Penrose Singularity Theorem

[PLuz]

Can anyone help me with this?

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 anyone explain why?

Thanks

 

[mark726]

for posting this question and seeking clarification! I am a scientist and I would be happy to help you understand why Penrose's singularity theorem does not apply to each of the given spacetimes.

Firstly, it is important to understand what a singularity is in the context of general relativity. A singularity is a point in spacetime where the gravitational field becomes infinite and the laws of physics as we know them break down. In other words, it is a point where our current understanding of the universe fails. Penrose's singularity theorem states that if certain conditions are met, then a singularity must exist.

Now, let's look at each spacetime individually:

(a) Minkowski's spacetime: This is a flat spacetime with no gravitational field, so there can be no singularities. Therefore, Penrose's theorem does not apply.

(b) Einstein's universe: This is a closed, expanding universe with a positive curvature. It satisfies the null energy condition, but there are no closed trapped surfaces. Therefore, Penrose's theorem does not apply.

(c) de Sitter's universe: This is a closed, expanding universe with a positive cosmological constant. It satisfies the null energy condition and has closed trapped surfaces. However, it is not globally hyperbolic. This means that there are certain regions in the spacetime where we cannot define a unique evolution of the system. Therefore, Penrose's theorem does not apply.

(d) Anti-de Sitter spacetime: This is a closed, contracting universe with a negative cosmological constant. It does not satisfy the null energy condition, and it is not globally hyperbolic. Therefore, Penrose's theorem does not apply.

I hope this helps to clarify why Penrose's singularity theorem does not apply to these specific spacetimes. If you have any further questions, please don't hesitate to ask.