Finkelstein's unidirectional membrane paper

[bcrowell]

Finkelstein's "unidirectional membrane" paper

The "unidirectional membrane" interpretation of black-hole event horizons originated with this paper:

Finkelstein, Phys. Rev. 110, 965–967 (1958), "Past-Future Asymmetry of the Gravitational Field of a Point Particle," downloadable from his web page at https://www.physics.gatech.edu/user/david-finkelstein

The basic idea is that if you impose a bunch of reasonable conditions on the Schwarzschild spacetime, and try to eliminate all coordinate singularities, you have to break time-reversal symmetry. This was a point that I hadn't understood properly before -- I'd thought that the Schwarzschild spacetime was time-reversal symmetric, as it appears to be when you write the metric in the Schwarzschild coordinates.

I have a few questions about the paper:

1. He seems to be claiming uniqueness of the solution, but I don't see where he proves that...? He uses the term "analytic," and certainly for an analytic function defined in part of the complex plane, any analytic extension to the whole plane is unique. But this is calculus on a manifold, so ...?

2. He describes two classes of spacetimes that differ by time reversal, and speculates that "it is possible that the gravitational equations imply that all particles in one universe belong to the same class." Was his conjecture right? He also speculates about particle-antiparticle interpretations...was there any validity to that?

3. He ignores negative-mass solutions. Is there any other physical ground for ignoring them, besides the fact that they would violate energy conditions? I guess they couldn't form by collapse of known forms of matter. Could they conceivably have been produced in the Big Bang?

Thanks in advance!

-Ben

 

[Bill_K]

you have to break time-reversal symmetry

Finkelstein's coordinates break time-reversal symmetry, but the complete analytic extension of Schwarzschild is given by Kruskal coordinates, which are time symmetric.

He seems to be claiming uniqueness of the solution,

Solutions of elliptic equations have to be analytic, but solutions of hyperbolic ones such as Einstein's equations do not. Discontinuities can occur along the characteristics, which in our case are light rays. For example there could be matter inside the hole which does not escape, which would modify the inner solution. When talking about extensions we restrict ourselves to analytic extensions, otherwise the extension would not be unique.

He ignores negative-mass solutions.

Aside from other arguments, the negative mass Schwarzschild solution has a naked singularity at r = 0 and cannot be extended further.

 

[TrickyDicky]

The "unidirectional membrane" interpretation of black-hole event horizons originated with this paper:

Finkelstein, Phys. Rev. 110, 965–967 (1958), "Past-Future Asymmetry of the Gravitational Field of a Point Particle," downloadable from his web page at https://www.physics.gatech.edu/user/david-finkelstein

The basic idea is that if you impose a bunch of reasonable conditions on the Schwarzschild spacetime, and try to eliminate all coordinate singularities, you have to break time-reversal symmetry. This was a point that I hadn't understood properly before -- I'd thought that the Schwarzschild spacetime was time-reversal symmetric, as it appears to be when you write the metric in the Schwarzschild coordinates.

Time-reversal symmetry is a coordinate-independent feature of static manifolds such as Schwarzschild's, its irrotational timelike killing vector assures this since imposes that the Lie derivative of the metric wrt this Killing vector field vanishes, and this condition is expressed in covariant form.
So how could Finkelstein possibly want to break that symmetry? He would have to change differential geometry rules first, wouldn't he?

 

[bcrowell]

Finkelstein's coordinates break time-reversal symmetry, but the complete analytic extension of Schwarzschild is given by Kruskal coordinates, which are time symmetric.

I see...maybe...So did I mischaracterize Finkelstein's result, or did he mischaracterize it, or ...?

 

[Bill_K]

He mentions it as a "Note added in proof" at the end of his paper.

 

[bcrowell]

He mentions it as a "Note added in proof" at the end of his paper.

"Note added in proof: Schild points out that M is still incomplete, it possesses a nonterminating geodesic of finite length in one direction. Kruskal has sketched for me a manifold M* that is complete and contains M. M* is time-symmetric and violates one of the conditions on M: it does not have the topological structure of all of 4-space less a line. Kruskal obtained M* some years ago (unpublished)."

So visualizing this on a conformal diagram like http://www.pitt.edu/~jdnorton/teaching/HPS_0410/chapters/black_holes_picture/index.html", did Finkelstein basically find possibilities like I+II and I+IV?

What's confusing me about the conformal diagram is the interpretation of the two singularities. The white hole singularity is in the past light cone of every event, so it must have existed for an infinite time in the past. Also, it can emit photons that can be observed by any observer arbitrarily far in the future, so it seems like it must exist for an infinite time in the future. The same seems to be true for the black hole singularity. But then that doesn't make sense, because you can draw a spacelike hypersurface that separates them from each other...? I guess this would explain the part of the note added in proof about the topology. But I don't understand how it can make sense that each singularity is permanent, but you can separate them with a spacelike surface ??

-Ben

[EDIT] Ah, I see. I think this explains it: http://en.wikipedia.org/wiki/White_hole#Origin

 

[PeterDonis]

What's confusing me about the conformal diagram is the interpretation of the two singularities. The white hole singularity is in the past light cone of every event, so it must have existed for an infinite time in the past. Also, it can emit photons that can be observed by any observer arbitrarily far in the future, so it seems like it must exist for an infinite time in the future. The same seems to be true for the black hole singularity.

Are you sure these deductions are valid? I agree they would be in a "normal" flat spacetime, but this isn't one. The presence of the horizons and the second exterior region "extends" the spacetime so that the white and black hole singularities can be spacelike and still be in the past/future light cones of every event. In a "normal" flat spacetime, that wouldn't be possible; only a timelike object can be in the past/future light cones of every event. That may be what the bit about the topology was referring to.

 

[bcrowell]

Are you sure these deductions are valid? I agree they would be in a "normal" flat spacetime, but this isn't one. The presence of the horizons and the second exterior region "extends" the spacetime so that the white and black hole singularities can be spacelike and still be in the past/future light cones of every event. In a "normal" flat spacetime, that wouldn't be possible; only a timelike object can be in the past/future light cones of every event. That may be what the bit about the topology was referring to.

Good point. In the conformal diagram, the singularities are horizontal, right? I guess operationally it might not make sense to talk about how long a singularity exists, since you can't stand close to it with a clock. Maybe it makes more sense to say that the white hole's horizon has existed for infinite time in the past, and the black hole's will exist for infinite time in the future? But even that doesn't quite make sense, because the horizons are lightlike. Gah, this makes my head hurt.

 

[PeterDonis]

Good point. In the conformal diagram, the singularities are horizontal, right?

Yes. This page has a decent image:

http://www.pitt.edu/~jdnorton/teaching/HPS_0410/chapters/black_holes_picture/index.html

I guess operationally it might not make sense to talk about how long a singularity exists, since you can't stand close to it with a clock.

Yes, since both the white and black hole singularities are spacelike, not timelike, it makes no sense (at least, not to me) to talk about "how long" they exist.

Maybe it makes more sense to say that the white hole's horizon has existed for infinite time in the past, and the black hole's will exist for infinite time in the future? But even that doesn't quite make sense, because the horizons are lightlike. Gah, this makes my head hurt.

The best way I've found to look at it comes from reading Hawking and Ellis a while back. (At least, I think that's the original source of the terms I'm about to use.) A "black hole" is present in a spacetime if the causal past of future null infinity is not the entire spacetime. The "future horizon" (or black hole horizon) is then the boundary of the region that is *not* in the causal past of future null infinity. Similarly, a "white hole" is present if the causal future of past null infinity is not the entire spacetime, and the "past horizon" is the boundary of the region that is not in the causal future of past null infinity. (One of the nice things about conformal diagrams is that all this is obvious just from looking at them.)

So really both singularities, and both horizons, are simply not "in" the region where it makes sense to talk about "how long" something exists, because that term implies (at least to me) that you're in a region of spacetime where "how long" can be any real number, of any magnitude, which implies that you're in a region which can causally communicate with both future and past infinity. The singularities and horizons are in a separate region of spacetime where the physics doesn't quite match our intuitions about space and time.

 

[bcrowell]

Thanks, Peter, that's very helpful!

OK, here's a far-out question. The maximally extended Schwarzschild spacetime has two copies of Minkowski space in it, and you can't get from one to the other. However, an event inside the black hole's event horizon can have both copies of Minkowski space in its light cone. So does this mean that if you dropped through a black hole's event horizon, you could meet another suicidal explorer who had dropper through from the other universe? Or is this an unrealistic feature of black holes that didn't form by gravitational collapse?

Also, isn't the central axis of the diagram an axis of rotational symmetry...? So in that case, why aren't the two copies of Minkowski space actually the same?

 

[yuiop]

This image

[PLAIN]http://www.mathpages.com/rr/s6-04/6-04_files/image043.gif

casts an interesting angle on the "unidirectional" aspect of the membrane. At time -35 the particles passes up towards the event horizon and out through the horizon and at a later time passes back through the horizon, while all the while its proper time advances in the positive direction. This does not appear unidirectional at all. Now, while the passing out event is from a white hole and the passing back in event is into a black hole, the diagram makes the them appear as one object and the events are only separated by time. However there is the issue of time and space swapping roles below the event horizon so the the particle moving outward is spatially separated from the the particle moving inwards. Note that we have the same particle moving outward is simultaneous with itself moving inward so the single particle is in two places at the same time. (Below the event horizon, two coordinates with the same radial coordinate are simultaneous.) Also note that we can draw the trajectories of other particles on the same chart so that we can have other particles going outward in the same region as particles going inwards, so there is no clear demarcation between black hole and white hole here. You cannot point to one region of the chart and say "the white hole is here" or "the black hole is there" because they are completely arbitrary. I can for example draw the path of an outgoing particle starting out at coordinate time +30 which crosses the path of the incoming particle in the chart.

 

[PeterDonis]

isn't the central axis of the diagram an axis of rotational symmetry...? So in that case, why aren't the two copies of Minkowski space actually the same?

I'll answer this one first because the answer to your other question kind of builds on the answer to this one. No, the central axis of the diagram is *not* an axis of rotational symmetry in the case of the maximally extended Schwarzschild spacetime. It is in the case of normal Minkowski spacetime, because the central axis is the worldline of the point r = 0 (the spatial origin). In the extended Schwarzschild spacetime, though, the central axis does not correspond to a single value of r. The point on the central axis where the two horizons cross is r = 2M (since that's the value of r everywhere on both horizons). As you go up or down the central axis from that central crossing point, the value of r decreases to zero. So actually the "axis of rotational symmetry" of the spacetime is represented by the two horizontal singularity lines! (At least, I'm pretty sure that's right, although the idea of two spacelike lines representing the axis of rotational symmetry makes my head hurt too.)

an event inside the black hole's event horizon can have both copies of Minkowski space in its light cone. So does this mean that if you dropped through a black hole's event horizon, you could meet another suicidal explorer who had dropper through from the other universe? Or is this an unrealistic feature of black holes that didn't form by gravitational collapse?

I would lean towards the latter, because the spacetime of a black hole that forms by gravitational collapse does not have the same conformal diagram as the maximally extended Schwarzschild spacetime, as you can see by looking at the example of one on the page I linked to in my earlier post. A gravitational collapse diagram only includes the first exterior region (region I) and the black hole region (region II), plus a non-vacuum portion representing the collapsing matter. Also, as you can see from the diagram, the central axis in the gravitational collapse case *is* an axis of rotational symmetry. (Actually it's a "left axis" in the example, which actually makes more sense to me, since the radial coordinate has a minimum value at zero, so a strictly correct conformal diagram for, say, Minkowski spacetime, would only have a region to the right of the "axis", since a region to the left of the "axis", such as appears in the diagram for Minkowski spacetime on the page I linked to, would represent negative values of r, which don't exist). However, there's still a twist: the horizontal singularity line (the upper left of the diagram) is *also* an axis of rotational symmetry! (It also represents r = 0, after the collapsing matter has vanished into the singularity.)

 

[PeterDonis]

Note that we have the same particle moving outward is simultaneous with itself moving inward so the single particle is in two places at the same time. (Below the event horizon, two coordinates with the same radial coordinate are simultaneous.)

This can't be correct, because for two events to be "simultaneous" (I think you meant to say "two events with the same radial coordinate are simultaneous"), they must be spacelike separated, and the two events in question (one inside the horizon at a given r-value on the "outgoing" leg, and the other inside the horizon at a given r-value on the "ingoing" leg) are timelike separated. This indicates that things are not as simple as your diagram makes them appear to be.

If you draw the trajectories you are talking about on the maximally extended conformal diagram, you will see that the horizon the object passes on its outgoing leg is *not* the same as the horizon it passes on its ingoing leg. The first horizon goes up and to the left at 45 degrees; the second goes up and to the right at 45 degrees. Similarly, the "r = 0" singularity line the object emerges from is *not* the same as the "r = 0" singularity the object goes into. The first singularity is at the bottom of the conformal diagram, and the second is at the top.

So your diagram leaves something out: there should be a "break" in the bottom axis (the r = 0 line) to indicate the separation between the two horizons.

 

[PeterDonis]

there should be a "break" in the bottom axis (the r = 0 line) to indicate the separation between the two horizons.

Oops, meant to say "between the two singularities".

 

[yuiop]

(I think you meant to say "two events with the same radial coordinate are simultaneous")

Yes, that is a better way of phrasing what I meant. :)

This can't be correct, because for two events to be "simultaneous" ..., they must be spacelike separated, and the two events in question (one inside the horizon at a given r-value on the "outgoing" leg, and the other inside the horizon at a given r-value on the "ingoing" leg) are timelike separated. This indicates that things are not as simple as your diagram makes them appear to be.

Below the event horizon, spacelike becomes timelike and vice versa. In the diagram below, events B and D (or B-C or C-D) are timelike separated and events A and E are spacelike separated. Agree?

Of course this leaves a problem because Wikipedia http://en.wikipedia.org/wiki/Spacetime states

For a particle traveling through space at less than the speed of light, any two events which occur to or by the particle must be separated by a time-like interval.

and we have a contradiction here. This implies that the particle rising out of the white hole is not the same as the particle entering the black hole, which seems reasonable, but there is the siren call of the particle's proper time progressing smoothly and continuously along the entire path. Lesson is, proper time progressing smoothly and continuously is not proof in itself of anything.

If you draw the trajectories you are talking about on the maximally extended conformal diagram, you will see that the horizon the object passes on its outgoing leg is *not* the same as the horizon it passes on its ingoing leg. The first horizon goes up and to the left at 45 degrees; the second goes up and to the right at 45 degrees.

That does not prove anything. The first horizon has coordinates r= 2m and t= -infinity and the second horizon has coordinates r= 2m and t= +infinity. They only differ in time, but not place. Of course that depends on how you interpret time and space. Above and below the event horizon timelike and spacelike swap roles, but happens exactly at the event horizon? Undefined I guess?

Also consider this KS chart:

The green r=constant line goes first up and to the left and then curves up and two the right, just like the two event horizons you mentioned so while its direction and time coordinate is changing, its spatial location does not. The two event horizons you mentioned are just an extreme version of the green r= constant curve.

So your diagram leaves something out: there should be a "break" in the bottom axis (the r = 0 line) to indicate the separation between the two horizons.

In the first chart at the top of this post, I have added the "break" between the two singularities that you desire. It does not fix anything. For example I have placed the break at "now" (Year 2011). Now consider the particle that has its apogee (event C) at year 1911. It rises and falls back into the white hole contradicting the fact that particles are only supposed to exit from white holes. Other trajectories have particles rising out of the black hole contradicting the fact that particles can only enter but not leave a black hole. The only way to cure these contradictions is to insist that all particles have their apogees at year 2011, but of course we know from experience that that is not realistic.

Any rational person looking deep into the scenario (and not starting out with a "Finkelstein said it must be right,so I won't look too deep into it" attitude) would agree there are multiple problems with his interpretation of black hole interiors. I can offer an alternative interpretation which makes all the problems go away, but the rules of PF forbid me from challenging the conventional interpretation, even though I or others might learn something from the discussion. As far as I know Finkelstein is one of the few well known people from the golden relativity era that is still alive and active in physics and it would be really cool if he came here to discuss it.

P.S. Have a look at the red and orange light cones I put in the top diagram. They imply light can travel directly from the white hole to the black hole (or vice versa) without crossing an event horizon.

 

[PeterDonis]

yuiop, I should have been a bit clearer about one point in my previous response: your diagrams are *not* coordinate charts, and you simply can't use them to reason about how events are related the way you can with (properly drawn) coordinate charts. In particular, your diagram does not properly represent the causal structure of the spacetime (which events are timelike/spacelike related).

In the diagram below, events B and D (or B-C or C-D) are timelike separated and events A and E are spacelike separated. Agree?

No, I do not. Both pairs are timelike separated. As you quoted from the Wikipedia page, both pairs of events lie on the same worldline, which is the worldline of a timelike object. Any pair of events lying on the worldline of the same timelike object must be timelike separated.

What is going on here is, as I said in my previous post, there are two *separate* regions in the spacetime which have r < 2M (i.e., two separate "interior" regions inside the horizon). Event A is in the first region, and event E is in the second. Thus, they are timelike separated even though they have the same r coordinates. If they were both in the same interior region, then you would be correct that they would have to be spacelike separated (since r is a timelike coordinate inside the horizon). But they're not.

The reason your diagram has difficulty representing this is that, as I said above, your diagram is not a proper coordinate chart, and it does not properly represent the causal structure of the spacetime. If you want to see how things are actually working, try drawing the "Year 2011" worldline represented in your top diagram on the Kruskal chart (I pick that one because your diagram shows that one emerging from the white hole and going back into the black hole, which is correct). You will see that the analogue of event A on that worldline is in region IV while the analogue of event E is in region II. You will also note that there are two "copies" of each r = constant curve; for r < 2M, one copy is in region IV and the other is in region II. So two events can both have the same r coordinate < 2M but lie on different r = constant curves, in different regions.

Now try drawing the 1911 and 2111 curves, also, on the Kruskal chart. You will see that there is *no* way to have a curve both emerge from and go back into the *same* "hole" (white or black). All three curves *have* to emerge from the white hole and go back into the black hole. So they all work the same as the 2011 curve did above. That's how A and E can be timelike separated.

This implies that the particle rising out of the white hole is not the same as the particle entering the black hole

No, it doesn't. It only implies, as I said above, that the white hole and the black hole are separate regions of the spacetime, each with their own copies of every r = constant curve for r values < 2M. For r > 2M, the particle is in region I the whole time, which is why your deduction that events B and D are timelike separated is correct (if B were in region I but D were in region III of the Kruskal chart--in the other exterior "universe"--then that deduction would *not* hold; the two events would be spacelike separated).

That does not prove anything. The first horizon has coordinates r= 2m and t= -infinity and the second horizon has coordinates r= 2m and t= +infinity. They only differ in time, but not place.

Incorrect; t = minus infinity, r = 2M corresponds to the line going up and to the left at 45 degrees, while t = plus infinity, r = 2M corresponds to the line going up and to the right at 45 degrees. So they really are two different horizons; they do differ in "place" as well as in time. Schwarzschild coordinates do not represent this properly because of the coordinate singularity at r = 2M; that's one of the reasons why Kruskal coordinates are useful.

The two event horizons you mentioned are just an extreme version of the green r= constant curve.

Yes, in a sense they are. So what? If you look at a chart of Minkowski spacetime with a pair of "Rindler horizon" light rays (just like the r = 2M lines on the Kruskal chart), and a set of hyperbolas representing accelerating observers all sharing the same Rindler horizon, the pair of Rindler horizon light rays is similarly the limit of the accelerating hyperbolas. That doesn't make the two light rays the same.

In the first chart at the top of this post, I have added the "break" between the two singularities that you desire.

That is *not* where the "break" belongs, except if you remove the 1911 and 2111 curves and only look at the 2011 curve. For that one, yes, the "break" is roughly where I would have put it. But remember that "coordinate time" goes to infinity at both ends, so if you want to add 1911 and 2111 trajectories to the diagram, given that the 2011 trajectories are correct, then you would have to have all three "coordinate time" curves going to the far left and right ends of the chart at r = 2M, and they would therefore have to straddle the break just as the 2011 curve does. (For example, your chart has the r = 2M, t = plus infinity point of the 1911 "coordinate time" line to the *left* of the r = 2M, t = minus infinity point of the 2111 "coordinate time" line. That's not possible; t = plus infinity can't occur before t = minus infinity.)

Again, I suggest that you try to draw all three of these trajectories (1911, 2011, and 2111) on the Kruskal chart. You will not be able to do it in such a way that the 1911 one both emerges from and goes back into the white hole, nor will the 2111 one be able to both emerge from and go back into the black hole.

Any rational person looking deep into the scenario (and not starting out with a "Finkelstein said it must be right,so I won't look too deep into it" attitude) would agree there are multiple problems with his interpretation of black hole interiors.

The only problem I'm aware of is that he thought his coordinate chart covered the entire spacetime, but then found out it didn't (as he notes in his "note added in proof"). But as a representation of just regions I and II, the "ingoing" Finkelstein chart works fine.

I can offer an alternative interpretation which makes all the problems go away, but the rules of PF forbid me from challenging the conventional interpretation, even though I or others might learn something from the discussion.

Isn't there a forum for speculative posts? Post it there and I, for one, will be glad to read it and comment.

As far as I know Finkelstein is one of the few well known people from the golden relativity era that is still alive and active in physics and it would be really cool if he came here to discuss it.

I can't guarantee that, of course. But I agree it would be cool.

P.S. Have a look at the red and orange light cones I put in the top diagram. They imply light can travel directly from the white hole to the black hole (or vice versa) without crossing an event horizon.

How so? I don't understand. (Not that it would prove anything anyway, since as I've already noted, your diagram isn't a proper coordinate chart and doesn't represent the causal structure of the spacetime correctly.)

 

[yuiop]

yuiop, I should have been a bit clearer about one point in my previous response: your diagrams are *not* coordinate charts, and you simply can't use them to reason about how events are related the way you can with (properly drawn) coordinate charts.

It is not my diagram, it is a diagram by Prof Brown on his mathpages website. See http://www.mathpages.com/rr/s6-04/6-04.htm

His diagram is consistent with the standard Schwarzschild coordinate chart. Also see this diagram from MTW:

which is also the standard Schwarzschild coordinate chart. All I have done is added some sketch notes to the standard diagrams.

Now try drawing the 1911 and 2111 curves, also, on the Kruskal chart. You will see that there is *no* way to have a curve both emerge from and go back into the *same* "hole" (white or black). All three curves *have* to emerge from the white hole and go back into the black hole. So they all work the same as the 2011 curve did above. That's how A and E can be timelike separated.

The Kruskal-Szekeres chart is derived from Schwarzschild cordinates by substituting new coordinates into the the Schwarzschild metric. If the Schwarzschild solution is wrong then so is the KS solution. Also, any valid coordinate transformation should have a one to one relationship between events in the two coordinate systems. Any unique event on a coordinate system should have one and only one corresponding event on the transformed system. KS coordinates introduce a whole new parallel universe that was not there in the original Schwarzschild coordinates.

No, it doesn't. It only implies, as I said above, that the white hole and the black hole are separate regions of the spacetime,

The Schwarzschild coordinate system is two dimensional. One is space (the radial x coordinate) and the other is time. There is no y or z coordinate, so saying two events with the same r coordinate are in two different places does not make a lot sense on a one dimensional radial line, unless we invoke the parallel universe of course, which is what K-S have done.

Again, I suggest that you try to draw all three of these trajectories (1911, 2011, and 2111) on the Kruskal chart. You will not be able to do it in such a way that the 1911 one both emerges from and goes back into the white hole, nor will the 2111 one be able to both emerge from and go back into the black hole.

That only illustrates what I said before, there should be a one for one relationship between the two charts.

Isn't there a forum for speculative posts? Post it there and I, for one, will be glad to read it and comment.

Yes there is https://www.physicsforums.com/showthread.php?t=82301, but there are lot of hurdles to jump through to get a post in there, but I might give it a shot as time allows.

 

[bcrowell]

Yes there is https://www.physicsforums.com/showthread.php?t=82301, but there are lot of hurdles to jump through to get a post in there, but I might give it a shot as time allows.

The IR forum has been shut down: https://www.physicsforums.com/showthread.php?t=506643

 

[PeterDonis]

It is not my diagram, it is a diagram by Prof Brown on his mathpages website. See http://www.mathpages.com/rr/s6-04/6-04.htm

Understood, thanks for the source. I believe there is a similar figure somewhere in MTW. I didn't mean to imply by the phrase "your diagram" that you had originated it, just that you were using it to illustrate your points. Sorry if that caused confusion.

His diagram is consistent with the standard Schwarzschild coordinate chart.

"Consistent" in the sense that it shows curves which have relationships between coordinate/proper time and radius that are allowed by the Schwarzschild coordinate chart, yes (but with a caveat I'll get to in a moment). But that does not mean that his diagram is itself a proper coordinate chart. It isn't, and it isn't meant to be.

The caveat is that the "Schwarzschild coordinate chart" is not really a single chart. There are actually three of them. The first is the "Schwarzschild exterior chart" (I don't know if my terms are exactly the "standard" ones, but I'll try to make clear what I mean by them). It covers the region r > 2M ("region I" in the Kruskal chart). The second is what is normally referred to (more or less--again I don't know if my terms are exactly the "standard" ones) as the "Schwarzschild interior chart", which covers the region r < 2M that can be reached by freely falling through the horizon from the exterior region ("region II" in the Kruskal chart, the "black hole" region). The third could also be called a "Schwarzschild interior chart", but it covers the region r < 2M from which outgoing freely falling observers can cross into the exterior region ("region IV" in the Kruskal chart, the "white hole" region). I don't know that I've seen this third chart discussed explicitly, but it should be obvious that it is a valid chart and that it is distinct from the other two. (If any experts on the forum want to weigh in on this, please do.)

So when you say that the diagram is "consistent with the Schwarzschild coordinate chart", that's only true if you put a "break" in the region r < 2M, as I said before, to reflect the fact that that region of the diagram is really two separate, distinct regions that do not "touch" or overlap; or, equivalently, that region does not represent a single Schwarzschild interior chart, but two disjoint ones, so the disjointness needs to be included on the diagram to make it fully consistent. And, of course, when you do that the issues you're raising go away, because it's clear that the "outgoing" and "ingoing" portions of the worldlines pass through two separate interior regions.

You'll note, by the way, that Prof. Brown's page, which you linked to, discusses the two interior regions in the paragraphs after the diagram you refer to appears. What I'm saying is consistent with what he says. The only possible curve ball he throws is this statement:

Hence if we observe objects falling into the inner region, and other object emerging from the inner region, we seem forced to conclude that there are two physically distinct inner regions, or else that there exist closed spacetime loops if we insist on a single interior region.

I haven't checked his later discussion of black holes and cosmology to see whether he talks about closed timelike curves (which is what I think he means by "closed spacetime loops"), but I believe it's been shown that there are no CTCs in Schwarzschild spacetime.

Also see this diagram from MTW:

You'll note that the text accompanying this diagram, and the correspondence between parts (a) and (b), makes it clear that in the (a) diagram, the interior region r < 2M corresponds to region II on the Kruskal chart, and that region only. It does *not* correspond to region IV. In other words, in the (a) diagram in MTW, the interior region is the "black hole" region, and that region only. If you tried to draw the entire worldline represented on Prof. Brown's diagram in part (a) of the MTW diagram, you would not be able to do it; the "outgoing" portion coming from the white hole can't be put anywhere on part (a) of the MTW diagram.

The Kruskal-Szekeres chart is derived from Schwarzschild cordinates by substituting new coordinates into the the Schwarzschild metric. If the Schwarzschild solution is wrong then so is the KS solution.

I didn't say the Schwarzschild solution was "wrong". As I clarified above, the Schwarszschild "solution" or "chart" is not a single chart, and none of the three charts that can be called a "Schwarzschild chart" covers the entire spacetime. The Kruskal chart *does* cover the entire spacetime, and makes clear the relationship between the different Schwarzschild charts. It's all consistent.

Also, any valid coordinate transformation should have a one to one relationship between events in the two coordinate systems. Any unique event on a coordinate system should have one and only one corresponding event on the transformed system.

This is true, and it holds for each individual Schwarzschild chart. I made clear which region of the Kruskal chart each Schwarzschild chart covers above. Within each of those regions, there is indeed a one-to-one relationship as you describe. The *appearance* of a two-to-one relationship is only because you are not recognizing that region II and region IV are two separate regions, each with its own Schwarzschild chart.

KS coordinates introduce a whole new parallel universe that was not there in the original Schwarzschild coordinates.

If you mean region III, you are correct. That region would actually require a *fourth* Schwarzschild chart (a second "exterior" chart). Mathematically, I believe this region has to be there to make the full analytic extension work right (but my math-fu is not good enough to give a proof of this; maybe one of the experts on the forum can give more detail about how this works). Physically, as far as I know, neither that region nor region IV is present in an spacetime that anyone believes is applicable to the real universe; in the spacetime of a black hole formed from a collapsing star, for instance, only region I and region II are present (plus a non-vacuum portion representing the collapsing matter). For regions III and IV to actually be there physically, the entire spacetime would have to be a vacuum spacetime (i.e., no actual matter present anywhere) but still somehow have a black hole (and white hole) present. As far as I know, nobody believes this is actually physically possible; there *has* to be some matter somewhere for a black hole to form, and once you have a non-vacuum region, you have the spacetime I just referred to, where the only vacuum regions are regions I and II. So in any actual physical spacetime, there would be no "parallel universe" and no "white hole".

The Schwarzschild coordinate system is two dimensional. One is space (the radial x coordinate) and the other is time.

Actually, the full coordinate system also has two angular coordinates, theta and phi. Those are often left out because the spacetime is spherically symmetric, so nothing of physical interest depends on those coordinates. Their presence does not affect the issues we've been discussing, because there are certainly such things as radial geodesics that have constant values of the angular coordinates for their entire length; basically we've just been restricting attention to those.

There is no y or z coordinate, so saying two events with the same r coordinate are in two different places does not make a lot sense on a one dimensional radial line, unless we invoke the parallel universe of course, which is what K-S have done.

As I noted above, mathematically, regions III and IV are there in the maximal analytic extension, and are separate from regions I and II. But in any actual physical spacetime, they would not be there, because physically it's not reasonable to have a curved spacetime with no matter present anywhere (as I noted above). So in an actual physical spacetime, you are correct; there is only one interior region, so any value of the r coordinate corresponds to only one "place". But also, of course, in this case there is no "white hole", so there are no freely falling worldlines that pass through the same value of r twice.

 

[yuiop]

As far as I know, nobody believes this is actually physically possible; there *has* to be some matter somewhere for a black hole to form, and once you have a non-vacuum region, you have the spacetime I just referred to, where the only vacuum regions are regions I and II. So in any actual physical spacetime, there would be no "parallel universe" and no "white hole".

Well I am glad you think that regions III and IV are unphysical, but that means we have no need for "maximally extended spacetime".

Another problem with the maximal KS chart is illustrated below (in your favourite chart):

Mass or light from the white hole (region IV) can move freely into our universe and mass and light from the other universe (region III) can also move freely into the black hole (region II) in our universe. Matter or light cannot pass from our universe to any part of the "other universe" so over time, if any mass is in the other universe it will end up in our universe and gradually our universe will be inexplicably increasing in mass and energy. Has anyone detected that?

Actually, the full coordinate system also has two angular coordinates, theta and phi. Those are often left out because the spacetime is spherically symmetric, so nothing of physical interest depends on those coordinates. Their presence does not affect the issues we've been discussing, because there are certainly such things as radial geodesics that have constant values of the angular coordinates for their entire length; basically we've just been restricting attention to those.

Fully agree and thanks for picking me up on that fine point.

As you expressed an interest only I will show you an interesting mathematical oddity that falls right out of the Schwarzschild equations (courtesy of Prof Brown)

Copy and paste the quoted function below (based on Prof Brown's equation ref http://www.mathpages.com/rr/s6-04/6-04.htm )

realonly((ax/2+2)*sqrt(ax/2-1)*acos(x*2/ax-1)+(ax/2)*sqrt(ax/2-1)*sin(acos(x*2/ax-1))+2*ln((sqrt(ax/2-1)+tan(acos(x*2/ax-1)/2))/(sqrt(ax/2-1)-tan(acos(x*2/ax-1)/2))))

into function box (f) of this online complex function plotter java applet: http://www.digitalhermit.com/math/calc/index.html#Math ref: Complex representation: rectangular form and then type (2,t) in function box (g) and adjust the zoom to 24 or less. Drag the small box marked a (the apogee) to a value of x=+3. This is the normal in-falling curve for a particle falling towards a black hole. Note that there is no curve below the event horizon (the red vertical line) because the "realonly" function only plots curves with a zero imaginary component and the path below the horizon is complex. (This is fixed by prof Brown by subtracting and arbitrary imaginary constant or equivalently arbitrarily adding an absolute function to the log part of his equation. OK, now drag the small (a) box so that the apogee is at x=1 (below the event horizon) and observe where the real valued curve is now. That might be a bit of an eye opener. To play around with the graph, change "realonly" to "re" or "im" to see the real and imaginary parts of the curve independently.

Note that when the apogee is above the event horizon, that:

1)the curve above the apogee is imaginary (so non-physical).
2)the curve below the apogee and above the EH is pure real (so physically real).
3)the curve below the event horizon is complex (probably not physical).
4)the curve for negative r is complex (probably not physical).

When the apogee is below the event horizon:

5)The curve above the event horizon is complex (probably not physical).
6)the curve above the apogee and below the EH is pure real (so physically real?).
7)the curve below the apogee and above r=0 is imaginary (so non-physical).
8)the curve for negative r is complex (probably not physical).

(Here, complex means the real and imaginary components are both non zero, imaginary means only the imaginary component is non zero and real means only the real component is non zero.)

With some analysis and thought there is enough information there to see that the Schwarzschild equation do not "absolutely require" that timelike becomes spacelike and vice versa below the event horizon. That only happens if you add arbitrary constants to make it happen. It would seem that if a person where to accept only real results and reject imaginary or complex results, they would remove the time-space swap below the event horizon and also remove the infinite density central singularity where the "laws of physics break down". However by adding arbitrary imaginary constants, they can conclude that space and time "must" swap over and that a singularity of infinite density that breaks the laws of physics "must" exist.

It is also worth noting that one of the justifications given here on PF in the past for the "maximal extended KS solution" is that
a)all wordlines should either start or end at a real singularity (black or white) or at past/future infinity. (We will gloss over that the event horizon IS at future infinity)
b)all wordline should be smooth and continuous.

That seems reasonable, but if you carefully plot worldlines in regions I,II, III, and IV on a KS chart, it does not actually achieve those objectives. I did it a long time ago and arrived at 16 different permutations of swapping time and space around and never achieved that objective (if I did it right).

 

[PeterDonis]

Well I am glad you think that regions III and IV are unphysical, but that means we have no need for "maximally extended spacetime".

Physically, I agree with you. However, the same objection applies to Finkelstein's paper referenced in the OP. He is explicitly considering an "idealized" black hole spacetime where there is no non-vacuum region anywhere. Once you are considering that, you *have* to extend the spacetime to the maximal Kruskal extension; otherwise you will have geodesics that stop after a finite length without hitting a singularity, which means that there is a further region of spacetime that your current coordinate chart does not cover. That was what Finkelstein acknowledged in his "note added in proof". (I see you comment on this particular issue at the end of your post; see my further comments below.)

Mass or light from the white hole (region IV) can move freely into our universe and mass and light from the other universe (region III) can also move freely into the black hole (region II) in our universe. Matter or light cannot pass from our universe to any part of the "other universe"

Mass or light can't pass from region I to region III, true, but it *can* pass into region II from region I. That could in principle balance out the mass or light entering region I from region IV. So I don't think this by itself is a problem.

the path below the horizon is complex. (This is fixed by prof Brown by subtracting and arbitrary imaginary constant or equivalently arbitrarily adding an absolute function to the log part of his equation.

Prof. Brown goes through all these gyrations only because he is working around the coordinate singularity at r = 2M. Doing an analytic extension to the complex plane, and then taking the real part at the end, is just a mathematical way of working around that singularity. But he could just as easily have transformed to, say, Painleve coordinates (or ingoing Finkelstein coordinates, for that matter), and dealt with the full trajectory of an infalling object without having to use any complex numbers at all.

With some analysis and thought there is enough information there to see that the Schwarzschild equation do not "absolutely require" that timelike becomes spacelike and vice versa below the event horizon.

There is a sense in which this is sort of true. If you transform to coordinates which are not singular at r = 2M, such as the ones I just mentioned (Painleve or ingoing Finkelstein), you will find, indeed, that there is not a "swap" of timelike and spacelike inside the horizon. But things still don't match one's natural intuition. With Painleve coordinates, what you find is that inside the horizon, *all four* of the coordinates are spacelike! With ingoing Finkelstein coordinates, the usual formulation has one coordinate being null (replacing the "time" coordinate), and the radial coordinate r staying spacelike inside the horizon. An alternate formulation (since it's nicer to have null lines as 45-degree lines instead of horizontal planes on a spacetime diagram) is to define a "Finkelstein time" coordinate which is timelike outside the horizon; but this coordinate *also* becomes spacelike inside the horizon, so this formulation works just like Painleve, where all four coordinates are spacelike inside the horizon.

So you're right that the structure of the spacetime around a black hole does not *force* you to adopt coordinates where "time" and "radius" swap roles, so to speak, inside the horizon. But that certainly doesn't imply, as you seem to think it does, that the central singularity at r = 0 doesn't exist. That's still there regardless of the coordinates you use.

a)all wordlines should either start or end at a real singularity (black or white) or at past/future infinity.

Yes, this is true, provided you have the correct definition of "past/future infinity". See next comment.

(We will gloss over that the event horizon IS at future infinity)

No, it isn't; the fact that the event horizon's Schwarzschild time coordinate is plus infinity does not mean it is actually, physically, at "future infinity", because Schwarzschild coordinates are singular at the horizon and so don't properly represent it. The correct definition of "future infinity" actually defines *two* such infinities: "future timelike infinity" and "future null infinity". Neither of those contain the event horizon. Look up Penrose diagrams or conformal diagrams, or see Hawking and Ellis for the gory details. "Past infinity" is defined similarly, but at the other end of time, so to speak.

Btw, I'm assuming that by "worldline" you mean "the worldline of an object with mass, or a light ray", i.e., a timelike or null curve. If not, you need to add one more infinity, "spacelike infinity", which is where spacelike lines end.

b)all wordline should be smooth and continuous.

Yep.

That seems reasonable, but if you carefully plot worldlines in regions I,II, III, and IV on a KS chart, it does not actually achieve those objectives.

You'll need to post specific counterexamples if you want to convince me of this. MTW, for one, plot a number of worldlines on KS charts, and all of them meet the objectives.

 

[pervect]

Here are some sample geodesics (some timelike, some spacelike, some null) in the schwarzschild space-time on both the schw and KS charts from MTW. They are continuous and smooth on the KS chart,and a real mess on the Schwarzschild chart.

 

[TrickyDicky]

What I don't fully understand is why is an admittedly nonphysical extended spacetime (Kruskal) used to theoretically justify the existence of a claimed physical entity (Black hole).
If that extended spacetime is not physical, which everyone seems to accept, doesn't it mean that the original Schwarzschild spacetime -the abridged version ;) - is the only one that is physical?
I'm puzzled when I see people "proving" the existence of black holes thru the Kruskal four quadrant chart and the next minute acknowledging that it is a purely nonphysical extension of the original Schwarzschild solution and that nobody seriously believes that regions III and IV of Kruskal exist. But then where is the extension of the original Schwarzschild coordinates (that only cover K-S region I ) that shows what would be the regions I and II in Kruskal? Anybody has a reference to a "physical" metric that thru a coordinate transformations from the original metric shows the equivalent to regions I and II?

 

[pervect]

What I don't fully understand is why is an admittedly nonphysical extended spacetime (Kruskal) used to theoretically justify the existence of a claimed physical entity (Black hole).
If that extended spacetime is not physical, which everyone seems to accept, doesn't it mean that the original Schwarzschild spacetime -the abridged version ;) - is the only one that is physical?
I'm puzzled when I see people "proving" the existence of black holes thru the Kruskal four quadrant chart and the next minute acknowledging that it is a purely nonphysical extension of the original Schwarzschild solution and that nobody seriously believes that regions III and IV of Kruskal exist. But then where is the extension of the original Schwarzschild coordinates (that only cover K-S region I ) that shows what would be the regions I and II in Kruskal? Anybody has a reference to a "physical" metric that thru a coordinate transformations from the original metric shows the equivalent to regions I and II?

There's quite a bit of uncertanity about what a real black hole will look like. But all the issues that I am aware of are on the inside of the event horizon, not the outside.

Birkhoff's theorem in particular gives a pretty good reason to believe that the exterior Schwarzschild solution is the one of physical interest in the exterior region.

Issues include the fact that a real black hole will rotate, and the fact that even if you managed to form one that didn't by some huge, artificial engineering effort, you'd have small imperfections - "clumps" in the matter that would spoil the perfect radial symmetry.

These imperfections aren't so much an issue under normal conditions,but when gravity gets extreme, as it does in the interior, even a small "clump" tends to grow and accumulate more matter, so the assumption of a perfectly symmetrical radial collapse, while much easier to analyze mathematically, is unlikely to actually happen. People who've studied it seem to think that the BKS class of singularities are more likely to represent what an actual black hole would look like if it could somehow be made non-rotating.

I'm not sure what you mean by "proving black holes exist", I'm taking the liberty of interpreting it as thinking of it as "you can't avoid having an event horizon if you've got enough matter in one place". Are you imagining that you can avoid this somehow? If so, how.

You might also mean proving that a singularity exists, given than an event horizon exists. The singularity theorems show this,though of course you have to assume GR is valid.

 

[PeterDonis]

If that extended spacetime is not physical, which everyone seems to accept, doesn't it mean that the original Schwarzschild spacetime -the abridged version ;) - is the only one that is physical?

Not quite. The "physical" black hole spacetime I described earlier, which includes the collapsing matter that forms the hole, contains both an exterior vacuum region (corresponding to region I in the Kruskal diagram) and an interior vacuum region (corresponding to region II in the Kruskal diagram), plus the non-vacuum region that contains the collapsing matter.

 

[bcrowell]

People who've studied it seem to think that the BKS class of singularities are more likely to represent what an actual black hole would look like if it could somehow be made non-rotating.

I tried googling on "bks black hole" and various similar things, but didn't find much. Is there a reference you could point me to, preferably one that's free online?

 

[atyy]

I tried googling on "bks black hole" and various similar things, but didn't find much. Is there a reference you could point me to, preferably one that's free online?

I think it's just a typo, probably BKL?
http://arxiv.org/abs/gr-qc/9805008
http://arxiv.org/abs/gr-qc/0304052

 

[pervect]

My memory was a bit faulty, it was a BKL singularity, not a BKS singularity. As for my source, see "Black Holes and Time Warps, Einstein's Outrageous Legacy", p 474.

Belinsky, Khalatnikov, and Lifgarbagez have given us yet another answer to our questions, and this one, being totally stable against small pertubations,is probably the right answer, the answer that applies to the real black holes that inhabit our Universe.
The star that forms the hole and everything that falls into the hole when the hole is young gets torn apart by the tidal gravity of a BKL singularity.

(emphaisis in original)

This was Thorne's "best guess" as of 1993, I can't really guarantee he hasn't changed his mind since then.

Wiki has an assortment of papers with more detail, which I haven't looked at for the most part.

 

[TrickyDicky]

Not quite. The "physical" black hole spacetime I described earlier, which includes the collapsing matter that forms the hole, contains both an exterior vacuum region (corresponding to region I in the Kruskal diagram) and an interior vacuum region (corresponding to region II in the Kruskal diagram), plus the non-vacuum region that contains the collapsing matter.

Dscriptions are nice, but what is the metric of that "physical" spacetime?

 

[PeterDonis]

Dscriptions are nice, but what is the metric of that "physical" spacetime?

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.

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.

For a non-idealized case, the "BKL singularity" referred to in pervect's post is, I believe, a vacuum geometry, which more or less looks like the spherically symmetric Schwarzschild geometry far away from the singularity (so probably the exterior region, at least, would still be approximately Schwarzschild, at least provided the black hole wasn't rotating appreciably), but has chaotic curvature fluctuations as you get close to r = 0. I'm not sure exactly what non-vacuum geometry would "match" to that to describe the collapsing matter; again, while the matter is still far away from r = 0 (i.e., in the early stages of the collapse), I think it would still be close to the FRW geometry (with small perturbations), but it would have to also have chaotic fluctuations in the late stages of the collapse, and I'm not sure that such a non-vacuum geometry has a name (unless the "BKL singularity" solutions are supposed to cover thise case too).

 

[PAllen]

The caveat is that the "Schwarzschild coordinate chart" is not really a single chart. There are actually three of them. The first is the "Schwarzschild exterior chart" (I don't know if my terms are exactly the "standard" ones, but I'll try to make clear what I mean by them). It covers the region r > 2M ("region I" in the Kruskal chart). The second is what is normally referred to (more or less--again I don't know if my terms are exactly the "standard" ones) as the "Schwarzschild interior chart", which covers the region r < 2M that can be reached by freely falling through the horizon from the exterior region ("region II" in the Kruskal chart, the "black hole" region). The third could also be called a "Schwarzschild interior chart", but it covers the region r < 2M from which outgoing freely falling observers can cross into the exterior region ("region IV" in the Kruskal chart, the "white hole" region). I don't know that I've seen this third chart discussed explicitly, but it should be obvious that it is a valid chart and that it is distinct from the other two. (If any experts on the forum want to weigh in on this, please do.)

While I'm no expert, I have a difficulty with this. I've always seen region IV of the Kruskal chart mapped to an extension of the Schwarzschild geometry, rather than re-mapping the interior region. While overlapping coordinate charts are fine, this one seems equivalent to saying every Minkowski space is represented by two coordinate charts, one the time reversal of the other. No physical laws would be violated, but most would consider it not meaningful to do so.

 

[PeterDonis]

While I'm no expert, I have a difficulty with this. I've always seen region IV of the Kruskal chart mapped to an extension of the Schwarzschild geometry, rather than re-mapping the interior region.

I think this is consistent with what I was saying. I did not mean to imply that region IV of the Kruskal chart is just a "re-mapping" of the Schwarzschild interior region. I was saying that there are *two* regions of the full maximally extended spacetime that can be described as "interior" regions, region II and region IV. Region IV is not just a "re-mapping" of region II; it is a separate region of the spacetime in its own right.

Normally, when people talk about the "Schwarzschild interior chart", they mean a chart using Schwarzschild coordinates that covers region II, only, and correspondingly, when people talk about the "interior region of the Schwarzschild geometry", they mean region II. I think this is how you are using the terms; you view region II as "part of" the Schwarzschild gometry, while region IV is an "extension" of it. I'm fine with this usage of terms, and I didn't intend to preclude it in my previous post.

I *was* trying to point out, however, that you could have a chart using "interior" Schwarzschild coordinates that covers region IV, only. See next comment for more about that.

While overlapping coordinate charts are fine, this one seems equivalent to saying every Minkowski space is represented by two coordinate charts, one the time reversal of the other. No physical laws would be violated, but most would consider it not meaningful to do so.

It is true that an observer moving through region IV, emerging from the "white hole" singularity and heading for the "white hole" horizon that leads into region I, would see his Schwarzschild "time" coordinate t, in the Schwarzschild "interior" chart that covers region IV, *decreasing* as his proper time increased, because t = minus infinity at the white hole horizon leading into region I. This has to be true for continuity with the exterior chart that covers region I, which also has t = minus infinity at the white hole horizon. However, since region IV does not overlap with region II, there is no single region of the spacetime that is covered by two charts which are time reverses of each other. You simply have two different regions, each covered by its own Schwarzschild "interior" chart, such that the two charts happen to have time coordinates that are "reverses" of each other relative to an observer's proper time.

 

[PAllen]

I think this is consistent with what I was saying. I did not mean to imply that region IV of the Kruskal chart is just a "re-mapping" of the Schwarzschild interior region. I was saying that there are *two* regions of the full maximally extended spacetime that can be described as "interior" regions, region II and region IV. Region IV is not just a "re-mapping" of region II; it is a separate region of the spacetime in its own right.

Normally, when people talk about the "Schwarzschild interior chart", they mean a chart using Schwarzschild coordinates that covers region II, only, and correspondingly, when people talk about the "interior region of the Schwarzschild geometry", they mean region II. I think this is how you are using the terms; you view region II as "part of" the Schwarzschild gometry, while region IV is an "extension" of it. I'm fine with this usage of terms, and I didn't intend to preclude it in my previous post.

I *was* trying to point out, however, that you could have a chart using "interior" Schwarzschild coordinates that covers region IV, only. See next comment for more about that.



It is true that an observer moving through region IV, emerging from the "white hole" singularity and heading for the "white hole" horizon that leads into region I, would see his Schwarzschild "time" coordinate t, in the Schwarzschild "interior" chart that covers region IV, *decreasing* as his proper time increased, because t = minus infinity at the white hole horizon leading into region I. This has to be true for continuity with the exterior chart that covers region I, which also has t = minus infinity at the white hole horizon. However, since region IV does not overlap with region II, there is no single region of the spacetime that is covered by two charts which are time reverses of each other. You simply have two different regions, each covered by its own Schwarzschild "interior" chart, such that the two charts happen to have time coordinates that are "reverses" of each other relative to an observer's proper time.

Ok, I agree with all of this. Just terminology confusion. Now the last point of clarification. By arguments of continuity across the horizon, I've always seen it motivated that if you set things up so that an infalling world line goes from region I to region to region II, then an outgoing worldline from region IV does not reach region I, instead it reaches a parallel copy of it (region III).

 

[PeterDonis]

Ok, I agree with all of this. Just terminology confusion.

Ok, good.

Now the last point of clarification. By arguments of continuity across the horizon, I've always seen it motivated that if you set things up so that an infalling world line goes from region I to region to region II, then an outgoing worldline from region IV does not reach region I, instead it reaches a parallel copy of it (region III).

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.

 

[bcrowell]

By arguments of continuity across the horizon, I've always seen it motivated that if you set things up so that an infalling world line goes from region I to region to region II, then an outgoing worldline from region IV does not reach region I, instead it reaches a parallel copy of it (region III).

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

Like PeterDonis, I'm having trouble following your argument.

 

[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.