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To perform numerical relativity simulations one almost universally adopts the so called "3+1" approach: spacetime is divided up into spacelike slices, each representing a "moment in time". After some wrangling, it is possible to reformulate the field equations into a well-posed initial value problem on the slices.

This implicitly assumes the slicing is possible: that it is possible to draw smooth, everywhere spacelike slices through spacetime, such that every event lies on one and only one on them. Implicitly it assumes we can find a universal time function on the spacetime. This right away stops us from simulating spacetimes with either closed timelike lines (CTCs) or timelike singularities.

My question is whether some alternative to the slicing approach exists which could in principle do this. For example, can the field equations be somehow solved self-consistently "at once" over some compact volume using Monte Carlo or other methods?

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I can't say that I have ever seen any attempts at simulating non-causal spacetimes (the closest I've seen is the simulation of fields upon such spacetimes). A few non-causal spacetimes do admit a time slicing, by the way, although by definition not all of these slices are achronal.

Just solving it like any other PDE might be an avenue worth exploring, but there is a trick to keep in mind : for an initial (partial) Cauchy surface, if this initial surface has an extension that contains closed timelike curves, it also has an extension that is causal, as shown by Krasnikov. As you might know, matter fields lack unique solutions generally on closed timelike curves, but the same is true of the metric itself.

As it is not unique, I am not quite sure what a numerical simulation might produce. Although it might be the CTCs since the development of those spacetimes will include singularities otherwise. The only method to check would probably be to just test the method, for instance on some spacetime with an initial partial Cauchy surface, ideally perhaps a vacuum one. Ori described the following spacetime with initially causal slices that develop CTCs with a vacuum metric, of the form

\begin{equation} ds^2 = -dt d\varphi + dx^2 + dy^2 + (f(x,y,\varphi) - t) d\varphi^2 \end{equation}

(on the manifold $\Bbb R^3 \times S^1$), with the condition that $f_{,xx} + f_{,yy} = 0$ for it to be flat. The spacetime will develop CTCs when $f(x,y,\varphi) - T < 0$ for all values of $\varphi$.

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  • $\begingroup$ Not to impose on your time too much, but could you show me an example of a field simulated upon a non-causal spacetime? This sounds interesting. $\endgroup$ – AGML Nov 26 '15 at 20:04
  • $\begingroup$ "As it is not unique, I am not quite sure what a numerical simulation might produce. Although it might be the CTCs since the development of those spacetimes will include singularities otherwise." I don't understand this point fully. Why couldn't the development be regular? $\endgroup$ – AGML Nov 26 '15 at 20:04

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