A note on the Gannon–Lee theorem

Abstract

We prove a Gannon–Lee theorem for non-globally hyperbolic Lorentzian metrics of regularity $$C^1$$ C 1 , the most general regularity class currently available in the context of the classical singularity theorems. Along the way, we also prove that any maximizing causal curve in a $$C^1$$ C 1 -spacetime is a geodesic and hence of $$C^2$$ C 2 -regularity.

The codimension 2 null cut locus with applications to spacetime topology

“…The results we present on spacetime topology, in both the horizon and no horizon cases, use a proof strategy similar to a number results in the literature; cf. [9,13,6,7,20,4,16]. One of our main results, first considered by Costa e Silva [6] in the no horizon setting (see also [7]), is to allow S (as in the beginning of Section 2.2) to have nontrivial fundamental group.…”

“…The following theorem generalizes a well known result of Gannon ([9, Corollary 1.20]; see also Lee [13,Theorem 5]). Indeed, it is an adaptation of [6, Theorem 2.1] (see also [7,20]). Our modification of the covering space argument in [6] (see the remark after Proposition 15) has required a strengthening of the assumption on i 2 * ; see Proposition 16.…”

“…Our results fall into two classes, depending on whether or not one assumes the presence of horizons. The results in which we do not assume the presence of horizons (as in [9,13,6,7,20]) may be interpreted as singularity results: Nontrivial topology (in a suitable sense) leads to future incompleteness. The results in which we do assume the presence of horizons (as in [4,16]) pertain to the notion of topological censorship, which has to do with the idea that, outside of all black holes, one expects the topology to be simple (in suitable sense) at the fundamental group level.…”

“…An essential step in the proof is to construct a certain covering spacetime. An issue arises in the covering construction in [6] (see also [20]) which we address here (see section 3.1).…”

The singularity theorems of General Relativity and their low regularity extensions

“…In this section we wish to provide an overview of the extension of the classical singularity theorems to metrics of low regularity that have emerged over the last couple of years. Indeed the three key theorems discussed above have first been generalised to Lorentzian metrics of regularity C 1,1 [KSSV15, KSV15, GGKS18] and then in a further effort to C 1 -metrics [Gra20, KOSS22] with a Gannon-Lee theorem proved in [SS21]. We shall review the main mathematical advances that were developed to arrive at these results, mainly concentrating on the analytic side of the arguments, but we shall also comment on the recent advancements of the causality parts of the theorems [Min19a].…”