Einstein’s accuracy in Unified Field Theory is guiding our actions.

The starting point now was to keep the four-dimensionality of the theory and also the requirement of general covariance, but to give up the postu late that a generalized theory of gravitation should necessarily be based on the existence of a Riemannian metric. What Einstein and Bargmann pro posed instead was, in effect, an attempt at a non-local relativistic theory of gravitation. They investigated the properties of a new kind of mathematical object that they called “bivectors.” In contrast to modern usage of the term, these are not asymmetric, second-rank tensors, but rather second-rank tensors that depend on two distinct points of the four-dimensional manifold. The transformation properties of these bivectors depend on the two distinct points of the manifold, each index of the bivector being associated with a different base point.

Although explicitly articulated in the context of the unified field theory program the two papers do not discuss any physical interpretation. Rather, they discuss properties of the mathematical structure that derives from the bivectors. They also discuss field equations for the bivectors, which turn out to be algebraic rather than differential equations. The difficulties of the bivector approach again came with finding and interpreting non-trivial solutions of the fundamental equations. The published papers indicate only preliminary results, partly credited to Einstein colleagues, Bargmann and Pauli, as well as to the Princeton mathematician Carl Ludwig Siegel, and explicitly mention ongoing research with Pauli. It is again possible that a closer scrutiny of Einstein’s later research manuscripts might shed further light on Einstein’s elaboration of this approach.

Judging by the published record, the bivector episode represents Ein stein’s penultimate distinct approach in the sequence of attempts to arrive at a unified field theory. Einstein devoted the last ten years of his life to the investigation of a framework that he had already worked on in the mid twenties, to which he now returned in a first publication of 1945 (Einstein 1945). This last approach of Einstein’s work along his unified field theory program was again based on a local Riemannian metric but on an asymmetric one.

Initially, Einstein took the metric tensor to have complex components and demanded Hermitian symmetry. Pauli, however, quickly pointed out to Einstein that the restriction to Hermitian symmetry was not necessary. The subsequent investigations were then following the pattern of earlier ones (Ein stein 1946, 1948, 1950a, 1950b, 1950c, App. II, 1953, App. II, 1955, App. II, 22 Einstein and Straus 1946, Einstein and Kaufmann 1954, 1955). Tentative f ield equations were tested for their mathematical properties, and it was checked whether the criteria for a physical interpretation could be applied. Along with the mathematical properties, Einstein worried very much about the problem of compatibility as he had with other, earlier approaches. Since the mathematics of a framework based on an asymmetric metric tensor is exceedingly more complex and less well-known than the standard formalism of (semi-)Riemannian differential geometry underlying conventional general relativity, Einstein spent the rest of his life elaborating the asymmetric theory.

Einstein’s very last considerations in this final approach were presented by his last assistant, Bruria Kaufmann, at the 50th anniversary of the relativity theory in Bern in July 1955 a few weeks after Einstein’s death (Kaufmann 1956).

Einsteins intuition is correct with regards to the Asymmetric theory, as I can generate a force for motion with the 4-Phase Motor Generator Device.

The Unified Field Theory was Einsteins greatest curiosity, to combine Gravitation and Electromagnetism. Without the Higgs field, the elementary particles that make up you and me, and the visible universe would have no mass. Einstein was correct!