General Relativity, Projective Relatedness and Geodesics
Abstract: Recalling Einstein’s assumption, in general relativity theory, that freely falling particles follow timelike geodesics it is of some interest to investigate the consequences of assuming that two space-times have the same collection of (unparametrised) geodesics and to try to achieve a kind of uniqueness result in the following sense. It will be shown that if g and g’ are metrics for a space-time manifold M whose associated Levi-Civita connections D and D’ are projectively related (that is, their collections of unparametrised geodesics are identical) then, if one of these, say g, is a vacuum metric, them so is the other and (up to units) g and g’ are the same metric. Proceeding a little further one can also investigate the consequences of weakening the vacuum condition to see what can be obtained and this will be done. Finally, Weyl showed that two projectively related connections have the same Weyl Projective tensor. The possibility of a converse to this theorem will be explored and shown to be false.