Physical meaning of Hawking's Singularity theorem

Question

I'm studying O'Neill's "Semi-Riemannian Geometry with applications to Relativity".

I know that the following theorems are related to the Big Bang, but I don't understand how.

Let $M$ be a semi-Riemannian manifold. Suppose $Ric(v,v)>0$ for every timelike $v$ tangent to $M$. Let $S$ be a spacelike Cauchy hypersurface with future convergence $k\ge b>0$. Then, every inextendible future-pointing timelike curve starting from $S$ has length $\le \frac{1}{b}$.

A similar result holds if we weaken the hypotheses.

Let $M$ be a semi-Riemannian manifold. Suppose $Ric(v,v)>0$ for every timeline $v$ tangent to $M$. Let $S$ be a compact spacelike hypersurface with future convergence $k>0$. Then, $M$ is geodesically-time-incomplete in the future.

I am not sure this is the right place to post this question, since it's rather involved with physics.

Answer 1

If you have ready access to O'Neill's book (a great one, by the way) he provides an extra comment in page 433, and I'll quote,

When time-orientation is reversed in Hawking’s theorem, past convergence (future expansion) implies past singularities.

and a little further

Since our universe seems to be at least approximately Robertson-Walker, the hypotheses of Theorem 55A [The first one in your question] are not unreasonable, and this result strongly suggests that our universe is catastrophically singular in the past. This conclusion is thus freed from the specific Robertson-Walker model ; in particular, the global hypothesis of exact spatial isotropy is no longer needed.

In other words, the physical idea is the following: current observations of galaxies suggest that astronomical bodies are receding from us, when you account for peculiar velocities. One would say that the timelike curves of this bodies worldines have future convergences $k<-b<0$, where $b$ is related to the Hubble constant. By using the Copernican Principle, which roughly states that we do not occupy a special position in the universe, it is reasonable to assume that a Cauchy hypersurface containing us we'll have negative future convergence, at least in a first approximation.

Then you just replace future with past in the theorems and read that if the universe has a Cauchy hypersurface $S$ with positive past convergence (i.e. negative future convergence, or in physical terms, the universe is expanding) then every past directed timelike curve starting in $S$ has finite lenght, and in particular all past directed timelike geodesics are incomplete.

In the absence of a satisfactory definition of a singularity in General Relativity we just take causal geodesic incompleteness as a proxy for the presence of singularities. In the context of the universe that's the Big Bang. The second theorem shows that with a compact spatial section one can do away with some causal requirements.