The Principle of Equivalence and Projective Structures in Spacetimes

This paper discusses the extent to which one can determine the spacetime metric from a knowledge of a certain subset of the ( unparametrized) geodesics of its Levi-Civita connection, that is, from the experimental evidence of the equivalence principle. It is shown that, if the spacetime concerned is known to be vacuum, then the Levi-Civita connection is uniquely determined and its associated metric is, generically, uniquely determined up to a choice of units of measurement, by the specification of these geodesics. A special case arises here concerning pp-waves and is dealt with. It is further demonstrated that if two spacetimes share a similar subset of unparametrized geodesics they share all geodesics ( that is they are projectively related). Furthermore, if one of these spacetimes 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.