Globally hyperbolic spacetimes can be defined as ‘causal’ instead of ‘strongly causal’

Abstract

The classical definition of global hyperbolicity for a spacetime (M, g) comprises two conditions: (A) compactness of the diamonds J + (p) ∩ J − (q), and (B) strong causality. Here we show that condition (B) can be replaced just by causality. In fact, we show first that the classical definition of causal simplicity (which impose to be distinguishing, apart from the closedness of J + (p), J − (q)) can be weakened in causal instead of distinguishing. So, the full consistency of the causal ladder (recently proved by the authors in a definitive way) yields directly the result.2000 MSC: Primary 53C50, Secondary 53C80, 83C75.

Generalised hyperbolicity in spacetimes with string-like singularities

“…In examples such as the Kerr and Reissner-Nordström spacetimes there is a loss of global hyperbolicity instead of a loss of regularity. Mathematically a spacetime region N is said to globally hyperbolic if the causality condition is satisfied and for any two points p, q ∈ N the causal diamond J + (p) ∩ J − (q) is compact and contained in N [19]. One demands this condition because it is sufficient to guarantee the global well-posedness of the wave equation and other physical fields [20].…”

The causal hierarchy of spacetimes

“…Following [8,Sect. 3], the next result yields directly that a globally hyperbolic spacetime (according to our Definition 3.70) is causally simple.…”

“…(1) As stressed in [8], the full consistency of the causal ladder yields that any globally hyperbolic spacetime is not only causally simple but also strongly causal. This last hypothesis is usually imposed in the definition of global hyperbolicity, instead of causality, but becomes somewhat redundant.…”

Recent progress on the notion of global hyperbolicity

“…Item (2) is a typical definition of global hyperbolicity, which is used in standard books such as [4,29,37,40,48]. Item (1) is an even simpler definition in [8]. In fact, (1) ⇒ (2) because the assumptions (1) imply that M is causally simple (as J ± (p) is closed for all p) and, therefore, strongly causal, as required.…”

“…The equivalences are based in the central article by Geroch [23], and include a recent conceptual simplification in [8] (see Theorem 2.1). Section 3 is devoted to the implications of the folk problems of smoothability on the differentiable and metric structure of globally hyperbolic spacetimes (Theorem 3.1).…”