Gravitation and electromagnetism

Abstract

The theory of general relativity unifies gravitation with the geometry of spacetime by replacing the scalar Newtonian gravitational potential with the symmetric metric tensor gµν of a four-dimensional general Riemannian manifold by means of the equivalence principle. As is well known, the electromagnetic field has resisted all efforts to be interpreted in terms of the geometrical properties of space-time as well. In this investigation, we show that the electromagnetic field may indeed be given a geometrical interpretation in the framework of a modified version of general relativity - unimodular relativity. According to the theory of unimodular relativity developed by Anderson and Finkelstein, the equations of general relativity with a cosmological constant are composed of two independent equations, one which determines the null-cone structure of space-time, another which determines the measure structure of space-time. The field equations that follow from the restricted variational principle of this version of general relativity only determine the null-cone structure and are globally scale-invariant and scale-free. We show that the electromagnetic field may be viewed as a compensating gauge field that guarantees local scale invariance of these field equations. In this way, Weyl’s geometry is revived. However, the two principle objections to Weyl’s theory do not apply to the present formulation: the Lagrangian remains first order in the curvature scalar and the non-integrability of length only applies to the null-cone structure.