A symmetric-Tensor-antisymmetric-tensor theory of gravitation and electromagnetism from a quaternion representation of general relativity

Abstract

The factorization of Einstein’s formalism into a pair of simultaneous quaternion field equations in general relativity entails the enlargement of the original formalism from 10 to 16 independent relations. By iteration, the quaternion field equations are shown to be physically equivalent to a symmetric-tensor-antisymmetric-tensor formalism. The symmetric-tensor part is in one-to-one correspondence with the 10 relations of Einstein’s original theory of gravitation. The remaining 6 (antisymmetric tensor) relations have no counterpart in the earlier gravitational theory. In addition to the generalization of the metrical field (and therefore the description of gravitational forces) that follows from the incorporation of the antisymmetric-tensor contribution, the covariant divergence of the latter formalism automatically leads to a system of field equations whose structure is in one-to-one correspondence with the Maxwell theory for electromagnetism. It is shown that Einstein’s original formalism entails 10, rather than 16 relations, because, in addition to its covariance under the group of general relativity (aconnected topological group) it is also covariant under time and space reflection transformations. The latter is an undue restriction since it is not required by the principle of general relativity alone. When these discrete symmetry elements are dropped, the (more general) quaternion formalism results. With the expression of the latter (which is not sensitive to the «handedness» of space or the direction of time) as thesum of two formalisms—one that is even and the other odd under space or time reflections—the original 10 relations of Einstein’s equationsplus the 6 relations that lead to the Maxwell field equations follow. Finally, with the application of the Schwarzschild conditions in the anti-symmetric-tensor part of the field equations, they reduce (in orderv/c) to a scalar-field formalism. With this approximation then, the full exploitation of the principle of general relativity—in terms of a symmetric-tensor-antisymmetric-tensor formalism—reduces to a scalar-tensor formalism, such as the one that has been studied by Brans and Dicke. It follows that to within the approximations that have been considered in this comparison (which would be applicable to a derivation of planetary motion) the present theory and a trulyscalar-tensor theory would be experimentally indistinguishable. In the derivation presented in this paper, however, the generalization of the metrical field to incorporate with the usual formalism an antisymmetric-tensor part (and therefore a scalar part in the application to planetary problems) follows in a natural way from the lowest-dimensional irreducible representations of the group of general relativity and no new fundamental constants need be introduced.