Spacetime extensions II

Research output: Contribution to journalArticle

2 Citations (Scopus)

Abstract

The global extendibility of smooth causal geodesically incomplete spacetimes is investigated. Denote by γ one of the incomplete non-extendible causal geodesics of a causal geodesically incomplete spacetime (M, gab). First, it is shown that it is always possible to select a synchronized family of causal geodesics and an open neighbourhood μof a final segment of γ in M such that comprises members of Γ, and suitable local coordinates can be defined everywhere on provided that γ does not terminate either on a tidal force tensor singularity or on a topological singularity. It is also shown that if, in addition, the spacetime (M, g ab) is globally hyperbolic, and the components of the curvature tensor, and its covariant derivatives up to order k - 1 are bounded on , and also the line integrals of the components of the kth-order covariant derivatives are finite along the members of Γ - where all the components are meant to be registered with respect to a synchronized frame field on - then there exists a Ck - extension so that for each , which is inextendible in (M, gab), the image, , is extendible in . Finally, it is also proved that whenever γ does terminate on a topological singularity (M, g ab) cannot be generic.

Original languageEnglish
Article number155007
JournalClassical and Quantum Gravity
Volume27
Issue number15
DOIs
Publication statusPublished - 2010

Fingerprint

tensors
curvature

ASJC Scopus subject areas

  • Physics and Astronomy (miscellaneous)

Cite this

Spacetime extensions II. / Rácz, I.

In: Classical and Quantum Gravity, Vol. 27, No. 15, 155007, 2010.

Research output: Contribution to journalArticle

@article{937d19c4c389447fa3092a6c66c0820b,
title = "Spacetime extensions II",
abstract = "The global extendibility of smooth causal geodesically incomplete spacetimes is investigated. Denote by γ one of the incomplete non-extendible causal geodesics of a causal geodesically incomplete spacetime (M, gab). First, it is shown that it is always possible to select a synchronized family of causal geodesics and an open neighbourhood μof a final segment of γ in M such that comprises members of Γ, and suitable local coordinates can be defined everywhere on provided that γ does not terminate either on a tidal force tensor singularity or on a topological singularity. It is also shown that if, in addition, the spacetime (M, g ab) is globally hyperbolic, and the components of the curvature tensor, and its covariant derivatives up to order k - 1 are bounded on , and also the line integrals of the components of the kth-order covariant derivatives are finite along the members of Γ - where all the components are meant to be registered with respect to a synchronized frame field on - then there exists a Ck - extension so that for each , which is inextendible in (M, gab), the image, , is extendible in . Finally, it is also proved that whenever γ does terminate on a topological singularity (M, g ab) cannot be generic.",
author = "I. R{\'a}cz",
year = "2010",
doi = "10.1088/0264-9381/27/15/155007",
language = "English",
volume = "27",
journal = "Classical and Quantum Gravity",
issn = "0264-9381",
publisher = "IOP Publishing Ltd.",
number = "15",

}

TY - JOUR

T1 - Spacetime extensions II

AU - Rácz, I.

PY - 2010

Y1 - 2010

N2 - The global extendibility of smooth causal geodesically incomplete spacetimes is investigated. Denote by γ one of the incomplete non-extendible causal geodesics of a causal geodesically incomplete spacetime (M, gab). First, it is shown that it is always possible to select a synchronized family of causal geodesics and an open neighbourhood μof a final segment of γ in M such that comprises members of Γ, and suitable local coordinates can be defined everywhere on provided that γ does not terminate either on a tidal force tensor singularity or on a topological singularity. It is also shown that if, in addition, the spacetime (M, g ab) is globally hyperbolic, and the components of the curvature tensor, and its covariant derivatives up to order k - 1 are bounded on , and also the line integrals of the components of the kth-order covariant derivatives are finite along the members of Γ - where all the components are meant to be registered with respect to a synchronized frame field on - then there exists a Ck - extension so that for each , which is inextendible in (M, gab), the image, , is extendible in . Finally, it is also proved that whenever γ does terminate on a topological singularity (M, g ab) cannot be generic.

AB - The global extendibility of smooth causal geodesically incomplete spacetimes is investigated. Denote by γ one of the incomplete non-extendible causal geodesics of a causal geodesically incomplete spacetime (M, gab). First, it is shown that it is always possible to select a synchronized family of causal geodesics and an open neighbourhood μof a final segment of γ in M such that comprises members of Γ, and suitable local coordinates can be defined everywhere on provided that γ does not terminate either on a tidal force tensor singularity or on a topological singularity. It is also shown that if, in addition, the spacetime (M, g ab) is globally hyperbolic, and the components of the curvature tensor, and its covariant derivatives up to order k - 1 are bounded on , and also the line integrals of the components of the kth-order covariant derivatives are finite along the members of Γ - where all the components are meant to be registered with respect to a synchronized frame field on - then there exists a Ck - extension so that for each , which is inextendible in (M, gab), the image, , is extendible in . Finally, it is also proved that whenever γ does terminate on a topological singularity (M, g ab) cannot be generic.

UR - http://www.scopus.com/inward/record.url?scp=77953793616&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=77953793616&partnerID=8YFLogxK

U2 - 10.1088/0264-9381/27/15/155007

DO - 10.1088/0264-9381/27/15/155007

M3 - Article

VL - 27

JO - Classical and Quantum Gravity

JF - Classical and Quantum Gravity

SN - 0264-9381

IS - 15

M1 - 155007

ER -