Less Than Unification

During the time span considered here, there also were those who did not adhere to the program of unified field theory, or changed their mind about it. The mathematician G. D. Birkhoff belonged to the first group. Instead of removing the matter tensor in favour of some general geometry, he just remained in the old setting of Einstein’s theory: To the energy-momentum tensors of a perfect fluid and of the electromagnetic field on the r.h.s. of Einstein’s field equations, he added a contribution, named the “atomic potential tensor” describing a scalar wave function $ψ$ à la Schrödinger: $Aij = ψ gij$ with the “potential” $ψ$ . He proposed to apply this theory

“to the consideration of the small oscillations of the proton and single electron forming the hydrogen atom.” [18 , 17 ].

Another one was the Russian physicist G. Rumer, whose somewhat exotic suggestion, within a framework of Kantian philosophy, was to remain in Riemannian geometry and keep to Einstein’s vacuum field equations but raise the number of the dimension of the underlying manifold. He named such a manifold an $Fn$ and considered a three-dimensional Riemannian subspace $V3$ embedded, locally and isometrically, into $Fn$ . Then, by use of the Gauss–Codazzi and Ricci–Codazzi equations, he decomposed the Ricci tensor of $Fn$ into its part in $V3$ and the rest. His classification went as follows:

“We have seen that the $V 3$ itself is either an $F 3$ (in this case it is empty), or a subspace of an $F4$ (thus it contains a gravitational field), or a subspace of $F5$ (then also an electrical field is present). In the case in which ‘something’ exists which is neither a gravitational nor an electrical field, the $V3$ must be a subspace of $Fn$ ( $n ≥ 6$ ). However, geometry shows that every $V3$ is a subspace of a particular $F6$ , i.e., the Euclidean or pseudo-Euclidean space. This shows us that transition to $F 6$ is the final step.”²⁷⁴ ([285 ], p. 277)

The unidentified parts in the decomposition of the Ricci tensor into its piece in $V3$ and the rest Rumer ascribed to “matter.” He acknowledged Born’s

“[...] stimulation and his interest extended toward the completion of this paper.”²⁷⁵

“[...] a young Russian surfaced here who brought with him a 6-dimensional relativity theory. As I already felt frightened by the various 5-dimensional theories, and had little confidence that something beautiful would result in this way, I was very skeptical.”²⁷⁷

After Lanczos had (mildly) criticised Einstein’s parallelism at a distance [201 ], he seemed to have lost confidence in Einstein’s program for unification and became a “renegade”. He developed a theory by which

“[...] the basic properties of the electromagnetic field may be derived effortlessly from the general properties of Riemannian geometry by use of a variational principle characterised by a very natural demand.”²⁷⁸ ([202 ], p. 168)

For his Lagrangian, he took $ℒ = RikRik + C (Rk )2 k$ , with $C$ being a constant. He first varied with respect to the metric $g ik$ and the Ricci tensor $R ik$ as independent variables, and then expressed the variation $δRik$ with $δgij$ . The resulting variation is then set equal to zero. In the process “spontaneously” a

“free vector appears for which, later, a restraining equation of the type of the equation for the [electromagnetic] potential results – as a consequence of the conservation laws for energy and momentum.”

Also Rainich’s approach [265 ] mentioned in Section 6.1, which, in the case of Maxwell’s equations without sources, and for non-null electromagnetic fields, did substitute a set of algebraic conditions on the Einstein tensor for Maxwell’s equations, might be seen as an alternative for the unification of gravity and electromagnetism. According to L. Witten:

“The only criterion for a unified field theory that these equations do not satisfy is that they are not derived from a variational principle by means of a Lagrange’s function involving geometric quantities alone.” ([422 ], p. 397)

Finally, van Dantzig’s program after 1934, which we briefly met in Section 1, might be considered. It aimed at showing, eventually, that the

“metric should turn out finally to be a system of some statistical mean values of certain physical quantities.” ([363 ], p. 522)

8 Less Than Unification