### Abstract

Let (M, g) be a time-oriented Lorentzian manifold and d the Lorentzian distance on M. The function τ(q) := sup
_{p<q} d(p, q) is the cosmological time function of M, where as usual p < q means that p is in the causal past of q. This function is called regular iff τ(q) < ∞ for all q and also τ → 0 along every past inextendible causal curve. If the cosmological time function r of a spacetime (M, g) is regular it has several pleasant consequences: (i) it forces (M, g) to be globally hyperbolic; (ii) every point of (M, g) can be connected to the initial singularity by a rest curve (i.e. a timelike geodesic ray that maximizes the distance to the singularity); (iii) the function τ is a time function in the usual sense; in particular, (iv) τ is continuous, in fact, locally Lipschitz and the second derivatives of τ exist almost everywhere.

Original language | English (US) |
---|---|

Pages (from-to) | 309-322 |

Number of pages | 14 |

Journal | Classical and Quantum Gravity |

Volume | 15 |

Issue number | 2 |

DOIs | |

State | Published - Feb 1998 |

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### ASJC Scopus subject areas

- Physics and Astronomy(all)

### Cite this

*Classical and Quantum Gravity*,

*15*(2), 309-322. https://doi.org/10.1088/0264-9381/15/2/006

**The cosmological time function.** / Andersson, Lars; Galloway, Gregory J; Howard, Ralph.

Research output: Contribution to journal › Article

*Classical and Quantum Gravity*, vol. 15, no. 2, pp. 309-322. https://doi.org/10.1088/0264-9381/15/2/006

}

TY - JOUR

T1 - The cosmological time function

AU - Andersson, Lars

AU - Galloway, Gregory J

AU - Howard, Ralph

PY - 1998/2

Y1 - 1998/2

N2 - Let (M, g) be a time-oriented Lorentzian manifold and d the Lorentzian distance on M. The function τ(q) := sup p d(p, q) is the cosmological time function of M, where as usual p < q means that p is in the causal past of q. This function is called regular iff τ(q) < ∞ for all q and also τ → 0 along every past inextendible causal curve. If the cosmological time function r of a spacetime (M, g) is regular it has several pleasant consequences: (i) it forces (M, g) to be globally hyperbolic; (ii) every point of (M, g) can be connected to the initial singularity by a rest curve (i.e. a timelike geodesic ray that maximizes the distance to the singularity); (iii) the function τ is a time function in the usual sense; in particular, (iv) τ is continuous, in fact, locally Lipschitz and the second derivatives of τ exist almost everywhere.

AB - Let (M, g) be a time-oriented Lorentzian manifold and d the Lorentzian distance on M. The function τ(q) := sup p d(p, q) is the cosmological time function of M, where as usual p < q means that p is in the causal past of q. This function is called regular iff τ(q) < ∞ for all q and also τ → 0 along every past inextendible causal curve. If the cosmological time function r of a spacetime (M, g) is regular it has several pleasant consequences: (i) it forces (M, g) to be globally hyperbolic; (ii) every point of (M, g) can be connected to the initial singularity by a rest curve (i.e. a timelike geodesic ray that maximizes the distance to the singularity); (iii) the function τ is a time function in the usual sense; in particular, (iv) τ is continuous, in fact, locally Lipschitz and the second derivatives of τ exist almost everywhere.

UR - http://www.scopus.com/inward/record.url?scp=0032340442&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0032340442&partnerID=8YFLogxK

U2 - 10.1088/0264-9381/15/2/006

DO - 10.1088/0264-9381/15/2/006

M3 - Article

AN - SCOPUS:0032340442

VL - 15

SP - 309

EP - 322

JO - Classical and Quantum Gravity

JF - Classical and Quantum Gravity

SN - 0264-9381

IS - 2

ER -