1997/09/30 by L. Andersson, G. J. Galloway, R. Howard · 1 citation
Mathematics · Physics and Astronomy · #dg-ga #gr-qc #math.DG
paper · pdf · doi:10.1088/0264-9381/15/2/006
published as Class.Quant.Grav. 15 (1998) 309-322 · 19 pages, AEI preprint, latex2e with amsmath and amsthm
arxiv created 1997/09/30 · arxiv updated 2009/11/30
Let (M,g) be a time oriented Lorentzian manifold and d the Lorentzian distance on M. The function τ(q):=supp< 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 τ of a space time (M,g) is regular it has several pleasant consequences: (1) It forces (M,g) to be globally hyperbolic, (2) 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), (3) the function τ is a time function in the usual sense, in particular (4) τ is continuous, in fact locally Lipschitz and the second derivatives of τ exist almost everywhere.