# Numerical relativity in causally pathological spacetimes

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?

$$ds^2 = -dt d\varphi + dx^2 + dy^2 + (f(x,y,\varphi) - t) d\varphi^2$$
(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 $\varphi$.