Singularity Theorems

[Matterwave]

TL;DR

Discussion on some finer details of the singularity theorems.

My (very) rough sketch of the set up of the time-like singularity theorems as presented by Wald:

1. Get Raychaduri eqns and find that caustics form within finite proper time (given SEC and initially negative expansion).
2. Find conjugate points corresponding to those caustics between points in spacetime or hyper surface + point in spacetime.
3. Prove geodesics no longer maximize proper time beyond conjugate points.
4. Prove (in chapter 8 in Wald or Hawking+Ellis) that global hyperbolic spacetimes have compact casual diamonds (and also in the space of curves $C(q,\Sigma)$)

Finally, compactness+semi-upper-continuity of proper time guarantees curves of maximal proper time exist. This, combined with points 2+3, is going to lead to a contradiction and the way to get out is by saying geodesics are incomplete.

If there's some stuff grossly wrong in my understanding, please correct me!

But I have a major question:

That geodesics no long maximize proper time beyond a conjugate point -- the proof shows there are curves with more proper time which exists which aren't geodesics. I guess, why doesn't the compactness+upper-semi-continuous argument simply select out one of these non-geodesic curves? In my head, there seems to need a statement like "the maximal proper time curves (in C) must be geodesics" to make the proof solid (otherwise, instead of getting geodesic incompleteness, you'd just pick out some non geodesic curve). But I'm getting confused here. Maybe I have simply forgotten too much about my calculus of variations and I am unable to see in my head whether that procedure guarantees geodesics are global maximizers vs local maximizers.

I realize the singularity theorems have been significantly strengthened from this one (and Wald points to them) which requires global hyperbolicity.

I might have other follow up questions but I'll just start here for now.

 

[PeterDonis]

why doesn't the compactness+upper-semi-continuous argument simply select out one of these non-geodesic curves?

First, even if this happened, you would still have incomplete geodesics. So it wouldn't change the conclusion that incomplete geodesics must exist.

Second, there must be geodesics passing through every event in a spacetime. (The simplest way to see this is to observe that the tangent space at every event is Minkowski space, which obviously has geodesics, so there must be tangent vectors in the tangent space at every event that are tangent to geodesics.) So even if you imagine some non-geodesic curve beyond the conjugate point, there could not be any geodesics passing through any events on that curve, because geodesics can't extend beyond the conjugate point. So those events can't be part of the spacetime--which means your imagined non-geodesic curve can't either.

 

[PeterDonis]

why doesn't the compactness+upper-semi-continuous argument simply select out one of these non-geodesic curves?

I just looked up the theorem in Wald to see how it is explicitly stated. Theorem 9.5.1 states a maximum length to the past of the given Cauchy surface for all timelike curves, not just geodesics. The geodesics are the only such curves that must be incomplete, because a complete geodesic must be extendible to arbitrary values of its affine parameter. There is no such requirement for non-geodesic curves, so it's impossible to say that the non-geodesic timelike curves to the past of the Cauchy surface are incomplete. But the theorem still says they also can't be longer than the maximum length.

 

[Matterwave]

Let me digest and reread the section when I get a chance. I'll be back if I have more questions! :)