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! :)

 

[PeterDonis]

there seems to need a statement like "the maximal proper time curves (in C) must be geodesics"

Theorems 9.4.2 and 9.4.3 in Wald prove this, the first for the family of curves between fixed points $p$ and $q$, the second for the family of curves between a spacelike surface $\Sigma$ and a point $p$.

 

[martinbn]

A curve that has extrmal length between two points has to be geodesic. It follows from the first variation of arc-lenght. Wald does the calculation in the section for geodesics, around page 45.

 

[PeterDonis]

A curve that has extrmal length between two points has to be geodesic.

Extremal doesn't always mean maximal. An example is a free-fall geodesic orbit around a planet; its proper time is extremal but not maximal--the geodesic of maximal proper time is the radial one that intersects the orbit first going up and then coming back down.

 

[Matterwave]

I reread the sections carefully. I *think* I have resolved my confusion.

The core confusion indeed came from my reading of "extremal" as "maximal" and my inherent intuition to keep that around in my head.

Wald states several theorems (e.g. 9.3.5 and 9.4.2/9.4.3 as pointed out by Peter) which says the necessary (and sufficient -- this is left out of the versions in 9.4) condition for a curve to *maximize* proper time between p and q is that it is a geodesic "and there is no conjugate point between p and q (or Sigma)".

In my head, the conditional ("and there is no...") meant that if that conditional is false then "some other curve" could maximize proper time. There was no space in my head to also understand "some other curve" would also be a geodesic and the original geodesic would be "only" a stationary curve (i.e. extremal but not max or min).

So, sure, "the proof shows there are curves with more proper time which exists which aren't geodesics" -- BUT the maximal curve will necessarily still be a geodesic. And hence "why doesn't the compactness+upper-semi-continuous argument simply select out one of these non-geodesic curves?" -- because a maximal curve *must be a geodesic*.

Sorry if my explanation of my confusion is illogical. Confusions are illogical lol.

But I think I got it?

 

[Matterwave]

Extremal doesn't always mean maximal. An example is a free-fall geodesic orbit around a planet; its proper time is extremal but not maximal--the geodesic of maximal proper time is the radial one that intersects the orbit first going up and then coming back down.

This helped me visualize different geodesics between two points one which maximizes and one which only extremizes. Thanks! It was crucial to clearing up my confusion.