The Principle of Equivalence and Projective Structures in Spacetimes

Graham Stanley Hall, D. P. Lonie

Research output: Contribution to journalArticlepeer-review

40 Citations (Scopus)


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.

Original languageEnglish
Pages (from-to)3617-3636
Number of pages20
JournalClassical and Quantum Gravity
Issue number14
Publication statusPublished - 21 Jul 2007


  • general-relativity
  • connections
  • curvature
  • collineations


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