Global hyperbolicity and completeness

Abstract

We prove global hyperbolicity of spacetimes under generic regularity conditions on the metric.We then show that these spacetimes are timelike and null geodesically complete if the gradient of the lapse and the extrinsic curvature K are integrable. This last condition is required only for the tracefree part of K if the universe is expanding.

Letter: Global Hyperbolicity of Sliced Spaces

“…(3.8), using (3.6), becomes, This result provides a partial converse to the completeness theorem given in [2] (Thm.…”

“…Consider a Cauchy sequence of numbers (t n ) which converges to T and the corresponding points (c n , t n ) of the curve C, where c n (with components C i (t n )) are points of M. It follows that the sequence c n cannot converge to the point c(T ). But this is impossible, since the estimates of [2], p.…”

On Curvature Decay in Expanding Cosmological Models

“…Furthermore, they stated that causal geodesic completeness should hold, though they did not prove it. However, this was shown for a larger class of spacetimes in [5], a paper which extends the results of [4] to the non-polarized case, using the results of [6]. Finally, they showed that the Cauchy surfaces should undergo a Cheeger-Gromov type collapse.…”

Singular Isotropic Cosmologies and Bel-Robinson Energy

“…On the other hand, one can formulate equally plausible and generic geometric criteria for the long time existence of geodesically complete, generic spacetimes in general relativity and also other theories of gravity, cf. [2]. Such criteria assume a globally hyperbolic spacetime in the so-called sliced form (cf.…”