Is Spacetime Hole-Free

Here, we examine hole-freeness – a condition sometimes imposed to rule out seemingly artificial spacetimes. We show that under existing definitions (and contrary to claims made in the literature) there exist inextendible, globally hyperbolic spacetimes which fail to be hole-free. We then propose an updated formulation of the condition which en-ables us to show the intended result. We conclude with a few general remarks on the strength of the definition and then formulate a precise question which may be interpreted as: Are all physically reasonable spacetimes hole-free? Hole-freeness is a condition sometimes imposed to rule out seemingly ar-tificial spacetimes. The idea, roughly, is that spacetime should be such that the Cauchy development of any spacelike surface is “as large as it can be”. The condition is of interest because, without it, it seems that a failure of determinism in general relativity threatens even before one considers more interesting questions of global structure such as the cosmic censorship hy-pothesis.1 The first precise formulation of hole-freeness was due to Geroch (1977):2 Definition. A spacetime (M, gab) is said to be hole-free if, for any spacelike surface S ⊂ M, and for any metric preserving embedding θ: D(S) → M ′ ∗I wish to thank John Earman and Robert Geroch for helpful discussions on this topic