A tensor theory of gravitation in a curved metric on a flat background

Abstract

A theory of gravity is proposed using a tensor potential for the field on a flat metric. This potential cannot be isolated by local observations, but some details can be deduced from measurement at a distance. The requirement that the field equations for the tensor potential shall be deducible from an action integral, that the action and field equations are gauge invariant, and, conversely, that the Lagrangian in the action integral can be integrated from the field equations leads to Einstein's field equations. The requirement that the field energy-momentum tensor exists leads to a constraint on the tensor potential. If the constraint is a differential gauge condition, then it can only be the Hilbert condition giving a unique background tensor, metric tensor and tensor potential. For a continuous field inside a solid sphere the metric must be homogeneous in the spatial coordinates, and the associated field energy-momentum tensor has properties consistent with Newtonian dynamics.