TrickyDicky said:
I entertained this idea as well. Do you know of any reference (or some heuristic explanation)that confirms that homogeneity requires 3 translation spacelike KVFs?
I don't have a reference handy right now but I'll check my copy of MTW as soon as I'm able. I'm pretty sure they go into it when they discuss FRW spacetime. In any case, the fact that FRW spacetime has 3 spacelike translation KVFs which Schwarzschild spacetime does not is certainly a fact.
TrickyDicky said:
That standard definition also says that the timelike KVF is global for the spacetime. Please explain what you think global means in this context. And if you consider the maximally extended Schwarzschild spacetime to be (globally) static or non-static.
Why do we keep having to harp on terminology instead of physics? I've already given the physics, several times, and so have others: the 4th KVF on Schwarzschild spacetime (the one in addition to the 3 that arise from spherical symmetry) is timelike outside the horizon, null on the horizon, and spacelike inside it. I haven't used the word "global", and whether that word appears in the standard definition depends on whose definition you read. I don't think MTW use the word (but I'll check when I can). This, once again, is why I have said it's no good just reading the words sources use about these things; you have to look at the actual math, which I have given.
If you want a guess as to why the word "global" appears, it's because the sources you're reading don't draw a clear distinction between an entire, maximally extended manifold, and a the region of that manifold that is covered by a particular coordinate chart without coordinate singularities. So when they say the 4th KVF is "globally" static, they really mean "static over the entire region covered by the exterior Schwarzschild chart", which is just the region outside the horizon. Someone who didn't realize that the exterior Schwarzschild chart doesn't cover the entire maximally extended manifold (and there have sure been plenty of them posting on PF) might think that static region was the entire manifold; and someone who didn't stop to think about that possible misinterpretation might use the word "global" in the sloppy sense I have described. But that's just a guess; I don't know what the people who used the word "global" were thinking.
TrickyDicky said:
You are confirming here that there is no frame-independent (here frame is used in both its meaning of coordinate system and observer state of motion senses) definition of non-static spacetime since it relies in a family of comoving observers, so I don't know in what sense you call it coordinate-independent.
The definition of "expansion" I gave applies to a family of timelike worldlines, but the definition of "static" that I gave does *not*; it applies to a region of spacetime, not a family of curves in that region, and whether or not a given region of spacetime is or is not static is a coordinate-independent fact.
It is perfectly possible to have a region of spacetime which is static but has families of timelike curves in it that have nonzero expansion, so there is no necessary connection between a region being static and the expansion of families of timelike curves within that region. If you insist on using the word "static" in a non-standard way, as meaning "zero expansion", I suppose I can't stop you, but please don't read *me* as using it that way.
(Also, once a family of timelike curves is defined, its expansion is coordinate-independent; it comes out the same regardless of which chart you use to describe the curves. But that's a secondary point.)
TrickyDicky said:
We were talking clearly about the 3-space volume.
Yes, but *which* 3-space?
TrickyDicky said:
If observers arrive at the singularity in a finite time i guess for them the volume is finite.
It depends on which 3-space volumes you pick. If you pick the 3-space volumes defined by constant r, theta, phi and the full range of Schwarzschild time t, then each such 3-space volume is infinite; and an infalling observer passes through a range of such 3-volumes between the horizon and the singularity (each one labeled by a different r--we're assuming a radially infalling observer). If you pick the 3-space volumes defined by constant Painleve time T, constant theta, phi, and 0 < r < 2m, then each such 3-space volume is finite, and an infalling observer also passes through a range of these 3-volumes (each one labeled by a different T) between the horizon and the singularity.
TrickyDicky said:
It's the first time you say that the Schwarzschild spacetime has a non-vacuum portion, are you sure?
I was talking there about the Oppenheimer-Snyder model, which has a non-vacuum portion representing spherically symmetric collapsing dust, joined to a portion of Regions I and II of the maximally extended vacuum Schwarzschild spacetime, representing the vacuum region outside the surface of the collapsing matter. I realize that's a switch of model, but I only mentioned it because you brought up matter that originally collapsed to form the BH. If you include such matter at all, then you aren't talking any more about the full maximally extended vacuum Schwarzschild spacetime, but only the portions that I just described. Everything I've said about vacuum Schwarzschild spacetime still applies to those portions of Regions I and II that appear in the Oppenheimer-Snyder model.
TrickyDicky said:
Are you talking about static patches within a nonstatic spacetime or to spacetimes globally defines as static.
I'm talking about static spacetime regions. See above.
TrickyDicky said:
Ok, but have you tried to compute the KVFs of the de Sitter spacetime using first the static coordinates and then the nonstatic ones including the dS slicing?
No, but you're right, it's a good exercise. I'll take a look at it when I get a chance.