### 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 |
---|---|

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

**The Principle of Equivalence and Projective Structures in Spacetimes.** / Hall, Graham Stanley; Lonie, D. P.

Research output: Contribution to journal › Article

*Classical and Quantum Gravity*, vol. 24, no. 14, pp. 3617-3636. https://doi.org/10.1088/0264-9381/24/14/005

}

TY - JOUR

T1 - The Principle of Equivalence and Projective Structures in Spacetimes

AU - Hall, Graham Stanley

AU - Lonie, D. P.

PY - 2007/7/21

Y1 - 2007/7/21

N2 - 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.

AB - 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.

KW - general-relativity

KW - connections

KW - curvature

KW - collineations

U2 - 10.1088/0264-9381/24/14/005

DO - 10.1088/0264-9381/24/14/005

M3 - Article

VL - 24

SP - 3617

EP - 3636

JO - Classical and Quantum Gravity

JF - Classical and Quantum Gravity

SN - 0264-9381

IS - 14

ER -