### 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, g_{ab}). 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 C^{k -} extension so that for each , which is inextendible in (M, g_{ab}), 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 language | English |
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Article number | 155007 |

Journal | Classical and Quantum Gravity |

Volume | 27 |

Issue number | 15 |

DOIs | |

Publication status | Published - 2010 |

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### ASJC Scopus subject areas

- Physics and Astronomy (miscellaneous)

### Cite this

*Classical and Quantum Gravity*,

*27*(15), [155007]. https://doi.org/10.1088/0264-9381/27/15/155007

**Spacetime extensions II.** / Rácz, I.

Research output: Contribution to journal › Article

*Classical and Quantum Gravity*, vol. 27, no. 15, 155007. https://doi.org/10.1088/0264-9381/27/15/155007

}

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.

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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 -