Singularity Theorems

These theorems prove the existence of singularities in the spacetime in a very general way. The assumptions about the spacetime and the matter are not so restrictive.

Here we will explain in more detail the proof of the theorem 1 of [3, p 263,] . If we take the time to do so it is because the mechanism of that proof is useful for the Topological Censorship .

First we have to give some definitions about the hypotheses of that theorem which say:

A spacetime cannot be null geodesically complete if:

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  for all null vector (see )  

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there is a non-compact Cauchy surface  H in M

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there is a closed trapped surface  in M

When we assume that the spacetime has a Cauchy surface  we assume that it is globally hyperbolic  and causally simple .

It is equivalent to say that and are closed in U.

In order to understand the definition of causally simple one need to define the notation used (Hawking and Ellis [3] notation).

Now that thes definitions are set we can begin to see the proof in more detail.

The idea of this kind of proof is to study the boundary  of the causal future   of the trapped surface  (see section page ) and find a contradiction with the hypothesis! It is a rather general method of point set topology. A similar method is applied to prove the second lemma of the Topological Censorship .

As we assume that the spacetime is causally simple , the boundary of is . In other words the boundary of the causal future of the trapped surface is generated by null geodesics with end points on the trapped surface and perpendicular to it.

From the hypothesis and the existence of the trapped surface (which mean that on ) we know that there will be conjugate points because of the Raychaudhuri equation (see section page ).Because of the conjugate points geodesics need to stop. But if we look to the boundary of the causal future there is an end point!

Take the continuous map (where Q is the ensemble ) which associate to each point p on with an affine distance along one of the directed null geodesics from . As is compact there will be a limit on the value of   , so this gives an upper limit on .

Form this limit we have, is a compact region. Then the boundary of the causal future will be compact as a closed subset of a compact set.

There is not yet a contradiction but as the Cauchy surface  is not compact there will be a problem. One cannot cover all of a non-compact space by a compact set! It is like excising a part of the spacetime, so it is not geodesically complete! Therefore there is a contradiction and the assumption is false