Global Hyperbolicity for Spacetimes with Continuous Metrics

Abstract

We show that the definition of global hyperbolicity in terms of the compactness of the causal diamonds and non-total imprisonment can be extended to spacetimes with continuous metrics, while retaining all of the equivalences to other notions of global hyperbolicity. In fact, global hyperbolicity is equivalent to the compactness of the space of causal curves and to the existence of a Cauchy hypersurface. Furthermore, global hyperbolicity implies causal simplicity, stable causality and the existence of maximal curves connecting any two causally related points.

Green operators in low regularity spacetimes and quantum field theory

“…For C 1,1 spacetimes we will adopt the first definition. However for non-totally imprisoned [41] C 0 spacetimes (and hence in particular for globally hyperbolic C 1,1 spacetimes) these four definitions remain equivalent [49]. See also [43] Theorem 2.45 for a more general notion formulated in terms of closed cone structures.…”

The Hawking–Penrose Singularity Theorem for C 1,1-Lorentzian Metrics

“…Remark 1.1 (Regarding the regularity of causal curves). Note that in line with, e.g., [8,14,39,16] we only require causal and timelike curves to be Lipschitz, whereas classical accounts use smooth or piecewise smooth or piecewise C 1 curves instead (see, e.g., [21,35]). Another convention that is sometimes used is requiring timelike curves to be piecewise C 1 while causal curves only have to be Lipschitz (e.g., [33]).…”

“…These definitions are equivalent to the existence of a Cauchy hypersurface (i.e., a set that is met exactly once by every inextendible causal curve), see [39,Theorem 5.7 and 5.9]. From this it is clear that if g is globally hyperbolic, then any g ′ ≺ g (remember that this was defined as g ′ (X, X) ≤ 0 =⇒ g(X, X) < 0) must be globally hyperbolic too.…”

“…We first show that there always exists a timelike maximizing geodesic from p to q for p ≪ q. First fix a smooth globally hyperbolic metricĝ with g ≺ĝ (such a metric exists by [39,Theorem 4.5]). Further, note that there exists a sequence of metricsĝ n with g ≺≺ĝ n ≺ĝ (this follows essentially from [ Since theĝ n have narrower lightcones thanĝ, they must all be globally hyperbolic as well and Jĝ n (p, q) ⊆ Jĝ(p, q) ⋐ M. Let γ n : [0, 1] → M be aĝ n -maximizingĝ n -timelikê g n -geodesic from p to q (these exist becauseĝ n is globally hyperbolic and p ≪ĝ n q).…”

“…First note that by [39,Theorem 4.5] there exists a smooth metricĝ with g ≺ĝ and g globally hyperbolic. Given any suchĝ there always exists a C 0 -fine (and hence C 1 -fine) neighborhood U ⊆ Pseud 1 (M) around g such that all g ′ ∈ U are Lorentzian metrics and satisfy g ′ ≺ĝ (this essentially follows from [39,Lemma 1.4]). Thus all g ′ ∈ U are globally hyperbolic and for any K ⊆ M one has 4 Further, by Proposition 2.10 and skipping over some of the K i if necessary, we may assume that all that all g-causal g-geodesics γ starting in…”