### Abstract

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 language | English |
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Pages (from-to) | 3617-3636 |

Number of pages | 20 |

Journal | Classical and Quantum Gravity |

Volume | 24 |

Issue number | 14 |

DOIs | |

Publication status | Published - 21 Jul 2007 |

### Keywords

- general-relativity
- connections
- curvature
- collineations

## Cite this

*Classical and Quantum Gravity*,

*24*(14), 3617-3636. https://doi.org/10.1088/0264-9381/24/14/005