Lorentzian and Einstein Manifold

Question

I am studying for my Bachelor thesis (in Mathematics). I and my advisor agreed on the Penrose-Hawking singularity theorems.

My question is:

1) Which mathematical background should I focus on mastering in order to be able to approach the singularity theorems?

2) Is it the Lorentzian (Pseudo-Riemannian) manifolds which is flat (and there isn't very much to say about it) or the Einstein (Riemannian) manifolds (which for what I know now is in a important way different form the nature of space-time? Or both? Perhaps am I missing something important?

3) What is the role of Schwarzschild metric in this? How is it related to the manifolds above?

4) Is the next statement correct?

Schwarzschild metric is a particular solution of Einstein field equations (under which hypothesis?) in the sense that it models the space time of a portion of universe as a manifold with that particular metric. In this setting the singularity theorems can be proved.

5) Can anyone give me a complete statement of the singularity theorems so that I can see what mathematical objects are involved?

I still haven't looked for them because I taught that studying the math before even reading them could save me some time. But I have been studying Manifolds for two weeks now and I am getting more and more confused and I fear I am wasting time.

Answer 1

Concerning (1), I strongly suggest you B. O'Neill's textbook about semi-Riemannian geometry. It is written for mathematicians interested in GR without any particular background in physics. The last chapters focus on the causal structure of spacetime and singularity theorems. There is a large part completely devoted to analyze Schwarzschild's metric and Kruskal's manifold.

(2) I guess that for "Lorentzian manifold" you actually means Minkowski spacetime. It seems that you are mostly interested in General Realtivity, so what you should learn is the general theory of semi-Riemannian manifolds.

(3)-(4)-(5) Schwarzschild metric is the metric of a very important semi-Riemannian manifolds describing the spacetime in the presence of spherical symmetry. That metric is indeed a solution of Einstein field equations. Though the maximal extension of Schwarzschild manifold contains a metrical singularity, the singularities considered in Hawking-Penrse's theorems are of more general nature. There are several statements of the singularity theorems. However, all these statements say that:

Under some physically sensible hypotheses on:

(a) the global spacetime geometry (e.g. global hyperbolicity),

(b) its energy-momentum content (e.g., validity of some energy condition), and

(c) assuming that the Gravity is somewhere strong enough (typically, existence of trapped surfaces),

then there exists a maximally extended causal geodesic that is incomplete.

Answer 2

1)The singularity theorems are statements regarding the global properties of pseudo-riemannian manifolds. Since you mentioned you started studying manifold theory, then you're looking forward to general finite-dimensional manifolds, including differential forms and integration theory, then passing through the specific case of pseudo-riemannian (lorentzian) manifolds. From there you should focus on understanding Einstein Equations, and then pass to more advanced stuff such as causality and other topics in global structure.

I second wholeheartedly V. Moretti's suggestion of O'Neill's Semi-Riemannian Geometry. Otherwise I would recommend Wald's General Relativity (You would be interested in chapters 1-6 plus appendices A-C and then chapters 8 and 9) in parallel with Hawking & Ellis The Large Scale Structure of Space-time. Both of them have significant deviations with respect to O'Neill (primarily the use of index notation, much maligned by mathematicians in general) but are as rigorous as one should desire.

If you want a brief preview that does not discard math I would try to get my hands on Geroch's General Relativity Notes. Chapters 28-34 (all around five pages long) discuss what you're interested on.

2)A Lorentzian flat manifold is the Minkowski spacetime, the grounds of special relativity. This is not what you're looking for. A Einstein manifold (riemannian or pseudo-riemannian) is a manifold of constant Ricci Curvature. They are interesting in their own rights, but not specifically relevant to singularity theorems. But since this theorems concern the global nature there is not much to be learned by studying strictly riemannian manifold.

3)The Schwarzschild metric is in particular an Einstein (pseudo-riemannian) manifold of zero Ricci curvature. It was the first example of a singular manifold obtained, so in some sense it is a specific case of the singularity theorems.

4)The statement is partially correct. The Schwarzschild metric is the unique solution of Eintein Equations which have zero Ricci curvature, spherical symmetry and are stationary. But is not a "setting" for singularity theorems, rather a example of the theorems, as I mentioned above.

5)A complete statement (very restricted) taken from Geroch's book is

Let a space-time satisfy Einstein’s equation, with a stress-energy satisfying the energy condition. Let that space-time have a Cauchy surface $S$ on which $c\geq c_0$, where $c_0$ is a positive constant. Then no future-directed timelike curve from $S$ has length greater than $3/c_0$.

Here $c$ is the convergence of a timelike vector field orthogonal to the surface $S$.

Therefore, as indicated by V. Moretti, there are usually three conditions that must be satisfied (in order of appearance)

a)a curvature condition (which by Einstein Equations are also a restriction on the content of matter, called energy conditions)

b)a global structure causality condition (in this case the existence of Cauchy surfaces, which imply global hyperbolicity)

c)a Jacobi field condition (the $c\geq c_0$, that is the convergence of this particular vector field is everywhere bounded from below).

In this case the spacetime has no future inextendible timelike curve. This is bad because observers (like you and me) travel causal curves, and if one has finite affine lenght then the observer in this curve stops at a certain point. The theorem above applies to a contracting universe, implying that somewhere in the future all observers reach an endpoint. Reversing time direction gives, very sketchly, the Big Bang, that is a everywhere expanding, globally well-behaved with reasonable matter content unavoidably has a past point where every causal curve ends and therefore has no meaning considering the "past" of this point.

Just a addendum, note that the thoerem is about the completness of the geodesics, it says nothing about existence of a singularity such as those where there is a blow-up in curvature. In fact there is not a definition of what a singularity is in general relativity, so that from a rigorous point of view one should call them "incompletness theorems" and not "singularity theorems". For a physicist the idea is that incomplete geodesics should be a mark of appearance of a singularity, whatever such thing is