Causality theory of spacetimes with continuous Lorentzian metrics revisited

Abstract

We consider the usual causal structure (I +, J +) on a spacetime, and a number of alternatives based on Minguzzi’s D + and Sorkin and Woolgar’s K +, in the case where the spacetime metric is continuous, but not necessarily smooth. We compare the different causal structures based on three key properties, namely the validity of the push-up lemma, the openness of chronological futures, and the existence of limit causal curves. Recall that if the spacetime metric is smooth, (I +, J +) satisfies all three properties, but that in the continuous case, the push-up lemma fails. Among the proposed alternative causal structures, there is one that satisfies push-up and open futures, and one that has open futures and limit curves. Furthermore, we show that spacetimes with continuous metrics do not, in general, admit a causal structure satisfying all three properties at once.

Remarks on the cosmological constant appearing as an initial condition for Milne-like spacetimes

“…This is a growing field with many tantalizing problems to solve. For low regularity causal theory, generalizations, and various results, see [7,13,20,27,29,32,35]. For low regularity spacetime inextendibility results, see [8,14,16,19,[37][38][39].…”

Conformal Isometries and Curvature Collineations of an Impulsive Plane Wave: a distributional approach

“…The linear mapping (7) is continuous if and only if for each U with support in the domain (K, φ) of an admissible chart, there exists a constant C(K) and a non negative integer m(K) such that…”

“…Indeed, for the computation of the curvature of a spacetime of low regularity, one must resort to mathematical tools [1][2][3][4] not within the classical smooth Lorentzian geometry. On the other hand, issues of low regularity concerning to geodesics, singularity theorems and causality theory have been the object of intense investigation (see [5][6][7] and references therein).…”

Remarks on the cosmological constant appearing as an initial condition for Milne-like spacetimes

“…This is a growing field with many tantalizing problems to solve. For low regularity causal theory, generalizations, and various results, see [7,13,20,27,28,31,34]. For low regularity spacetime inextendibility results, see [8,14,16,19,[36][37][38].…”