metric⁵⁰ :

“The disturbance, which is generated by the electrons and which forces us to adopt a coordinate system different from the usual one, is interpreted as the electromagnetic six-vector, as is known.”⁵¹ ([270 ], p. 136)

By his “coordinate rotation”, or, as he calls it in ([269

], p. 174), “electromagnetic rotation”, he tries to geometrize the electromagnetic field. As Weyl’s remark in Raum–Zeit–Materie ([398

], p. 267, footnote 30) shows, he did not grasp Reichenbächer’s reasoning; I have not yet understood it either. Apparently, for Reichenbächer the metric deviation from Minkowski space is due solely to the electromagnetic field, whereas gravitation comes in by a single scalar potential connected to the velocity of light. He claims to obtain the same value for the perihelion shift of Mercury as Einstein ([269 ], p. 177). Reichenbächer was slow to fully accept general relativity; as late as in 1920 he had an exchange with Einstein on the foundations of general relativity [271 , 71 ].

After Reichenbächer had submitted his paper to Annalen der Physik and seemingly referred to Einstein,

“Planck was uncertain to which of Einstein’s papers Reichenbächer appealed. He urged that Reichenbächer speak with Einstein and so dissolve their differences. The meeting was amicable. Reichenbächer’s paper appeared in 1917 as the first attempt at a unified field theory in the wake of Einstein’s covariant field equations.” ([262 ], p. 208)

In this context, we must also keep in mind that the generalisation of the metric tensor toward asymmetry or complex values was more or less synchronous with the development of Finsler geometry [126 ]. Although Finsler himself did not apply his geometry to physics it soon became used in attempts at the unification of gravitation and electromagnetism [274 ].

3.1 First steps in the development of unified field theories