# Given a null surface, calculate the manifold it resides in

This problem is related to General Relativity and specifically Black Holes.

The manifold is a 4-dimensional space-time with a Minkowski inner product (i.e. if $||v|| = 0$, $v$ is not necessarily $0$), A null surface is an embedded 3 dimensional surface where the normal vector has length 0 everywhere on the surface (I.e. a light-like closed surface or horizon). The null surface is determined by the geometry of the space time, specifically it is a consequence of the curvature of the manifold.

My question is, given a horizon or null surface shape , is it possible to calculate the metric tensor that describes a space-time where such a null surface will reside in. Given some boundary conditions (like, the metric approaches the minkowski metric at large distances from the null surface)

Specifically, I was wondering if, given a black hole horizon (some funky shape - like a beveled cube) it was possible to calculate the shape of the collapsing mass that would gives such a horizon.

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When you say "a 4-dimensional space-time with a Minkowski inner product", I presume you mean a metric with $(1,3)$ signature, i.e. a Minkowski inner product on the tangent spaces, and not literally a Minkowski inner product on the space-time? – joriki Apr 11 '11 at 5:47
@joriki this is correct. I'll edit my question to be more clear. – crasic Apr 11 '11 at 16:34