Finkelstein's unidirectional membrane paper

[PeterDonis]

You have "from region I to region to region II." Did you just mean "from region I to region II?"

In my last post, I assumed this was what PAllen meant.

 

[TrickyDicky]

I think the term "geometry" would be better than the term "metric", because "geometry" makes clear that what we are talking about is the underlying, invariant spacetime, not any particular expression of its metric in particular coordinates.

Well, this looks like a kind way to avoid saying there is no such metric.

The geometry of the vacuum regions (exterior and interior) is the Schwarzschild geometry. The geometry of the non-vacuum region containing the collapsing matter is a contracting FRW geometry (the time-reverse of the expanding FRW geometry that is used in cosmology--more precisely, the k = 1, "closed universe" case of that geometry). The two geometries are "matched" at the boundary of the non-vacuum region, meaning (I think this is right) that the curvature and its spacetime derivatives need to approach the same values as you approach the boundary from either "side" (the vacuum or the non-vacuum side). This is for the idealized, precisely spherically symmetric case.

The way you use the term "geometry" in this paragraph is basically the opposite of talking about an the underlying, invariant spacetime. You are differentiating the geometry of vacuum and of non-vacuum as if they were different spacetimes, in fact Schwarzschild metric and (contracting) FRW metric are not the same spacetime at all, to start with, one is a static manifold and the other is not.

 

[PeterDonis]

Well, this looks like a kind way to avoid saying there is no such metric.

The way you use the term "geometry" in this paragraph is basically the opposite of talking about an the underlying, invariant spacetime. You are differentiating the geometry of vacuum and of non-vacuum as if they were different spacetimes, in fact Schwarzschild metric and (contracting) FRW metric are not the same spacetime at all, to start with, one is a static manifold and the other is not.

I think you're misunderstanding what I'm saying. I'll try to restate it more precisely. Take the idealized, exactly spherically symmetric case where we have matter collapsing to form a black hole. What I'm saying is that the actual, physical spacetime consists of three regions which are "stitched" together:

Region I is a vacuum "exterior region" which is isomorphic to a portion of the exterior Schwarzschild geometry.

Region II is a vacuum "interior" region which is isomorphic to a portion of the interior Schwarzschild geometry.

Region C is a non-vacuum region containing the collapsing matter, which is isomorphic to a portion of the closed contracting FRW geometry.

The boundaries between the regions are defined as follows:

(1) By the surface of the collapsing matter (between regions I and C outside the horizon, and between regions II and C inside the horizon). At this boundary, the curvature of the FRW geometry and its spacetime derivatives, at the current (i.e., at that instant of the collapsing matter's proper time) physical radius of the collapsing matter, have to smoothly match the curvature and derivatives of the (exterior or interior, as appropriate) Schwarzschild geometry at the same physical radius.

(2) By the event horizon (between regions I and II) that forms when the collapsing matter has collapsed to the point where its radius is less than 2M, where M is the externally measured mass of the collapsing matter (in geometric units). Across the event horizon, the exterior and interior geometries have to match up in the "usual" way for the exterior and interior vacuum regions of the complete Schwarzschild geometry; the only difference is that the event horizon forms at a finite time in the past and does not extend all the way back to t = minus infinity in terms of the Schwarzschild time coordinate.

(3) Once the collapsing matter reaches r = 0, it forms a final singularity which is then part of the (future) boundary of region II. This singularity, once formed, works exactly like the final singularity in the complete Schwarzschild geometry; the only difference, once again, is that it extends only a finite "distance" in the "past" direction (I have to use quotes because it's a spacelike surface; hopefully it's clear what I mean).

So I am talking about a single, invariant geometry, but it's not a geometry which is described by a single solution of the equations of General Relativity. Instead, it's portions of different such solutions, stitched together at boundaries defined as above.

 

[TrickyDicky]

So I am talking about a single, invariant geometry, but it's not a geometry which is described by a single solution of the equations of General Relativity. Instead, it's portions of different such solutions, stitched together at boundaries defined as above.

You must be referring to the global topology when you say "geometry". GR is considered a "local geometry" theory, and in this sense it wouldn't constrain the global topology of the manifold. According to this , you have a point that the spacetime geometry(global topology) of the manifold would not be described by a single solution (metric) of the GR equations. Instead each patch would have a different metric.
So the original Schwarzschild metric would only describe the patch up to the EH, and the rest of the manifold (the other portions) would be described by different metrics. Is this moreless what you meant?

 

[PeterDonis]

You must be referring to the global topology when you say "geometry".

Not *just* the global topology. There could be many other geometries that would have the same global topology as the one I'm describing (just as, for example, a sphere and an irregularly shaped blob of genus 0 have the same global topology, but they are not the same geometry). I mean to include actual physical metrical relationships (distances and times) in "geometry". I just wanted to make clear that I'm not specifying any particular system of coordinates (or systems, if you want different ones for each region) in which to describe the geometry. I'm trying to focus attention on the physical invariants--the curvatures.

GR is considered a "local geometry" theory, and in this sense it wouldn't constrain the global topology of the manifold.

In the general sense, this is true. But particular solutions in GR do specify the global topology. (At least, they usually do; I can't say for certain that they *always* do, since I don't know enough about all the possible solutions. But certainly the geometry I'm describing is meant to include a specification of the global topology.)

According to this , you have a point that the spacetime geometry(global topology) of the manifold would not be described by a single solution (metric) of the GR equations. Instead each patch would have a different metric.

As I noted above, the geometry is more than just the global topology, so read what follows with that caveat in mind.

I would rather say that each patch has a different "local geometry"--a different pattern of curvature from event to event within the patch. For each patch, there are multiple ways of writing the metric, using different coordinates, so the word "metric" is ambiguous. If by "metric" you mean just a general reference to the fact that we are including metrical relationships (distances and times) in the "geometry", and are not intending to specify any particular coordinate system in which to write the metric, then I agree with what you're saying.

So the original Schwarzschild metric would only describe the patch up to the EH, and the rest of the manifold (the other portions) would be described by different metrics. Is this moreless what you meant?

Well, the Schwarzschild interior geometry (or "metric" in the general sense I gave above) is also a "Schwarzschild metric", so both region I and region II are described by a Schwarzschild metric in that sense. But in general, and with caveats as above, yes, that's more or less what I was getting at.

 

[TrickyDicky]

Not *just* the global topology. There could be many other geometries that would have the same global topology as the one I'm describing (just as, for example, a sphere and an irregularly shaped blob of genus 0 have the same global topology, but they are not the same geometry). I mean to include actual physical metrical relationships (distances and times) in "geometry". I just wanted to make clear that I'm not specifying any particular system of coordinates (or systems, if you want different ones for each region) in which to describe the geometry. I'm trying to focus attention on the physical invariants--the curvatures.

Then what you call geometry is what the metric (the line element) determines and what I called "local geometry", independently of the coordinates used.

In the general sense, this is true. But particular solutions in GR do specify the global topology. (At least, they usually do; I can't say for certain that they *always* do, since I don't know enough about all the possible solutions. But certainly the geometry I'm describing is meant to include a specification of the global topology.)

How is that specification implemented? Are you referring to the "unphysical" Kruskal metric?

 

[PeterDonis]

Then what you call geometry is what the metric (the line element) determines and what I called "local geometry", independently of the coordinates used.

Ok, I understand your terminology now and how it relates to mine. The only minor quibble I would have is that I would say the line element "describes" the local geometry in terms of a particular set of coordinates, not that the line element "determines" the geometry. What "determines" the geometry is the curvature: the physical invariants. The line element describes how coordinate differentials translate to actual physical distances and times, which is a consequence of the curvature.

How is that specification implemented? Are you referring to the "unphysical" Kruskal metric?

Not in the case I was discussing, because the spacetime I've been discussing, with the collapsing matter in it, has no extension to the full Kruskal spacetime; the three regions I described (I, II, and C) are the entire spacetime. The "unphysical" Kruskal spacetime, as I noted in an earlier post, assumes that you can have a curved spacetime with no matter anywhere, which everyone appears to agree is not physically reasonable.

As far as the specification of the global topology, I haven't specified it explicitly, but I believe it is implicitly specified by my description of the three regions and how they're stitched together. I believe the key things to note are that the "exterior" vacuum region is asymptotically flat (so the "topology at infinity" is the same as that of Minkowski spacetime); that the "interior" vacuum region is bounded by the event horizon and the final singularity, and is continuous between them; that the non-vacuum region has the topology of closed FRW spacetime (i.e., a spatial 3-sphere x a timelike line), and that the "stitching" across each boundary between regions is continuous. I realize this is hand-wavy; my topology-fu is not very strong, but hopefully I haven't misdescribed things too badly. I welcome input and/or correction from experts, however.

 

[PAllen]

Ok, good.



Can you be more specific about the argument for this, or point me to a reference? Just looking at the Kruskal diagram, it seems like a worldline from region IV could go into either region I or region III, just as infalling worldlines from both region I and region III can reach region II. So there are two possible types of free-falling trajectories, emerging from the white hole singularity and ending at the black hole singularity: either region IV -> region I -> region II, or region IV -> region III -> region II. It looks to me like these are two distinct families of trajectories.

My mistake. I reviewed some material on this, and I had just confused myself about the topology of the maximally extended geometry.

 

[TrickyDicky]

Ok, I understand your terminology now and how it relates to mine. The only minor quibble I would have is that I would say the line element "describes" the local geometry in terms of a particular set of coordinates, not that the line element "determines" the geometry. What "determines" the geometry is the curvature: the physical invariants. The line element describes how coordinate differentials translate to actual physical distances and times, which is a consequence of the curvature.

In fact n GR the metric determines the curvature thru the Levi-Civita connection that preserves the metric and is symmetric (so no contribution from torsion) and therefore if the Riemann curvature is uniquely determined by the metric and as you say the curvature determines the geometry, the only logic conclusion is that the metric determines the geometry .
Now we need to describe the metric somehow and the usual way is to use the line element that inevitably has to be in terms of a particular set of coordinates, but it doesn't mean a certain coordinate system is privileged, there is freedom to change the coordinate system.
So it is not the particular coordinates of a specific line element that determines the geometry, it is just that the metric is represented by a line element that has to be expressed in terms of diferent coordinate systems. The fact that the GR equations are expressed in terms of tensors assures that the specific coordinates used don't determine the physics.

From Hartle's Gravity: " A geometry is specified by the line element... The form of the line element for a geometry varies from coordinate system to coordinate system"

 

[PeterDonis]

In fact n GR the metric determines the curvature thru the Levi-Civita connection that preserves the metric and is symmetric (so no contribution from torsion) and therefore if the Riemann curvature is uniquely determined by the metric and as you say the curvature determines the geometry, the only logic conclusion is that the metric determines the geometry.

This may be just a difference in terminology again, but just to be clear about where I'm coming from, when I say that curvature "determines" the geometry (including the metric), I mean that physically, the curvature is what is directly *caused* by the source of gravity, which is matter and energy (the stress-energy tensor). More precisely, the Ricci tensor, which describes part of the curvature, is directly caused by stress-energy, and the other part of the curvature, described by the Weyl tensor, is caused by curvature propagating from one region of spacetime to another.

Mathematically, you are correct that if you know the metric, and you specify that the connection has no torsion, you can compute a unique Riemann curvature tensor. But to the best of my knowledge, it's equally true that if you know the Riemann curvature tensor everywhere, and you specify that the connection associated with it has no torsion, you can compute the metric. So I guess we could both be right, because ultimately what I'm calling "curvature" and what you're calling the "metric" could just be different ways of describing the same thing.

From Hartle's Gravity: " A geometry is specified by the line element... The form of the line element for a geometry varies from coordinate system to coordinate system"

No argument here.