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Can anyone help me with this?

Explain why Penrose's singularity theorem does not apply to each of the following geodesi-

cally complete Lorentzian manifolds:

(a) Minkowski's spacetime;

(b) Einstein's universe;

(c) de Sitter's universe;

(d) Anti-de Sitter spacetime.

Here is what the theorem says:

"Let (M,g) be a globally hyperbolic Lorentzian manifold. If (M,g) has a noncompact connected Cauchy surface, the null energy condition is satisfied and exists a closed trapped null surface, then (M,g) is singular."

Well, I believe for a) the answer is that there aren't any closed trapped surfaces.

b) I believe it's the same as a).

d)It isn't globally hyperbolic.

but for c) am not sure. There are closed traped surfaces, right? And it is globbaly hyperbolic then that leaves, there aren't any noncompact connected Cauchy surfaces? Can any one explain why?

Thanks

## Homework Statement

Explain why Penrose's singularity theorem does not apply to each of the following geodesi-

cally complete Lorentzian manifolds:

(a) Minkowski's spacetime;

(b) Einstein's universe;

(c) de Sitter's universe;

(d) Anti-de Sitter spacetime.

**2. Relevant Information**Here is what the theorem says:

"Let (M,g) be a globally hyperbolic Lorentzian manifold. If (M,g) has a noncompact connected Cauchy surface, the null energy condition is satisfied and exists a closed trapped null surface, then (M,g) is singular."

## The Attempt at a Solution

Well, I believe for a) the answer is that there aren't any closed trapped surfaces.

b) I believe it's the same as a).

d)It isn't globally hyperbolic.

but for c) am not sure. There are closed traped surfaces, right? And it is globbaly hyperbolic then that leaves, there aren't any noncompact connected Cauchy surfaces? Can any one explain why?

Thanks

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