The principle of equivalence and projective structure in space-times

TL;DR

This paper explores how the equivalence principle and geodesic data can determine the space-time metric in general relativity, especially in vacuum conditions, establishing uniqueness of the Levi-Civita connection and metric from geodesic information.

Contribution

It demonstrates that in vacuum space-times, the Levi-Civita connection and metric are uniquely determined by unparametrised geodesics, and extends this to compare different space-times sharing the same geodesics.

Findings

Unique determination of Levi-Civita connection in vacuum space-times from geodesics

Equivalence of connections for space-times sharing the same geodesics

Recovery of the vacuum metric from geodesic data

Abstract

This paper discusses the extent to which one can determine the space-time metric from a knowledge of a certain subset of the (unparametrised) geodesics of its Levi-Civita connection, that is, from the experimental evidence of the equivalence principle. It is shown that, if the space-time concerned is known to be vacuum, then the Levi-Civita connection is uniquely determined and its associated metric is uniquely determined up to a choice of units of measurement, by the specification of these geodesics. It is further demonstrated that if two space-times share the same unparametrised geodesics and only one is assumed vacuum then their Levi-Civita connections are again equal (and so the other metric is also a vacuum metric) and the first result above is recovered.