The Relation between Physical and Gravitational Geometry

TL;DR

This paper investigates the relationship between physical and gravitational geometries, proposing that if the physical geometry is Finslerian, it must reduce to a Riemannian form related by a generalized conformal transformation.

Contribution

It demonstrates that under the weak equivalence principle and causality, Finslerian physical geometries must simplify to Riemannian geometries related through a generalized conformal transformation.

Findings

Finslerian geometries reduce to Riemannian geometries under physical constraints

Physical metric relates to gravitational metric via generalized conformal transformation

Supports the universality of Riemannian geometry in gravitational theories

Abstract

The appearance of two geometries in one and the same gravitational theory is familiar. Usually, as in the Brans-Dicke theory or in string theory, these are conformally related Riemannian geometries. Is this the most general relation between the two geometries allowed by physics ? We study this question by supposing that the physical geometry on which matter dynamics take place could be Finslerian rather than just Riemannian. An appeal to the weak equivalence principle and causality then leads us the conclusion that the Finsler geometry has to reduce to a Riemann geometry whose metric - the physical metric - is related to the gravitational metric by a generalization of the conformal transformation.