# Geroch's (Splitting) Theorem & Minkowski spacetime

Reference: Topology in General Relativity (R. P. Geroch, J. Math. Phys. 8(4) April 1967)

Geroch's "Splitting" Theorem (Theorem 2) applies to spacetimes with compact geometries with boundary. Minkowski space is not a compact geometry.

Can Geroch's result nonetheless be extended to Minkowski spacetime by adding conformal infinity as boundaries, on the bounding spacelike 3-manifolds etc.?

A pedagogical answer either way would be much appreciated (as would any info on how then to handle the isometries of such a Minkowski spacetime with conformal infinity boundary)

EDIT

Statement of the theorem (quote from reference)

Let $M$ be a compact geometry whose boundary is the disjoint union of two compact space-like 3-manifolds, $S$ and $S'$. Suppose $M$ is isochronous, and has no closed timelike curve. Then $S$ and $S'$ are diffeomorphic [Wikipedia says, incorrectly, merely homeomorphic], and further $M$ is topologically $S \times[0,1]$.

The interest is in the construction and use of the non-vanishing timelike vector fields used in the proof, which relates to Theorem 1 ibid. (Rohklin) and the equivalence of such vector fields, given that "the problem of placing a Lorentz signature metric is equivalent" (Steenrod cited) to that of placing a continuous, nowhere vanishing vector field on $M$ (and nowhere tangent to the boundaries).

• Your reference to the theorem assumes the spacetime as globally hyperbolic. Does not mention compact. I think hyperbolic can be non-compact. Still, Minkowski is not hyperbolic but I think, though I could be wrong, that AdS is. Why bother with Minkowski, isn't it trivially strongly causal, which is the conclusion of Geroch for hyperbolic? – Bob Bee Apr 25 '17 at 0:01
• @BobBee The Wikipedia link was added by a moderator; I didn't intend to refer to that article (it's both misleading & in a minor but relevant way incorrect). The theorem now quoted verbatim makes the reference to compactness explicit: one should prefer primary sources over secondaries such as Wikipedia. Minkowski space is indeed Globally Hyperbolic (a statement about causality) but not "hyperbolic" (a statement about geometry; it is of course flat). The Why does not related to global hyperbolicity per se; that Minkowski space is GH is what would allow an extended result useful to me – Julian Moore Apr 25 '17 at 6:22
• Thanks. I'll have to read and assimilate from a better source than wiki (usually I only look at wiki for things I know and need some quick details), or I know nothing about. Though I know of Geroch's work and some of it, I did not know this theorem. I do know about Cauchy surfaces didn't know the relationship or def. of global hyperbolic – Bob Bee Apr 26 '17 at 0:26
• The statement of the theorem should say "...has no closed timelike curve." – MBN Apr 26 '17 at 12:21