Is General Relativity a v∕c → 0 Limit of a Finsler Geometry?

Abstract

Gravity is understood as a geometrization of spacetime. But spacetime is also the manifold of the boundary values of the spinless point particle in a variational approach. The manifold of the boundary variables for any mechanical system, instead of being a Riemannian space it is a Finsler metric space such that the variational formalism can always be interpreted as a geodesic problem on this manifold. This manifold is just the flat Minkowski spacetime for the free relativistic point particle. Any interaction modifies its flat Finsler metric. In the spirit of unification of all forces, gravity cannot produce, in principle, a different and simpler geometrization than any other interaction. This implies that the basic assumption that what gravity produces is a Riemannian metric instead of a Finslerian one is a strong restriction so that general relativity can be considered as a low velocity limit of a more general gravitational theory.