The Principle of Equivalence and Projective Structures in Spacetimes

Graham Stanley Hall, D. P. Lonie

Research output: Contribution to journalArticle

35 Citations (Scopus)

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 languageEnglish
Pages (from-to)3617-3636
Number of pages20
JournalClassical and Quantum Gravity
Volume24
Issue number14
DOIs
Publication statusPublished - 21 Jul 2007

Keywords

  • general-relativity
  • connections
  • curvature
  • collineations

Cite this

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

In: Classical and Quantum Gravity, Vol. 24, No. 14, 21.07.2007, p. 3617-3636.

Research output: Contribution to journalArticle

Hall, Graham Stanley ; Lonie, D. P. / The Principle of Equivalence and Projective Structures in Spacetimes. In: Classical and Quantum Gravity. 2007 ; Vol. 24, No. 14. pp. 3617-3636.
@article{bd6d7f864cb5405aa0a23ad50b2ee9b8,
title = "The Principle of Equivalence and Projective Structures in Spacetimes",
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.",
keywords = "general-relativity, connections, curvature, collineations",
author = "Hall, {Graham Stanley} and Lonie, {D. P.}",
year = "2007",
month = "7",
day = "21",
doi = "10.1088/0264-9381/24/14/005",
language = "English",
volume = "24",
pages = "3617--3636",
journal = "Classical and Quantum Gravity",
issn = "0264-9381",
publisher = "IOP Publishing Ltd.",
number = "14",

}

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 -