### Abstract

Original language | English |
---|---|

Article number | 125009 |

Number of pages | 10 |

Journal | Classical and Quantum Gravity |

Volume | 26 |

Issue number | 12 |

Early online date | 27 May 2009 |

DOIs | |

Publication status | Published - 21 Jun 2009 |

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### Cite this

*Classical and Quantum Gravity*,

*26*(12), [125009]. https://doi.org/10.1088/0264-9381/26/12/125009

**Projective equivalence of Einstein spaces in general relativity.** / Hall, G S; Lonie, D P.

Research output: Contribution to journal › Article

*Classical and Quantum Gravity*, vol. 26, no. 12, 125009. https://doi.org/10.1088/0264-9381/26/12/125009

}

TY - JOUR

T1 - Projective equivalence of Einstein spaces in general relativity

AU - Hall, G S

AU - Lonie, D P

PY - 2009/6/21

Y1 - 2009/6/21

N2 - There has been some recent interest in the relation between two spacetimes which have the same geodesic paths, that is, spacetimes which are projectively equivalent (sometimes called geodesically equivalent). This paper presents a short and accessible proof of the theorem that if two spacetimes have the same geodesic paths and one of them is an Einstein space then (either each is of constant curvature or) their Lévi-Civitá connections are identical. It also clarifies the relationship between their associated metrics. The results are extended to include the signatures (+ + + +) and (- - + +), and some examples and discussion are given in the case of dimension n > 4. Some remarks are also made which show how these results may be useful in the study of projective symmetry.

AB - There has been some recent interest in the relation between two spacetimes which have the same geodesic paths, that is, spacetimes which are projectively equivalent (sometimes called geodesically equivalent). This paper presents a short and accessible proof of the theorem that if two spacetimes have the same geodesic paths and one of them is an Einstein space then (either each is of constant curvature or) their Lévi-Civitá connections are identical. It also clarifies the relationship between their associated metrics. The results are extended to include the signatures (+ + + +) and (- - + +), and some examples and discussion are given in the case of dimension n > 4. Some remarks are also made which show how these results may be useful in the study of projective symmetry.

U2 - 10.1088/0264-9381/26/12/125009

DO - 10.1088/0264-9381/26/12/125009

M3 - Article

VL - 26

JO - Classical and Quantum Gravity

JF - Classical and Quantum Gravity

SN - 0264-9381

IS - 12

M1 - 125009

ER -