Application to Black Holes

  This section is about two other propositions of the Hawking & Ellis book [3, p 311, p320,].

Proposition : If

(a)

is future asymptotically predictable from a partial Cauchy surface S

(b)

   for all null vectors

then a closed trapped surface  in cannot intersect , i.e. cannot be seen from .

This demonstration is constructed in the same way the preceeding theorem but is more what we need for the topological censorship and the application to black holes.

The beginning of the proof of that theorem needs to be explained. First let us define what is meant by

We can embed the spacetime M in a conformal space  and then glue the boundary   on it.

This gives us globally hyperbolic . It has a Cauchy slice   such that it match with the partial Cauchy slice on M.

This is a difficult demonstration but it is more general than the restriction of asking for a Cauchy slice. For simplicity let's assume we have a Cauchy slice (like in the Topological Censorship ).

  
Figure: Representation of the neighborhood of the boundary in one of the asymptotically flat regions. the rest of the spacetime is not represented here.

We take the neighborhood of the boundary as on the figure . Then the causal past   of the trapped surface  is a compact region! The generator intersects   so the generator of would leave (we eliminate the case that it hit ).

is a null surface because we made the hypotheses that the spacetime was asymptotically flat and empty.

We take a null generator, , of . If it hit it has an infinite length but, like we said before, we must have a conjugate point. So it could not remain in all way out to . Therefore does not intersect .

The difficult part of the proof is to show that intersects , after it the contradiction follow easily from what we did before.

The fact we take a full Cauchy surface is not a problem because we add a part of the spacetime which is not causally linked with the one we study. It cannot influence what we are doing.

Proposition has a similar demonstration. It states Let be a regular predictable space developing from a partial Cauchy surface S, in which    for any null vector . Then an outer trapped surface  P in   does not intersect . 

This kind of demonstration through point set topology will be used in the topological censorship for the lemma 2. The aim of this part was to understand this rather general method which can also be used to generalize to cosmology the Censorship theorem.