The principle of equivalence and projective structure in space-times

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.

23 pages, submitted to Classical and Quantum Gravity

References in corpus (2)

• Holonomy groups and spacetimes
• On the compatibility of Lorentz-metrics with linear connections on 4-dimensional manifolds

Cited by in corpus (13)

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• Geodesically equivalent metrics in general relativity
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• Geometric inequivalence of metric and Palatini formulations of General Relativity
• Projective Compactifications and Einstein metrics
• Proof of projective Lichnerowicz conjecture for pseudo-Riemannian metrics with degree of mobility greater than two
• Projective vs metric structures
• Projective Structure and Holonomy in 4-dimensional Lorentz Manifolds
• Holonomy and Projective Equivalence in 4-Dimensional Lorentz Manifolds
• General structure of Thomas$-$Whitehead gravity
• The degree of mobility of Einstein metrics
• Covariant and Manifestly Projective Invariant Formulation of Thomas-Whitehead Gravity
• Splitting and gluing constructions for geodesically equivalent pseudo-Riemannian metrics