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.
In 1972, Geroch, Kronheimer, and Penrose introduced what is now called the causal boundary of a spacetime. This boundary is constructed out of Terminal Indecomposable Past sets (TIPs) and their future analogues (TIFs), which are the pasts and futures of inextendible causal curves. The causal boundary is a key tool to understand the global structure of a spacetime. In this paper, we show that in a spacetime with compact Cauchy surfaces, there is always at least one minimal TIP and one minimal TIF, minimal meaning that it does not contain another TIP (resp. TIF) as a proper subset. We then study the implications of the minimal TIP and TIF meeting each other. This condition generalizes some of the "no observer horizon" conditions that have been used in the literature to obtain partial solutions of the Bartnik splitting conjecture.
Topology change is considered to be a necessary feature of quantum gravity by some authors, and impossible by others. One of the main arguments against it is that spacetimes with changing spatial topology have bad causal properties. Borde and Sorkin proposed a way to avoid this dilemma by considering topology changing spacetimes constructed from Morse functions, where the metric is allowed to vanish at isolated points. They conjectured that these Morse spacetimes are causally continuous (hence quite well behaved), as long as the index of the Morse points is different from 1 and n − 1. In this paper, we prove a special case of this conjecture. We also argue, heuristically, that the original conjecture is actually false, and formulate a refined version of it.
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