Hollow spherical blackhole thought experiment

[Instine]

A wonderful reply JesseM

But if worldlines do indeed terminate at the singular surface (and if they don't, then there'd be no problem with the two regions exchanging information), then the observer inside could still deduce the size of the singularity by sending probes out and seeing when they stop transmitting. And by using this method to see how quickly the singular surface was collapsing to a point, is it possible he could deduce its mass as well?

I love this. This is a good point. And I'll have to ponder it. But my initial feeling is, this is too conveluted. The probes could die for other reasons.

Anyway, from a more philosophical point of view, just because you can't measure something from where you are doesn't mean it can't affect you? ...

Again good point, but I'm saying you couldn't, no matter where you where in that universe. This is a big difference.

And none of this broaches the new issue of, IF there is even a EH to cloth our inner universe's singularity. ?... Curiouser and curiouser

 

[George Jones]

Hmm, I would have thought there wouldn't be a very close analogy to the classical case, since nothing new or unusual happens when you introduce point masses of infinite density into Newtonian physics or point charges of infinite charge density into electromagnetism, but singularities in GR are associated with phenomena that you don't see in extended non-collapsing masses, namely event horizons and the termination of worldlines which hit the singularities. Suppose we had a 2D spherical surface in GR whose radius was larger than than the Schwarzschild radius for that mass--would it still have an event horizon near the surface (if not, would it be a form of 'naked singularity'?) and would wordlines hitting it still be terminated? What does this mean in the case of an extended mass whose radius is larger than the Schwarzschild radius and whose pressure keeps it from collapsing, like a star? I assume it'd only be identical to the Schwarzschild solution beyond its surface, but not inside it?

The solution I referenced is Minkowski inside. The metric is continuous at the (hyper)surface, but it has sharp corner (think absolute value of x) there. Consequently, taking one derivative gives jump discontinuities in the connection, and a further derivative shows that the curvature tensor involves Dirac delta functions, just as you suspected.

Physically, I think this means that an extended body that falls through the surface experiences infinite tidal forces at the surface. However, if the surface is outside the Schwarzschild radius, there is no event horizon.

As for the geodesics of an infalling (non-extended) point particle - I don't know. Very interesting question.

 

[JesseM]

I love this. This is a good point. And I'll have to ponder it. But my initial feeling is, this is too conveluted. The probes could die for other reasons.

They could, but any measurements could go wrong for various reasons, including measurements of the curvature of spacetime. What we know about the universe is different from what is actually true about the universe...surely if probes keep failing when they travel a certain distance from you, there must be an objective truth about why this is happening, regardless of whether you can ever be sure of what that truth is?

Again good point, but I'm saying you couldn't, no matter where you where in that universe. This is a big difference.

But that's only because you've defined the word "universe" in a peculiar way, excluding everything that's exterior to the singular surface. In the case of the expanding vacuum bubble, if you defined "our universe" as only the false vacuum outside the bubble, then if the bubble is expanding at the speed of light it would also be true that you could never measure it before it annihilated you, no matter where you were in "our universe". Again, would you say this means it could never affect us, and rule out the possibility of our being destroyed by such a bubble a priori on "philosophical" grounds? The argument just doesn't make much sense to me.

 

[Instine]

Another diagram, with some axes to help:

 

[JesseM]

Physically, I think this means that an extended body that falls through the surface experiences infinite tidal forces at the surface. However, if the surface is outside the Schwarzschild radius, there is no event horizon.

As for the geodesics of an infalling (non-extended) point particle - I don't know. Very interesting question.

Interesting, thanks. I also wonder what effect infinite tidal forces would have on an electromagnetic wave hitting the surface, whether the wave would just be absorbed into the mass of the surface like with a wave falling into a point singularity, or whether it could actually pass through and allow communication between the inside and outside.

 

[Instine]

They could, but any measurements could go wrong for various reasons, including measurements of the curvature of spacetime. What we know about the universe is different from what is actually true about the universe...surely if probes keep failing when they travel a certain distance from you, there must be an objective truth about why this is happening, regardless of whether you can ever be sure of what that truth is?

As I say, I'm not denying you have a good point here. Like I say, I'll need to think on it some more.

But that's only because you've defined the word "universe" in a peculiar way, excluding everything that's exterior to the singular surface. In the case of the expanding vacuum bubble, if you defined "our universe" as only the false vacuum outside the bubble, then if the bubble is expanding at the speed of light it would also be true that you could never measure it before it annihilated you, no matter where you were in "our universe". Again, would you say this means it could never affect us, and rule out the possibility of our being destroyed by such a bubble a priori on "philosophical" grounds? The argument just doesn't make much sense to me.

This is where I disagree more substantially, though admittedly, its hard to explain why at such an abstract level.

But that's only because you've defined the word "universe" in a peculiar way, excluding everything that's exterior to the singular surface.

Yes and I very much stick by this. The shell singularity defines an aboslute barrier to the point that you must exclude that which 'is beyond' from any physics 'within'. Completely and without exception.

Interesting, thanks. I also wonder what effect infinite tidal forces would have on an electromagnetic wave hitting the surface, whether the wave would just be absorbed into the mass of the surface like with a wave falling into a point singularity, or whether it could actually pass through and allow communication between the inside and outside.

no chance.

 

[Instine]

Re your bubble, I don't know, is the short answer.

 

[Instine]

And yes, nice post George.

I have issue with the Dirac Delta function, but I guess its pretty much accepted orthodoxy, so I'll not go off on one.

So are you saying this precludes it from forming? As it does suggest a Censorship violation...

 

[Instine]

Apologies for the manic posting. Could you expand on your Minkowski reference Geroge. Another one that's news to me.
Cheers


The way I'm now reading it, is we will have a naked singularity in at least one of the universes.

 

[George Jones]

I have issue with the Dirac Delta function, but I guess its pretty much accepted orthodoxy, so I'll not go off on one.

A delta function can be approximated physically be a sequence of increasingly narrow and high peaks.

So are you saying this precludes it from forming? As it does suggest a Censorship violation...

Once the shell falls through its Schwarzschild radius, an event horizon forms.

By Minkowski, I mean that inside the shell, spacetime is perfectly flat - no gravitational forces.

 

[Instine]

So do you agree with the images above? i.e. do you think there will only be one event horizon? And thereby a naked singularity in one or other universe?

 

[Instine]

I've had a think about your bubble JesseM. Do they run out of juice, these things? Or do they keep growing for ever?

 

[JesseM]

I've had a think about your bubble JesseM. Do they run out of juice, these things? Or do they keep growing for ever?

Not sure, but if it's expanding at the speed of light, it seems unlikely it would slow down. But if the universe keeps expanding, there should be regions of the universe it will never reach, because the distance between them and the edge of the bubble would be increasing at faster than the speed of light thanks to the expansion of space.

 

[Instine]

Which means you could be in a time and place traveling at a real velocity, and view the effects of the vacuum bubble without being annihilated.

 

[Instine]

I believe in a kind "fighting chance principle". i.e. yes galaxy destroying powers may exist (I believe they do), but they must be finite, not only in their probability of occurring, but also in their 'reach'. Otherwise we'd have no universe to see before us.

If all that happens in our TE is the shell implodes at the speed of light, FP's situation breaches this principle for me. But as you say, your bubble (if they exist) must have finite reach, or the universe must be infinitely old, or have been created infinitely large. Which it may have been.

 

[pervect]

Hmm, I would have thought there wouldn't be a very close analogy to the classical case, since nothing new or unusual happens when you introduce point masses of infinite density into Newtonian physics or point charges of infinite charge density into electromagnetism, but singularities in GR are associated with phenomena that you don't see in extended non-collapsing masses, namely event horizons and the termination of worldlines which hit the singularities.

Basically, I was suggesting that to find out if there is a black hole, you look for a "trapped null surface" - a region from which light can't escape.

Without a trapped null surface, you have no event horizon (and hence no black hole).

Unfortunately, while the presence of a trapped null surface proves that there must be a singularity by Penrose's theorem, the absence of a trapped null surface doesn't actually prove that there isn't a singularity if you want to get reallly technical.

So my argument doesn't really rule out a singularity, but it does rule out a black hole.

Suppose we had a 2D spherical surface in GR whose radius was larger than than the Schwarzschild radius for that mass--would it still have an event horizon near the surface (if not, would it be a form of 'naked singularity'?) and would wordlines hitting it still be terminated? What does this mean in the case of an extended mass whose radius is larger than the Schwarzschild radius and whose pressure keeps it from collapsing, like a star? I assume it'd only be identical to the Schwarzschild solution beyond its surface, but not inside it?

You should be able to see that there isn't any trapped null surface for a sphere of mass M and radius R such that r is larger than the Schwarzschild radius of M.

The spherical symmetry of the problem guarantees that there will be a spherically symmetrical solution.

The 'r' coordinate can be defined as a radial coordinate where scaled such that the a circle of that radius has a circumference of 2*Pi*r, or alternately, an area of 4*Pi*r^2. (r is not a distance from the center, but is a function of distance from the center.)

Birkhoff's theorem says that the metric is going to be -(1-2M/r) dt^2 + 1/(1-2M/r) dr^2 for r > R, where 2*Pi*R is the circumference of the sphere.

The metric will be -(1 - 2M/R) dt^2 + 1/(1-2M/R) dr^2 for r<=R

This is essentially what George said earlier.

To calculate the curvature tensor for the above metric you do have to comit the minor sin of allowing delta functions, otherwise you have to insist that the sphere is really very thin, rather than having zero thickness.

You can see by inspection that for R > 2M the metric is well behaved - it's the same metric outside the sphere for anybody of mass M and radius r>R, basically. And inside the sphere, it's just a Minkowski metric.

 

[chronon]

The way I'm now reading it, is we will have a naked singularity in at least one of the universes.

I think that you're using the word singularity in two different ways. Firstly the usual way, the singularity contained within the event horizon of a black hole. Secondly, the surface of finite mass but zero thinkness, which might be called 'singular' in some sense, but the consensus seems to be that this isn't a gravitational singularity. Certainly if you approached it from the outside you wouldn't notice any great problems. Suppose that the sphere has the same mass and radius as the earth. Then the gravity outside would be just the same as that on Earth - however close you approached the shell. As George points out, on the inside of the shell the gravitational force will be zero.

 

[Instine]

Re singular 2d mass/ singularity/singularity shell, I didn't realize there is a consensus on what to call them. Any ideas? In this case, will Singular Shell do?

As George points out, on the inside of the shell the gravitational force will be zero.

I'm aware of the flat space-time in the shell (read earlier post explaining the paradox).

Certainly if you approached it from the outside you wouldn't notice any great problems. Suppose that the sphere has the same mass and radius as the earth. Then the gravity outside would be just the same as that on Earth - however close you approached the shell.

This is another good point. So this would be the 'Thin Crust' (I'm hungry again).

It's true enough, but what if it where massive enough to have it's event horizon beyond the shell.

 

[JesseM]

I think that you're using the word singularity in two different ways. Firstly the usual way, the singularity contained within the event horizon of a black hole. Secondly, the surface of finite mass but zero thinkness, which might be called 'singular' in some sense, but the consensus seems to be that this isn't a gravitational singularity. Certainly if you approached it from the outside you wouldn't notice any great problems. Suppose that the sphere has the same mass and radius as the earth. Then the gravity outside would be just the same as that on Earth - however close you approached the shell. As George points out, on the inside of the shell the gravitational force will be zero.

George also said he thought tidal forces would become infinite on the surface itself, and he wasn't sure whether or not all worldlines would terminate when they intersected it. I suppose that whether or not you call it a "singularity" is just a matter of terminology though...I've been calling it a "singular surface" to try to avoid confusion. The focus of Instine's argument seems to be on the fact that the inside and outside of this surface would be absolutely cut off from each other, with no possibility of exchanging information, so it would be interesting to know whether this is in fact true. I already brought up the issue of an electromagnetic wave hitting the singular surface, I also wonder whether a gravitational wave hitting it from the outside could have any effect on the curvature of spacetime inside.

 

[JesseM]

It's true enough, but what if it where massive enough to have it's event horizon beyond the shell.

I think what people are saying is that there would only be an event horizon if the surface's radius was smaller than or equal to the Schwarzschild radius for its mass, otherwise there would be none.

 

[Instine]

Great, we're all agreed on something then

 

[Instine]

So for our r < Sr situation?

We all agree there's no field inside, but an event horrizon outside.

At this stage I'd love it if all of you guys could post a quick diagram of what you believe is going on at all significant radii.

(I feel like I'm setting home work )

Me? Well as you can tell, I don't know, and I don't think you can say, which is why I call it a paradox, but if anyone of you can give a cohesive model, I'd be delighted.

Again thanks to all of you

 

[chronon]

I guess in the case where the system forms a black hole the observer inside will go on experiencing 0G until he gets crushed by the infalling shell. So where's the singularity if it's not at the centre? Well since time and space get swapped around, for observers inside a black hole the singularity is not at the centre its in the future

 

[chronon]

George also said he thought tidal forces would become infinite on the surface itself

I wouldn't read too much into this. The tidal effect is due to the gravitational force changing with distance, and in this case there's no great change in the force ( from e.g. 1G outside the shell to 0G inside), it's just that the distance over which it changes is zero.

 

[Instine]

I thought we were now only conciderng the shell radius being greater than its Schwarzschild. Again, sorry to be bossy, but as OP author, I'd like to corral this back to the main issues of the thread.

We're considering a situation where there is an event horizon outside, and [non?] inside. Thickness of the shell is zero.

Is there a naked singular surface presented to FP? At the very least he seems to be able to veiw the effects of infinite tidal forces. I'm sure this will through up an issue or three.

Its still proving a head scratcher, no?

 

[Instine]

BTW, re the collaspse at the speed of light, why do people think this? What suggests c as the radial velocity of encroaching doom (supposing for a moment it isn't paradoxical)?

And I don't see any pictures yet. Come on you guys, don't be lazy, I want to see diagrams!

Blow the cobwebs off MS paint if you don't have photoshop.

 

[JesseM]

I wouldn't read too much into this. The tidal effect is due to the gravitational force changing with distance, and in this case there's no great change in the force ( from e.g. 1G outside the shell to 0G inside), it's just that the distance over which it changes is zero.

But are you saying nothing would happen to you if you hit the singular surface, you'd just pass right through it? George Jones wasn't sure about this...I wonder if general relativity gives clear-cut rules for when worldlines are terminated and when they aren't, or if you could model this situation either way and it would be a valid solution. I remember one of the problems with naked singularities is that they allow for multiple solutions without a way to choose between them, because it would be consistent for anything to pop out of one at any moment...the last paragraph of section 4.3.2 from this article says:

The most problematic kinds of singularities, in terms of determinism, are naked singularities (singularities not hidden behind an event horizon). When a singularity forms from gravitational collapse, the usual model of such a process involves the formation of an event horizon (i.e. a black hole). A universe with an ordinary black hole has a singularity, but as noted above, (outside the event horizon at least) nothing unpredictable happens as a result. A naked singularity, by contrast, has no such protective barrier. In much the way that anything can disappear by falling into an excised-region singularity, or appear out of a white hole (white holes themselves are, in fact, technically naked singularities), there is the worry that anything at all could pop out of a naked singularity, without warning (hence, violating determinism en passant). While most white hole models have Cauchy surfaces and are thus arguably deterministic, other naked singularity models lack this property. Physicists disturbed by the unpredictable potentialities of such singularities have worked to try to prove various cosmic censorship hypotheses that show -- under (hopefully) plausible physical assumptions -- that such things do not arise by stellar collapse in GTR (and hence are not liable to come into existence in our world). To date no very general and convincing forms of the hypothesis have been proven, so the prospects for determinism in GTR as a mathematical theory do not look terribly good.

So maybe something similar is going on with the singular surface when it has no event horizon, it might be equally consistent with GR for an object hitting it from one side to come out the other side as if it had just passed right through, or to disappear, or to reappear after a delay, etc. I'm just speculating obviously, I have no idea if the singular surface would really behave like a naked singularity in this sense.

 

[chronon]

But are you saying nothing would happen to you if you hit the singular surface, you'd just pass right through it? .

Yes, as far as gravity is concerned. Suppose that there was such a sphere made of dark matter, so that we didn't interact with it via any other forces. The my intuition is that we would be pulled towards it and go straight through the surface without being squashed or pulled apart or anything else particularly drastic.

 

[JesseM]

Yes, as far as gravity is concerned. Suppose that there was such a sphere made of dark matter, so that we didn't interact with it via any other forces. The my intuition is that we would be pulled towards it and go straight through the surface without being squashed or pulled apart or anything else particularly drastic.

Well, what about the naked singularity comparison? If a point particle's worldline hit a naked singularity, then since there'd be no event horizon do you think it would pass through it and continue on its merry way, or be absorbed into it? Again, I think the point about indeterminism in the presence of naked singularities may mean that either is possible, and a singular 2D surface with no event horizon could perhaps be considered a form of naked singularity.

 

[chronon]

Well, what about the naked singularity comparison? If a point particle's worldline hit a naked singularity, then since there'd be no event horizon do you think it would pass through it and continue on its merry way, or be absorbed into it? Again, I think the point about indeterminism in the presence of naked singularities may mean that either is possible, and a singular 2D surface with no event horizon could perhaps be considered a form of naked singularity.

The point about the naked singularity is that GR doesn't say what happens to the particle after it hits it, whereas my feeling is that the behaviour of a particle at the 2D surface would be perfectly deterministic - it would just go through it. But I can see that we're going to need some maths here to decide one way or the other.