Black Holes - the two points of view.

[Dale]

I'm trying to tie up all the (for me) loose ends. That's all.

OK, but if it isn't too much hassle, I would appreciate if you could explain how any remaining loose ends connect back to the rest. Especially the KVF->geodesic connection.

 

[PAllen]

I think the idea of contracting space in the SC interior comes from the waterfall analogy. You can view the dynamics (loosely) as spacetime collapsing and being consumed at the singularity - and being created at the horizon. A key point is that there are always a complete sequence of 2-spheres for all area defined radii > 0 and < Rs, shown most easily using Lemaitre coordinates. Thus, the volume inside the EH remains constant.

Similarly, as Peter described in an earlier post, a ball of free falling dust maintains its volume (while getting stretched and squeezed), up to the singularity.

The SC singularity is described in the literature as a Weyl singularity, while that in FRW collapse is a Ricci singularity. And here, my knowledge stops - I do not claim to know the technicalities of what is the difference between a Weyl singularity and a Ricci singularity.

 

[PeterDonis]

It is just the educated guess that all 4-spacetimes that are not stationary (not including the cosmological constant in the EFE) must have an either expanding or contracting 3-space volume. But maybe this is not the case here.

It depends on how you slice the spacetime (or the region of spacetime, such as the region inside the EH) into 3-spaces. There are ways to slice the region inside the EH into 3-spaces such that the 3-volume of all the 3-spaces is constant; for example, use slices of constant Painleve "time" T. I put "time" in quotes because inside the EH the Painleve T coordinate is spacelike, even though surfaces of constant T are *also* spacelike.

However, I believe that the following statement, which may be what is underlying your educated guess here, is correct: given any family of *timelike* geodesics that covers the entire region inside the EH, the family of spacelike surfaces orthogonal to those timelike geodesics will be decreasing in volume as proper time increases along the geodesics. In this sense, you could say that space inside the EH is "contracting". However, that statement does not have all the same implications as it would in, for example, FRW spacetime.

 

[PeterDonis]

The SC singularity is described in the literature as a Weyl singularity, while that in FRW collapse is a Ricci singularity. And here, my knowledge stops - I do not claim to know the technicalities of what is the difference between a Weyl singularity and a Ricci singularity.

As I understand it, a Weyl singularity has the Weyl tensor components going to infinity, while a Ricci singularity has the Ricci tensor components going to infinity. In the cases you name, the other tensor (Ricci for the Weyl singularity, Weyl for the Ricci singularity) is zero (in the idealized case), so *all* of the curvature is described by the tensor that goes to infinity at the singularity.

 

[PeterDonis]

However, I believe that the following statement, which may be what is underlying your educated guess here, is correct: given any family of *timelike* geodesics that covers the entire region inside the EH, the family of spacelike surfaces orthogonal to those timelike geodesics will be decreasing in volume as proper time increases along the geodesics.

On thinking it over some more, I'm no longer convinced that even this statement is true, because the worldlines of ingoing "Painleve observers" are timelike geodesics, *and* they are orthogonal to each surface of constant Painleve "time" T that they pass through, and as I said, each surface of constant T has the same 3-volume inside the horizon. So the infalling Painleve observers do *not* see a decreasing 3-volume of the "spaces" they pass through.

In fact, thinking it over still more, I'm not even convinced that the statement as I gave it is true for observers falling in along a geodesic of constant Schwarzschild t inside the EH. Such observers do pass through 2-spheres of decreasing area; however, the 3-volume of each 3-space orthogonal to their worldline, which is just a 3-surface of constant r < 2m, is *infinite* (one 2-sphere at r for each value of t, with a fixed finite area of 4 pi r^2, but the 3-surface as a whole covers the full infinite range of t, so its total 3-volume is infinite).

So I think the only sense in which space can be said to be "contracting" inside the EH is that any timelike observer must pass through 2-spheres of strictly decreasing area.

 

[TrickyDicky]

OK, but if it isn't too much hassle, I would appreciate if you could explain how any remaining loose ends connect back to the rest. Especially the KVF->geodesic connection.

As I said I forgot why I brought it up in the first place, then I came back to it from a stevendaryl's post I had not spotted previously. I agree that it is not related to what we are discussing and I won't mention it here again. It might make a good question for the Topology and geometry subforum though.

 

[TrickyDicky]

It depends on how you slice the spacetime (or the region of spacetime, such as the region inside the EH) into 3-spaces. There are ways to slice the region inside the EH into 3-spaces such that the 3-volume of all the 3-spaces is constant; for example, use slices of constant Painleve "time" T. I put "time" in quotes because inside the EH the Painleve T coordinate is spacelike, even though surfaces of constant T are *also* spacelike.
However, I believe that the following statement, which may be what is underlying your educated guess here, is correct: given any family of *timelike* geodesics that covers the entire region inside the EH, the family of spacelike surfaces orthogonal to those timelike geodesics will be decreasing in volume as proper time increases along the geodesics. In this sense, you could say that space inside the EH is "contracting". However, that statement does not have all the same implications as it would in, for example, FRW spacetime.

I see what you mean about the dependence on the slicing, and dealing with GR manifolds there should not be preferred slicing. So the slicing in which the 3-volumes are constant shouldn't be preferred over the "contracting" one or viceversa. See also that this can classify our coordinate patches in static and not static ones:look for instance to the example of de Sitter space where you can choose between static or non-static coordinates.
However, maybe you can also see that there is a certain ambiguity about this wrt 4-spacetimes in GR. In the sense that in order to decide whether a spacetime, or at least a patch of the spacetime is static or not we must "choose" a preferred way of slicing it into 3-spaces, in other words one must choose a preferred frame.
This can be clearly seen in FRW metric, being a GR spacetime we are in theory allowed to slice it whichever way we want, no preferred frame, however in practice there is only a slicing that allows us to consider it an expanding "space", the one with slices of homogeneous density. Other slicings may give us expanding "spacetimes" which make expansion lose its original meaning.
I guess when you say this situation cannot be compared with the Black hole inside because in this last case we are not demanding homogeneity(after all it is supposed to be a vacuum), you might be right. But as I said, as long as this interior is modeled as an S^2XR^2 space in which every point is a 2-sphere you have homogeneity, and given the finite volume inside the EH and that it might be full of matter that has fallen in the BH on its way to the singularity,I find it hard not to think of this inside region as one where the comparison with the FRW metric is valid to a certain extent.

On thinking it over some more, I'm no longer convinced that even this statement is true, because the worldlines of ingoing "Painleve observers" are timelike geodesics, *and* they are orthogonal to each surface of constant Painleve "time" T that they pass through, and as I said, each surface of constant T has the same 3-volume inside the horizon. So the infalling Painleve observers do *not* see a decreasing 3-volume of the "spaces" they pass through.

In fact, thinking it over still more, I'm not even convinced that the statement as I gave it is true for observers falling in along a geodesic of constant Schwarzschild t inside the EH. Such observers do pass through 2-spheres of decreasing area; however, the 3-volume of each 3-space orthogonal to their worldline, which is just a 3-surface of constant r < 2m, is *infinite* (one 2-sphere at r for each value of t, with a fixed finite area of 4 pi r^2, but the 3-surface as a whole covers the full infinite range of t, so its total 3-volume is infinite).

So I think the only sense in which space can be said to be "contracting" inside the EH is that any timelike observer must pass through 2-spheres of strictly decreasing area.

See my comments above.
There is something I can't fully understand about this dependence of spacetimes on the frame or the coordinate patch chosen to decide about their staticity or lack of it, when it is supposed to be something invariant and as you and other have insisted the KVFs shouldn't depend on the coordinates.
I know at least in Riemannian geometry KVFs are defined globally so in a space there cannot exist regions with different KVFs,that assures that they are truly coordinate independent entities, but then again in those spaces there is no timelike-null-spacelike distinction.
But in pseudoriemannian spacetimes I'm not sure this holds as simply as that. At least my discussion above suggests that there may be a dependence on the slicing, that is on the frame and coordinates chosen to have a KVF being timelike or spacelike.

 

[TrickyDicky]

I think the idea of contracting space in the SC interior comes from the waterfall analogy. You can view the dynamics (loosely) as spacetime collapsing and being consumed at the singularity - and being created at the horizon. A key point is that there are always a complete sequence of 2-spheres for all area defined radii > 0 and < Rs, shown most easily using Lemaitre coordinates. Thus, the volume inside the EH remains constant.

Similarly, as Peter described in an earlier post, a ball of free falling dust maintains its volume (while getting stretched and squeezed), up to the singularity.

Actually the waterfall image with the eternal spring and sink that mantains the volume in a time-invariant way has its beauty and is inspiring, but doesn't it look like a completely static image of a spacetime to you?
There's something of a catch-22 situation with the geometry of this region, it would seem to admit either a timelike or a spacelike KVF depending on the slicing in a kind of situation that reminds of the de Sitter space static and expanding coordinate patches.
See my reply to PeterDonis in my previous post.
Going back to the waterfall analogy and to reinforce the sometimes slippery distiction between static and non-static that I see in the inside region of a BH, consider this statement from the wikipedia page "spacetime symmetries" where it consideres Einsten static spacetime as a subcase of FRW metrics:
"For example, the Schwarzschild solution has a Killing algebra of dimension 4 (3 spatial rotational vector fields and a time translation), whereas the Friedmann-Lemaitre-Robertson-Walker (FLRW) metric (excluding the Einstein static subcase) has a Killing algebra of dimension 6 (3 translations and 3 rotations). The Einstein static metric has a Killing algebra of dimension 7 (the previous 6 plus a time translation). "

This is made clearer here, from the wikipedia page on de Sitter universe, where the Einstein static spacetime is made a special case of de Sitter spacetime (that is considered here as a FRW metric without privileged frame so that it is an expandig "spacetime" rather than "space" in accordance with the perfect cosmological principle) in a similar way to the waterfall description in which as you say the dynamics of collapsing at the singularity and "creation" at the EH are finely tuned too to produce a static volume:
"As a class of models with different values of the Hubble constant, the static universe that Einstein developed, and for which he invented the cosmological constant, can be considered a special case of the de Sitter universe where the expansion is finely tuned to just cancel out the collapse associated with the positive curvature associated with a non-zero matter density. "

 

[PeterDonis]

Iin order to decide whether a spacetime, or at least a patch of the spacetime is static or not we must "choose" a preferred way of slicing it into 3-spaces, in other words one must choose a preferred frame.

Not for the standard definition of "static", which is that the spacetime (or a region of it) has a timelike KVF. That definition is coordinate-independent. By that definition, Schwarzschild spacetime (the vacuum solution) is static only outside the EH; the region at and inside the EH is not static.

This can be clearly seen in FRW metric, being a GR spacetime we are in theory allowed to slice it whichever way we want, no preferred frame, however in practice there is only a slicing that allows us to consider it an expanding "space", the one with slices of homogeneous density. Other slicings may give us expanding "spacetimes" which make expansion lose its original meaning.

There is also a coordinate-independent definition of "expansion", but it applies to families of timelike curves, not "space" itself. The standard definition of an "expanding" or "contracting" FRW spacetime uses the family of timelike curves that describe the worldlines of "comoving" observers; the expansion of that family of curves, defined in the standard coordinate-independent way, is positive for an expanding FRW spacetime and negative for a contracting one.

But as I said, as long as this interior is modeled as an S^2XR^2 space in which every point is a 2-sphere you have homogeneity

Not by the standard definition of "homogeneity", the one that applies to FRW spacetimes. Can you give a reference for this different definition of "homogeneity"? The standard term for what you're describing, AFAIK, is simply "spherical symmetry".

and given the finite volume inside the EH

The "volume" inside the EH is not necessarily finite; as I said before, it depends on what "volume" you are looking at. The 4-volume inside the EH is infinite, since it covers an infinite range of the t coordinate. Spacelike 3-volumes cut out of that 4-volume may be finite or infinite, depending on how they are cut.

and that it might be full of matter that has fallen in the BH on its way to the singularity

The portion of the spacetime that is occupied by the matter that originally collapsed to form the BH *is* full of matter on its way to the singularity. And this portion (at least in the idealized spherically symmetric case) *is* isometric to a portion of a collapsing FRW spacetime. That is the model that Oppenheimer and Snyder described in their 1939 paper. However, that only applies to the non-vacuum portion of the spacetime. I don't really think an analogy between the *vacuum* portion of the spacetime inside the EH and FRW spacetime is useful, but that may be just me.

There is something I can't fully understand about this dependence of spacetimes on the frame or the coordinate patch chosen to decide about their staticity or lack of it, when it is supposed to be something invariant and as you and other have insisted the KVFs shouldn't depend on the coordinates.

Staticity, by the standard definition, *is* coordinate-independent. See above.

I know at least in Riemannian geometry KVFs are defined globally so in a space there cannot exist regions with different KVFs

The KVF $\partial / \partial t$ in Schwarzschild spacetime is not a "different KVF" in different regions. But any vector field on a manifold is a mapping between points in the manifold and vectors in a vector space, and different points may map to different vectors. In a manifold which has the timelike-spacelike-null distinction, i.e., where vectors have a "causal nature", that means the same vector field may map different points in the manifold to vectors with a different causal nature.

At least my discussion above suggests that there may be a dependence on the slicing, that is on the frame and coordinates chosen to have a KVF being timelike or spacelike.

No. Whether a KVF, or indeed *any* vector field, is timelike, spacelike, or null *at a given event* is an invariant, independent of coordinates. But the particular vectors which are mapped to different events by a vector field are different vectors, and may have a different causal nature.

consider this statement from the wikipedia page "spacetime symmetries" where it consideres Einsten static spacetime as a subcase of FRW metrics:
"For example, the Schwarzschild solution has a Killing algebra of dimension 4 (3 spatial rotational vector fields and a time translation), whereas the Friedmann-Lemaitre-Robertson-Walker (FLRW) metric (excluding the Einstein static subcase) has a Killing algebra of dimension 6 (3 translations and 3 rotations). The Einstein static metric has a Killing algebra of dimension 7 (the previous 6 plus a time translation). "

The terminology here is sloppy; the "time translation" KVF is only timelike outside the horizon. This is why I made a point of saying before that, for this topic of discussion, you can't just quote statements without looking at the actual definitions and math behind them. The actual math is perfectly clear: the 4th KVF in Schwarzschild spacetime is timelike outside the horizon, null on the horizon, and spacelike inside it. I'm sorry that so many sources are sloppy about describing this, but that's why I've spent so much effort in this thread explicitly showing the actual math.

 

[PeterDonis]

the Friedmann-Lemaitre-Robertson-Walker (FLRW) metric (excluding the Einstein static subcase) has a Killing algebra of dimension 6 (3 translations and 3 rotations).

I forgot to comment on this specifically. AFAIK, the standard definition of "homogeneity" for a spacetime is that there are 3 KVFs corresponding to spatial translations. The fact that those are *not* present in Schwarzschild spacetime is why I have been objecting to describing that spacetime as "homogeneous".

 

[PeterDonis]

"As a class of models with different values of the Hubble constant, the static universe that Einstein developed, and for which he invented the cosmological constant, can be considered a special case of the de Sitter universe where the expansion is finely tuned to just cancel out the collapse associated with the positive curvature associated with a non-zero matter density. "

This seems like sloppy terminology as well, since the de Sitter universe has *zero* matter density and a positive cosmological constant, so a model with *nonzero* matter density is not, properly speaking, a "special case" of it.

 

[Dale]

I see what you mean about the dependence on the slicing, and dealing with GR manifolds there should not be preferred slicing.

I wouldn't say that. If your GR manifold is everywhere vacuum then there should not be preferred slicing, but if your GR manifold has matter fields then the matter fields will disrupt the symmetry and may define a preferred slicing.

At least my discussion above suggests that there may be a dependence on the slicing, that is on the frame and coordinates chosen to have a KVF being timelike or spacelike.

The V in KVF is "vector". The timelike or spacelike character of a vector has nothing to do with the coordinates chosen, they remain timelike or spacelike in any coordinate chart. If it were not so, then they would not be vectors.

 

[erasrot]

I hope you find this question related with this thread topic. Let us assume that all the ideal conditions for the Oppenheimer-Snyder collapse are met, does it ever stop for any observer?

 

[PeterDonis]

I hope you find this question related with this thread topic. Let us assume that all the ideal conditions for the Oppenheimer-Snyder collapse are met, does it ever stop for any observer?

Yes.

 

[PAllen]

I hope you find this question related with this thread topic. Let us assume that all the ideal conditions for the Oppenheimer-Snyder collapse are met, does it ever stop for any observer?

A good trigger for a summary.

Anything you see (most noticeably for things far away) is an image from the past. That is all you can possibly know.

Talking about what is happening at a distance 'now' must be based on extrapolation and and some definition of 'now'.

Someone observing Oppenheimer-Snyder collapse from a distance never sees anything cross the event horizon. You can say that visually, the collapse appears to freeze. The is the physical observable. There are many plausible extrapolations to 'now' such that you may consider that a horizon has formed and the matter collapsed to a singularity. There are also definitions of 'now' such that this never happens.

For a free fall observer moving with the collapsing dust, the horizon is crossed and a singularity is reached after finite time on their wristwatch. In general, any radial free fall observer sees the collapse proceed to a singularity.

 

[TrickyDicky]

Not for the standard definition of "static", which is that the spacetime (or a region of it) has a timelike KVF. That definition is coordinate-independent. By that definition, Schwarzschild spacetime (the vacuum solution) is static only outside the EH; the region at and inside the EH is not static.

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.

There is also a coordinate-independent definition of "expansion", but it applies to families of timelike curves, not "space" itself. The standard definition of an "expanding" or "contracting" FRW spacetime uses the family of timelike curves that describe the worldlines of "comoving" observers; the expansion of that family of curves, defined in the standard coordinate-independent way, is positive for an expanding FRW spacetime and negative for a contracting one.

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.

Not by the standard definition of "homogeneity", the one that applies to FRW spacetimes. Can you give a reference for this different definition of "homogeneity"? The standard term for what you're describing, AFAIK, is simply "spherical symmetry".

Give me your standard definition of homogeneity or stop referring to it. Spherical symmetry only requires a foliation of concentric 2-spheres around an origin. A foliation in which each point is 2-sphere has an origin for each point. Can't you see that?

The "volume" inside the EH is not necessarily finite; as I said before, it depends on what "volume" you are looking at. The 4-volume inside the EH is infinite, since it covers an infinite range of the t coordinate. Spacelike 3-volumes cut out of that 4-volume may be finite or infinite, depending on how they are cut.

We were talking clearly about the 3-space volume. If observers arrive at the singularity in a finite time i guess for them the volume is finite.

The portion of the spacetime that is occupied by the matter that originally collapsed to form the BH *is* full of matter on its way to the singularity. And this portion (at least in the idealized spherically symmetric case) *is* isometric to a portion of a collapsing FRW spacetime. That is the model that Oppenheimer and Snyder described in their 1939 paper. However, that only applies to the non-vacuum portion of the spacetime. I don't really think an analogy between the *vacuum* portion of the spacetime inside the EH and FRW spacetime is useful, but that may be just me.

It's the first time you say that the Schwarzschild spacetime has a non-vacuum portion, are you sure? And how you separate the non-vacuum part of the BH from the vacuum part. I thought the whole spacetime was supposed to be a vacuum solution.

Staticity, by the standard definition, *is* coordinate-independent. See above.

Are you talking about static patches within a nonstatic spacetime or to spacetimes globally defines as static. What would you consider to be the case with Schwarzschild spacetime?

The KVF $\partial / \partial t$ in Schwarzschild spacetime is not a "different KVF" in different regions. But any vector field on a manifold is a mapping between points in the manifold and vectors in a vector space, and different points may map to different vectors.
No. Whether a KVF, or indeed *any* vector field, is timelike, spacelike, or null *at a given event* is an invariant, independent of coordinates. But the particular vectors which are mapped to different events by a vector field are different vectors, and may have a different causal nature.

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?

 

[TrickyDicky]

I forgot to comment on this specifically. AFAIK, the standard definition of "homogeneity" for a spacetime is that there are 3 KVFs corresponding to spatial translations. The fact that those are *not* present in Schwarzschild spacetime is why I have been objecting to describing that spacetime as "homogeneous".

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?

 

[erasrot]

Ok, let me be sure I understand this properly. Timelike geodesics are timelike everywhere in Schwarzschild spacetime (in the manifold, i.e., independent of the given chart), including the region inside the EH. This spacetime is geodesically incomplete, what implies that the affine parameter for any infalling observer is bounded from above so, after proper calculation, the difference between the bound for a given geodesic and any other value of the affine parameter along any other timelike geodesic is also finite. I am right?

 

[TrickyDicky]

I wouldn't say that. If your GR manifold is everywhere vacuum then there should not be preferred slicing, but if your GR manifold has matter fields then the matter fields will disrupt the symmetry and may define a preferred slicing.

Sure, but that would just give a special status to a certain frame, it would not break the general covariance of GR that demands coordinate -independence.

The V in KVF is "vector". The timelike or spacelike character of a vector has nothing to do with the coordinates chosen, they remain timelike or spacelike in any coordinate chart. If it were not so, then they would not be vectors.

Look up the static and non-static coordinates for de Sitter spacetime. Why would staticity (a property that includes admission of timelike Killing vector field) depend on the coordinates used?
Something similar happens with Schwarzschild vs Lemaitre coordinates for the region inside the EH.

 

[PeterDonis]

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.

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.

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.)

We were talking clearly about the 3-space volume.

Yes, but *which* 3-space?

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.

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.

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.

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.

 

[PeterDonis]

Look up the static and non-static coordinates for de Sitter spacetime. Why would staticity (a property that includes admission of timelike Killing vector field) depend on the coordinates used?

Static coordinates are just coordinates that make the timelike KVF (in the regions where it is timelike) manifest, by explicitly having one coordinate that the metric is independent of. In the case of Schwarzschild spacetime described by the Schwarzschild chart, that coordinate is t. So any region of spacetime that is static can be described by a static coordinate chart.

However, the converse is not true; there is no reason why a static region *has* to be described by a static chart. Sometimes there are good reasons not to, as with Lemaitre coordinates on Schwarzschild spacetime. However, even in a non-static chart on a static spacetime region, if you describe the timelike KVF in the non-static chart, it will still satisfy Killing's equation--i.e., it will still be a KVF. It just won't be as obvious.

(I'll defer further remarks on this until I've had a chance to explicitly do the computation for de Sitter spacetime.)

Edit: I should probably add that even in regions where a given KVF is *not* timelike, there will still be analogues of the "static" chart--i.e., a chart in which the metric is independent of some coordinate corresponding to the KVF--though of course "static" is not a good name for the chart in those regions. An example is the Schwarzschild chart on the interior vacuum region of Schwarzschild spacetime: the metric is still independent of t, but t is not timelike and $\partial / \partial t$, while still a KVF, is not a timelike KVF.

 

[TrickyDicky]

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.

I don't think so, my guess is that the language of the definitions comes from Riemannian geometry and there you don't have to make causal distinctions for vectors. When those definitions are applied to pseudoriemannian spacetimes certain conceptual problems appear due to the different causal vectors. And globality is one of those concepts that suffer with the introduction of these distinctions, the other I would say is the coordinate independence, not of the KVF itself, but of its causal nature.

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.

Fine, but then you are implying that staticity and non-staticity can coexist in the same region, that are not mutually excluding concepts, right?

 

[Dale]

Sure, but that would just give a special status to a certain frame, it would not break the general covariance of GR that demands coordinate -independence.

Yes, exactly.

Look up the static and non-static coordinates for de Sitter spacetime. Why would staticity (a property that includes admission of timelike Killing vector field) depend on the coordinates used?

Whether or not the spacetime is static doesn't depend in any way on the coordinates. Static coordinates are a very different thing from a static spacetime. If you have a static spacetime then there exists a set of coordinates where the components of the metric are not functions of time, but you can always choose different coordinates where some of the components are. For example, Minkowski spacetime is obviously static, but you can use non-static rotating coordinates if you want. Doing so doesn't change any of the KVF's, but of course the components of the KVF's are not as easy to figure out in those coordinates.

 

[PeterDonis]

I don't think so, my guess is that the language of the definitions comes from Riemannian geometry and there you don't have to make causal distinctions for vectors. When those definitions are applied to pseudoriemannian spacetimes certain conceptual problems appear due to the different causal vectors.

This certainly could be the case, yes.

And globality is one of those concepts that suffer with the introduction of these distinctions

If you think "globality" is an important concept, yes, I suppose it could. I've never really thought of "globality" as a concept of interest.

the other I would say is the coordinate independence, not of the KVF itself, but of its causal nature.

I've already addressed this, and so have others. The causal nature of any vector at a particular event in any spacetime is coordinate-independent. The causal nature of vectors at *different* events, but which are part of the same vector field, can be different, but whether or not they are, and if so, how, is also coordinate-independent, once the particular events in question are specified.

Fine, but then you are implying that staticity and non-staticity can coexist in the same region, that are not mutually excluding concepts, right?

I'm not sure what you mean by this, but I have already described the physics many times, and I have explicitly given you the definition I am using for the word "static", and explicitly told you how it applies to Schwarzschild spacetime. Whatever question you are asking here, you should be able to answer it for yourself from what I have already written.

 

[TrickyDicky]

de Sitter space metric in static coordinates:

$$ds^2=-(1-\frac{r^2}{\alpha^2})dt^2+(1-\frac{r^2}{\alpha^2})^{-1} dr^2+r^2d\Omega^2_{n-2}$$

Schwarzschild spacetime in Lemaitre coordinates:
$$ds^2=d\tau^2-\frac{2\mu}{r}dρ^2-r^2(d\theta^2+sin^2\theta d\phi^2)$$

In both cases none of the metric coefficients are a function of time.

 

[Dale]

Schwarzschild spacetime in Lemaitre coordinates:
$$ds^2=d\tau^2-\frac{2\mu}{r}dρ^2-r^2(d\theta^2+sin^2\theta d\phi^2)$$

In both cases none of the metric coefficients are a function of time.

For Lemaitre coordinates, if by "time" you mean τ then the metric coefficients of ρ θ and phi are all functions of time since r is a function of time.

 

[PAllen]

de Sitter space metric in static coordinates:

$$ds^2=-(1-\frac{r^2}{\alpha^2})dt^2+(1-\frac{r^2}{\alpha^2})^{-1} dr^2+r^2d\Omega^2_{n-2}$$

Schwarzschild spacetime in Lemaitre coordinates:
$$ds^2=d\tau^2-\frac{2\mu}{r}dρ^2-r^2(d\theta^2+sin^2\theta d\phi^2)$$

In both cases none of the metric coefficients are a function of time.

except that r=(3/2(ρ - τ ))^2/3 / (2M)^1/3

So, all non-vanishing components of the metric are time dependent except g00.

 

[PAllen]

As for the static de Sitter coordinates, for which metric given in recent post, the key point is that static coordinates only cover a quarter of the spacetime. Thus the issue is similar to complete SC geometry. The portion of de-Sitter covered by the static patch really is static per timelike KVF definition, but the rest of the spacetime is not static.

 

[PeterDonis]

As for the static de Sitter coordinates, for which metric given in recent post, the key point is that static coordinates only cover a quarter of the spacetime.

Just to clarify, I believe that the line element TrickyDicky wrote down is valid over the entire spacetime, but the t coordinate, which is the one the line element is independent of (i.e., $\partial / \partial t$ is the KVF) is only timelike in the region you mention, that covers a quarter of the spacetime. The rest of the spacetime is beyond the "cosmological horizon" and the t coordinate is spacelike there (and null on the horizon itself).

 

[PAllen]

Just to clarify, I believe that the line element TrickyDicky wrote down is valid over the entire spacetime, but the t coordinate, which is the one the line element is independent of (i.e., $\partial / \partial t$ is the KVF) is only timelike in the region you mention, that covers a quarter of the spacetime. The rest of the spacetime is beyond the "cosmological horizon" and the t coordinate is spacelike there (and null on the horizon itself).

Ok, that makes sense, like the SC coordinates. I've only read things where they use different coordinates in the non-static regions, using the 'static' coordinates only in truly static section.

 

[PeterDonis]

Ok, that makes sense, like the SC coordinates. I've only read things where they use different coordinates in the non-static regions, using the 'static' coordinates only in truly static section.

Yes, but when comparing with SC coordinates, a key thing to remember is that there are an infinite number of "static" coordinate charts on de Sitter spacetime, because the line element that TrickyDicky wrote down can be centered on *any* spatial point (i.e., any spatial point can be designated as r = 0). Of course in Schwarzschild spacetime there is only one KVF which is timelike on any portion of the spacetime, and the only charts I'm aware of whose "time" coordinate matches that KVF are the Schwarzschild and Painleve charts.

As the above statement indicates, there are also an infinite number of timelike KVFs on de Sitter spacetime, just as there are on Minkowski spacetime; but in de Sitter spacetime, each timelike KVF has its own "static region" in which it is timelike, and which is bounded by its own cosmological horizon on which the KVF becomes null (and then spacelike beyond the horizon).

Actually, it may be even more complicated than that, because de Sitter spacetime has the same isometry group as Minkowski spacetime, and Minkowski spacetime actually has *two* infinite families of timelike KVFs. The first corresponds to the worldlines of inertial observers (at rest in all the different possible global inertial frames), and the second corresponds to the worldlines of Rindler observers (with all the different possible Rindler horizons, based on which event in the spacetime is chosen as the "pivot point" where the past and future horizons intersect). I will have to do some computations to figure out which of those two types of observers in Minkowski spacetime corresponds to the observers who are at rest in the static chart that TrickyDicky wrote down (in the region where it is possible to have static observers); I suspect the closest analogue is actually Rindler observers, because it looks to me like any observer who remains at a constant r in the static de Sitter chart (where 0 < r < alpha) will have nonzero proper acceleration.

 

[TrickyDicky]

except that r=(3/2(ρ - τ ))^2/3 / (2M)^1/3

So, all non-vanishing components of the metric are time dependent except g00.

For Lemaitre coordinates, if by "time" you mean τ then the metric coefficients of ρ θ and phi are all functions of time since r is a function of time.

Sure, I was looking only at the explicit dependence.

Ok, so substitute Lemaitre's with the Gullstrand–Painlevé coordinates as PeterDonis suggests.

 

[Dale]

OK, and what is the point? Are you trying to claim that these different coordinate charts change the KVFs somehow? If so, simply citing them is insufficient.

 

[TrickyDicky]

Just to clarify, I believe that the line element TrickyDicky wrote down is valid over the entire spacetime, but the t coordinate, which is the one the line element is independent of (i.e., $\partial / \partial t$ is the KVF) is only timelike in the region you mention, that covers a quarter of the spacetime. The rest of the spacetime is beyond the "cosmological horizon" and the t coordinate is spacelike there (and null on the horizon itself).

My point was not to show that the t coordinate doesn't blow up a the cosmological horizon in the static coordinates, it does because they only cover a double wedge of the hyper-hyperboloid that represents the whole space.
My point was to show that this very region that is covered by the static coordinates and that admits a timelike KVF with this coordinates, when expressed in different coordinates (like the closed slicing that covers the entire space) doesn't admit the timelike KVF.

 

[Dale]

My point was to show that this very region that is covered by the static coordinates and that admits a timelike KVF with this coordinates, when expressed in different coordinates (like the closed slicing that covers the entire space) doesn't admit the timelike KVF.

You certainly didn't show that. To do that, you would have to write down the vector field and the metric in both coordinates, compute the Lie derivative of the metric wrt the vector field in each, and show that it is 0 in one and non-zero in the other.