# [Physics] Space-time Topologies

differential-geometrygeneral-relativityparticle-physicstopology

When it comes to questions of existence of bounds for PDE's and such, one must often make some assumptions regarding the topology of the space-time to use well known theorems.

My question is two-pronged:

i) I've often read, on Wikipedia (http://en.wikipedia.org/wiki/Spacetime) for example, that space-time is paracompact. I am aware of the mathematical definition and this seems counter-intuitive to me. Since the form of the stress-energy tensor entering Einstein's equations need only satisfy the conservation of energy $\nabla_{\mu} T^{\mu \nu} = 0$ condition from the Bianchi identities, how does one show this result? Can someone give me a reference for a proof?

ii) In (electro)-vacuum, what other topologies are permitted globally? I have seen some papers finding solutions that resemble black holes (in the sense they have singularities, event horizons…) but are topologically not 2-spheres at their cross section (seemingly in violation of Hawking's theorem, see e.g, http://arxiv.org/abs/hep-th/9808032, but perhaps the fact the space-time is asymptotically anti de-Sitter is why there is no violation). What space-times are compact? The spheres $S^{n}$ are all compact, but presumably even the Schwarzschild space-time is not globally? I think one can only have a compact space-time if its Euler characteristic $\chi = 0$, can this be translated into a demand on $T^{\mu \nu}$?

I am particularly interested in compact space-times, for one can then apply the Yamabe problem.

Thanks!

On point ii): if by "compact" you mean "compact without boundary" (like $S^n$), compact space-times indeed necessarily have vanishing Euler characteristic - conversely, any compact manifold with vanishing Euler characteristic admits a time oriented Lorentzian metric. However, such space-times are not physically interesting because they necessarily have closed timelike curves. The argument is simple: since any space-time $(\mathscr{M},g)$ may be covered by the chronological futures of all its points (which are open sets), using compactness one can pass to a finite subcover, say $\mathscr{M}=I^+(p_1)\cup\cdots\cup I^+(p_n)$. Therefore, $p_1$ must belong to $I^+(p_{j_1})$ for some $j_1=1,\ldots,n$, $p_{j_1}$ must belong to $I^+(p_{j_2})$ for some $j_2=1,\ldots,n$, and so on. Since we are dealing with a finite number of points, eventually one must have $p_{j_k}=p_1$ for some $k$ between $1$ and $n$, thus producing a closed timelike curve. Since such space-times are not globally hyperbolic, they are also unsuitable for the analysis of hyperbolic (i.e. wave-like) PDE's. Noncompact (Hausdorff, connected and paracompact, as in point (i)) manifolds, on the other hand, always admit a time oriented Lorentzian metric.