Hollow spherical blackhole thought experiment

[JesseM]

OK I see now. Its like to joining up lots of little black holes into a spherical shell.

I don't think that comparison works. To create a truly continuous 2D surface you'd have to join up an infinite number of pointlike singularities, and if each has a finite mass, the mass of the surface will be infinite. In contrast, a 2D surface of infinite density can have a finite mass, while each point on that surface will infinitesimal mass.

 

[Instine]

not if each singularity has a Plank scale radius.

 

[JesseM]

not if each singularity has a Plank scale radius.

I was just talking about what a 2D sheet-like singularity would have to be like if it appeared in general relativity, which is a classical theory which doesn't predict anything special happening at the Planck scale. If you want to bring quantum gravity into it, then it might be a different story...but my understanding is that most quantum gravity theories suggest that "singularities" of infinite density do not exist in the first place, so perhaps a hypothetical 2D singularity in GR would transform into something that is not quite as impenetrable in a theory of quantum gravity.

 

[Instine]

Good point. But I thought it was worth mentioning.

 

[Crothers]

I refer again to Schwarzschild's paper. I note that the issues I raised with respect to this have not been addressed. One cannot make the arbitrary moves on his variables from which the black hole has been conjured. Examine his equation (14), his arguments to his eq. (6) and note the points I made in my previous post. Clearly, the standard line-element by which the black hole is conjured up is inconsistent with Schwarzschild's true solution, for the fact that the manipulations of his variables are mathematically inadmissible. The standard metric is a corruption of Schwarzschild's solution, and is consequently geometrically invalid. Schwarzschild's true solution is regular on 0 < r < oo.

In addition, a geometry is completely determined by the form of the line-element. Only the intrinsic geometrical structure of the line-element and the consequent geometrical relations between the components of the metric tensor have any meaning. The black hole violates the intrinsic geometry of the spherically symmetric vacuum field line-element. This is clear from Schwarzschild's true solution.

As for emotional responses, I have not become emotional. I offer only citations of relevant papers and mathematical truths. You asked for the standard rebuttal of the black holers and I simply stated it. It is the black holers who respond with emotion by resorting to accusations of "crackpottery" instead of rigour, with few exceptions.

I ask now for a mathematically rigorous justification of the arbitrary corruption of Schwarzschild's solution, by which the black hole is alleged, addressing the points I have made in my previous postings and repeated above.

I have noted the remarks concerning the Kruskal-Szekeres "extension". This extension is based upon the very same corruption of Schwarzschild's solution, and is therefore invalid. I yet again refer you to the points I made in my previus postings and repeated have above. In relation to this alleged Kruskal-Szekeres "extension" I ask the black holers for a rigorous mathematical proof that General Relativity actually requires that a singularity must occur only where the Riemann tensor scalar curvature invariant (the Kretschmann scalar) is unbounded. I refer you to Kruskal's original paper wherein he has simply assumed that General Relativity requires singularity at an unbounded curvature scalar. In his paper, "Maximal Extension of Schwarzschild Metric", Phys. Rev. Vol. 119, No. 5, Sept. 1, 1960, Kruskal states

"That this extension is possible was already indicated by the fact that the curvature invariants of the Schwarzschild metric are perfectly finite and well behaved at r = 2m*."

Note that Kruskal never provided a proof that General Relativity requires the "curvature invariants" to be unbounded at a singularity. Indeed, no one has ever provided the required proof. Kruskal's remark is an unproved assumption. However, it is in fact easily proved that there are no curvature-type singularities in Einstein's gravitational field. This completely invalidates Kruskal's objective. I will not provide the simple proof of this just yet. I'm interested to see if the black holers will rigorously address the issues I have raised concerning the corruption of Schwarzschild's solution and the alleged requirement in General Relativity for singularity at an unbounded Kretschmann scalar, instead of diverging into other matters.

Wald was mentioned. Wald too does not address these issues and simply proceeds upon the very same unpoved assumptions. Consequently, his analysis is fatally flawed. His analysis is fundamentally no different to that given in any of the textbooks. These analyses routinely violate the geometrical structure of isotropic spherically symmetric type 1 Einstein spaces.

 

[chronon]

I don't think that comparison works. To create a truly continuous 2D surface you'd have to join up an infinite number of pointlike singularities, and if each has a finite mass, the mass of the surface will be infinite. In contrast, a 2D surface of infinite density can have a finite mass, while each point on that surface will infinitesimal mass.

My point is that you can't squash a surface down to a gravitational singularity. Either there's enough mass present to form a 'normal' black hole, or you don't get a gravitational singularity - that is gravity won't take over to continue the flattening of the surface down to zero thickness the way it does in the 3D case.

 

[pervect]

As for emotional responses, I have not become emotional. I offer only citations of relevant papers and mathematical truths. You asked for the standard rebuttal of the black holers and I simply stated it. It is the black holers who respond with emotion by resorting to accusations of "crackpottery" instead of rigour, with few exceptions.

Calm, non-emotional responses are good. If you can avoid attributing any particular motivation whatsoever to "black-holers" and simply recognize that they (we?) totally disagree with you it will be a good start.

Calling "black-holers" nasty names is not going to advance calm argumentation.

It will also be necessary to adhere to the PF guidelines about sources and the other PF guidlenes as well. The Abrams paper appears to me to meet those guidelines, though I still think it is misguided.

I ask now for a mathematically rigorous justification of the arbitrary corruption of Schwarzschild's solution, by which the black hole is alleged, addressing the points I have made in my previous postings and repeated above.

I have noted the remarks concerning the Kruskal-Szekeres "extension". This extension is based upon the very same corruption of Schwarzschild's solution, and is therefore invalid.

Could you go over, in more detail, why you think the Kruskal-Szerkes extension is invalid?

It is a "simple" algebraic manipulaiton of Schwarzschild'd solution in terms different variables.

It is hardly controversial - it is used in many textbooks, including the one I have right in front of me by Wald, "General Relativity", one of the standard textbooks.

(I see you acknowedge this point, though I'm not sure who previously mentioned Wald in this thread).

Are you claiming that the particular variables used to express a line element have some physical significance?

Let us start with the initial Schwarzschild metric:

ds^2 = -(1-2M/r) dt^2 + 1/(1-2M/r) dr^2 + r^2 d\Omega^2

Do you agree that this is a valid vacuum solution of Einsteins' Field equations? (EFE).

Now make the following variable substitutions. (These are from Wald, not that it particularly matters BTW).

(r/2M - 1) exp(r/2M) = X^2 - T^2
(t/2M) = ln [(X+T)/(X-T)]

Note that these are of the form

(X+T)(X-T) = f(r)
(X+T)/(X-T) = g(t)

hence we can solve them for X+T = sqrt(f*g) and X-T = sqrt(f/g)

We can then write:

<br /> dr = 4\, \left( 2\,X{\it dX}-2\,T{\it dT} \right) {M}^{2} \left( {e^{{<br /> \frac {r}{2M}}}} \right) ^{-1}{r}^{-1}<br />

<br /> dt = 2\,M \left( 2\,X{\it dT}-2\,T{\it dX} \right) \left( {\frac {r}{<br /> 2M}}-1 \right) ^{-1} \left( {e^{{\frac {r}{2M}}}} \right) ^{-1}<br />

Now we can re-write the Schwarzschild metric, with r(X,T) implicictly defined by

(r/2M - 1) exp(r/2M) = X^2 - T^2

as

<br /> \frac{32 M^3 e^{r/2M}}{r}(-dT^2 + dX^2) + r^2 d \Omega^2<br />

We see that the new expression is perfectly fininte in the new variables X,T at r=2M (which is at X=T), removing the coordinate singularity at the event horizon (r=2M, or X=T).

However, a singularity remains at r=0. We know we can't eliminate that because the curvature scalar diverges.

In short, a "simple" (it's simple with computer algebra, anyway) variable substitution eliminates the singularity at the event horizon.

I do not understand your objection to this procedure.

 

[pervect]

Here is another approach.

r can be written explicitly as a function of (X,T) using the LamabertW function in Maple

r := 2*(LambertW((X^2-T^2)/exp(1))+1)*M

One can then use GrTensor to explictly calculate the Ricci tensor for the metric

<br /> \frac{32 M^3 e^{r/2M}}{r}(-dT^2 + dX^2) + r^2 d \Omega^2<br />

and show that it (the Ricci) is is zero (applying the identity below where needed).

LambertW(x) * exp(LambertW(x)) = x

This explicitly demonstrates that the above metric is a vacuum metric.

Some notes on the LambertW function might be helpful:

As the equation y exp(y) = x has an infinite number of solutions y for each (non-zero) value of x, LambertW has an infinite number of branches. Exactly one of these branches is analytic at 0. In Maple this branch is referred to as the principal branch of LambertW, and is denoted by LambertW(x). The other branches all have a branch point at 0, and these branches are denoted in Maple by LambertW(k, x), where k is any non-zero integer. (The principal branch can also be referred to as LambertW(0, x)).

 

[Crothers]

Dear Pervect,

I appreciate your willingness to discuss the scientific issues concerning this topic. This is a rare attitude.

However, before I can address the Kruskal-Szekeres alleged coordinates, it is necessary that the questions I put to the forum be first answered, as they are central. If I deal directly with the K-S extension at this juncture the fundamental issues would be masked, and so I would need to go into a long explanation. My questions are actually motivated to answer the questions you have raised, amongst others.

Consequently, before going further I ask that you refer directly to Schwarzschild's paper and address the questions I have put to the forum. This will prove to be the most simple and most expedient way of getting to the crux of the matter. As I have already remarked, the K-S extension is based upon invalid assumptions. I have identified Kruskal's basic error in my citation of his paper. So, please first address the issues, mathematically, of the corruption of Schwarzschild's true solution and the matter of proof of the necessity of an alleged unbounded curvature scalar for a singularity in GR. The K-S extension relies upon the validity of this assumption, but it has never been proved. I therefore require first your attempts to rigorously prove the legitimacy of the arbitrary modification of Schwarzschild's true solution, which is regular on 0 < r < oo, in relation to the form you call Schwarzschild's solution and your, or anyone's proof (even Wald's, or Thorne's, or Hawking's, or Penrose's etc, but they have never given one) proof that GR necessarily requires singularity at an unbounded Kretschmann scalar.

I regard your arguments for the Kruskal-Szekers extension a la Wald as a diversion from the central issue, even if it is unintentional.

The line-element you call Schwarzschild's solution is indeed a solution to Einstein's field equations. After all, that line-element is Schwarzschild's form in his auxiliary quantity R.

Also, I have not claimed at any point that particular variables appearing in the line-element have physical significance. One can easily generate an infinite set of such "coordinates". I have claimed that a geometry is fully determined by the form of the line-element. That is something entirely different. Moreover, it is a fundamental mathematical fact. This is easily illustrated by the following:

Replace r in the line-element you call Schwarzschild's solution with sin^2 r. Then check that the resulting components of the metric tensor satisfy Einstein's field equations. Also check that the resulting line-element is Ricci flat. You will find that it satisfies both. In fact, you can replace your r with any analytic function of r and the resulting line-element will satisfy the field equations and be Ricci flat. Eddington knew this general fact. I refer you to his famous book for verification of his knowledge of this fact. However, using sin^2 r in the line-element in place of r does not produce a metric that satisfies for Einstein's gravitational field. Other factors must be applied to obtain a solution for Einstein's gravitational field. Thus, the form of the metric is, as I have said, of central importance, and its geometry must itself be used to ascertain the admissible form for the analytic function of r.

So, first provide rigorous answers to my previous questions.

 

[pervect]

Dear Pervect,

Consequently, before going further I ask that you refer directly to Schwarzschild's paper and address the questions I have put to the forum.

Which questions were those again?

Also, I have not claimed at any point that particular variables appearing in the line-element have physical significance. One can easily generate an infinite set of such "coordinates". I have claimed that a geometry is fully determined by the form of the line-element. That is something entirely different. Moreover, it is a fundamental mathematical fact. This is easily illustrated by the following:

Replace r in the line-element you call Schwarzschild's solution with sin^2 r. Then check that the resulting components of the metric tensor satisfy Einstein's field equations. Also check that the resulting line-element is Ricci flat. You will find that it satisfies both. In fact, you can replace your r with any analytic function of r and the resulting line-element will satisfy the field equations and be Ricci flat. Eddington knew this general fact. I refer you to his famous book for verification of his knowledge of this fact.

So far I am in complete agreement.

However, using sin^2 r in the line-element in place of r does not produce a metric that satisfies for Einstein's gravitational field.

Other factors must be applied to obtain a solution for Einstein's gravitational field. Thus, the form of the metric is, as I have said, of central importance, and its geometry must itself be used to ascertain the admissible form for the analytic function of r.

So, first provide rigorous answers to my previous questions.

What do you mean by "Einstein's gravitational field"? I have heard both Chrsitoffel symbols and the metric coeffcients referred to as Einstein's gravitational field, but it's entirely possible you have something else in mind.

Neither of the above quantities has any fundamental physical significance, for they depend entirely on the choice of coordinates. If they "blow up" at a point, it does not necessarily indicate a problem with the solution.

r->sin(r) is a diffeomorphism, and since GR is diffeomorphism invariant, one expects such a transformation to satisfy Einstein's field equations.

The problem with r->sin(r) is that the range of sin(r) is only -1..+1, so the resulting geometry is a subset of the original geometry (and not equivalent to the whole of the original geometry), i.e. the new geometry is the original geometry with -1<=r<=1. One also needs an invertible transformation, thus one must specify a range for r in the new geometry such as -pi < r < pi so that the inverse exists.

 

[Crothers]

Here are extracts from my previous posts.

"I refer again to Schwarzschild's paper. I note that the issues I raised with respect to this have not been addressed. One cannot make the arbitrary moves on his variables from which the black hole has been conjured. Examine his equation (14), his arguments to his eq. (6) and note the points I made in my previous post. Clearly, the standard line-element by which the black hole is conjured up is inconsistent with Schwarzschild's true solution, for the fact that the manipulations of his variables are mathematically inadmissible. The standard metric is a corruption of Schwarzschild's solution, and is consequently geometrically invalid. Schwarzschild's true solution is regular on 0 < r < oo."

"I'm interested to see if the black holers will rigorously address the issues I have raised concerning the corruption of Schwarzschild's solution and the alleged requirement in General Relativity for singularity at an unbounded Kretschmann scalar, instead of diverging into other matters."

"I therefore require first your attempts to rigorously prove the legitimacy of the arbitrary modification of Schwarzschild's true solution, which is regular on 0 < r < oo, in relation to the form you call Schwarzschild's solution and your, or anyone's proof (even Wald's, or Thorne's, or Hawking's, or Penrose's etc, but they have never given one) proof that GR necessarily requires singularity at an unbounded Kretschmann scalar."

These are the proofs I have asked for. Please provide them before proceeding to other things such as the K-S alleged extension.

As for Einstein's gravitational field, it is clear that satisfaction of the field equations is necessary but insufficient. For example, the solution must be asymptotically Minkowski. It must also satisfy the intrinsic geometry of the metric since a geometry is fully determined by the form of its metric. What is the intrinsic geometry of the metric? That will begin to become apparent when you or others attempt to provide the rigorous proofs I have asked for.

 

[Instine]

hmmm.

The thread is slipping here. As OP author, I'd like to draw a line under this issue.

1) I'm a 'black holer' and though your concerns are duly noted, Crothers, I'm not changing my view drasticly on this note
2)

I offer only citations of relevant papers and mathematical truths.

This is not so. Abstracted applied maths is not apriory, as it is applied! This is philosophy, and not the kind we need to be discussing. And again, I don't want the thread bumped.
3) It has little to do with the thought experiment at hand, as we are assuming the plausability, at least, of black holes, in order to discuss this topic.

So thanks for the input, but the the whole 'do black holes really exist' should be on another thread, possibly another forum.

So what of the question? How does an event horzon, governed by a singularity, behave, if no information exists about it (other than human memories) in the universe that contains it?

There's plenty to go at there without arguing very old ground.

Cheers

 

[Instine]

BTW - on a side note, I think I'm getting away with 'promoting a personal theory', as I'm not really, its a participatory thought experiment, which allows you guys to teach me something (which you are) and vice versa. So although this may bend forum rules I hope I don't break them.

However I do have another theory I'd like to discuss with others. And it doesn't fit into a thought experiment. And is much more likely to bend the rules beyond where the moderators are lilely to want it to go. So where can/should I discuss it? Is there somewhere on this forum? Or does anyone know somewhere else more suitable. It is a form of a unified theory, but is so far purely a model, with no work having been done to test it. Not only would I be promoting my personal theory, some may see it as a little crackpotish too. So wheres good to go. Obviously I don't just want to expound it to other 'crackpots' on MySpace, but folk who know what they're talking about, and this has been one of the best forums I've found that fits this discription. Sorry for the off topic.

All suggestion welcome, but again, back the the topic at hand

 

[selfAdjoint]

However I do have another theory I'd like to discuss with others. And it doesn't fit into a thought experiment. And is much more likely to bend the rules beyond where the moderators are lilely to want it to go. So where can/should I discuss it? Is there somewhere on this forum? Or does anyone know somewhere else more suitable. It is a form of a unified theory, but is so far purely a model, with no work having been done to test it. Not only would I be promoting my personal theory, some may see it as a little crackpotish too. So wheres good to go. Obviously I don't just want to expound it to other 'crackpots' on MySpace, but folk who know what they're talking about, and this has been one of the best forums I've found that fits this discription. Sorry for the off topic.

There is a place on PF that was created for exactly what you describe. It's called the Independent Research (IR) forum: https://www.physicsforums.com/forumdisplay.php?f=146. It has very strong guidelines for presentation of your theory, basically you have to "write it up" almost as if you were preparing it for a peer reviewed journal.

 

[Instine]

Thanks for the pointer. I'll check it out, although I'm more up for an open forum, than an electronic peer review system. Being dyslexic, I find 'writing up' prohibitively frustrating. Where as chatting more freely I make fewer mistakes, and can correct them as they are pointed out.

Any thoughts on where there may be such a place. If not, I'll likely give IR a bash anyway.

 

[Instine]

As a quick 'show of hands' who's still actually pondering this one, and who thinks its soluble?

And if insoluble, what do you think this suggests?

 

[pervect]

As a quick show of hands, I don't see a prima facie case for a paradox at all.

 

[pervect]

To avoid hijacking the OP's thread, I've moved my reply to Crother. I suggest that further discussions of the issue take place in this thread.

https://www.physicsforums.com/showpost.php?p=1019615

 

[George Jones]

There does exists a solution for a collapsing thin, pressureless, spherical shell of matter. Outside the shell, spacetime is Schwarzschild, inside the shell, spacetime is Minkowski.

References: Poisson's book, section 3.9; Israel, Nuovo cimento, 1966.

 

[JesseM]

There does exists a solution for a collapsing thin, pressureless, spherical shell of matter. Outside the shell, spacetime is Schwarzschild, inside the shell, spacetime is Minkowski.

Again, Instine isn't talking about a spherical shell of matter with finite volume. He's talking about a weird form of 2D singularity with zero volume but finite area and mass (and therefore infinite density, just like a point singularity or a ring singularity), shaped like the surface of a sphere. I still am not sure whether such a thing would even be allowed in GR, and even if it is there may not be any way it could form naturally.

 

[George Jones]

Again, Instine isn't talking about a spherical shell of matter with finite volume. He's talking about a weird form of 2D singularity with zero volume but finite area and mass (and therefore infinite density, just like a point singularity or a ring singularity), shaped like the surface of a sphere. I still am not sure whether such a thing would even be allowed in GR, and even if it is there may not be any way it could form naturally.

My short description was not very clear, but this is what I mean. It's similar to electromagnetism, where a surface charge density is an often-used and useful approximation to physical reality.

 

[pervect]

Again, Instine isn't talking about a spherical shell of matter with finite volume. He's talking about a weird form of 2D singularity with zero volume but finite area and mass

Suggestion: try thinking about the Newtonian case first.

Some suggested thought experiments:

1) Using Gauss's law, calculate the surface gravity of a spherical shell of total mass M and radius r. Show that the result depends only on the total mass M and the radius r - that it is not a function of the thickness of the shell, or the density. Show that the result for the surface gravity is finite if M and r are finite.

2) Because the surface gravity does not depend on the thickness t, taking the limit as t->0 is trivial. If the shell is very thin, show that the surface gravity remains finite, even as thickness approaches zero.

2a) If you want extra convincing, try figuring out the gravity of an infinite plane sheet of thickness t and density rho. Show that it depends only on the product of rho*thickness specifically

acceleration = 2*Pi*G* (rho*thickness)

where rho is the density / meter^3, and thickness is the thickness of the disk in meters.

3) Calculate the escape velocity for the sphere in addition to the surface gravity. As long as the Newtonian escape velocity is << c, one expect that Newtonian gravity will work just fine.

4) Make a rough order of magnitude estimate of when the object will become a black hole by computing when the escape velocity reaches the speed of light.

5) For a full GR treatment -- consider that Birkhoff's theorem says that any spherically symmetric vacuum solution of Einstein's equation will be the Schwarzschild solution. Re-read George's response with this in mind.

 

[JesseM]

My short description was not very clear, but this is what I mean. It's similar to electromagnetism, where a surface charge density is an often-used and useful approximation to physical reality.

Suggestion: try thinking about the Newtonian case first.

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?

5) For a full GR treatment -- consider that Birkhoff's theorem says that any spherically symmetric vacuum solution of Einstein's equation will be the Schwarzschild solution. Re-read George's response with this in mind.

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?

 

[Instine]

This is a very hard line to ballance upon, but now you guys are not being philisophical enough.

I'm not interested in the matematical solution. As it is not the maths I'm worried about. I'm worried that FP is in a universe surrounded by singularity, BUT there is not enough information in that universe to describe the singularity, and thereby proscribe what happens next.

So again, can anyone say what happens next, as seen by FP.

Again the silly graphic.

 

[Instine]

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?

I've forgotn who I was repliing to, but this was suggested by someone earlier. My reply is at https://www.physicsforums.com/showpost.php?p=1018820&postcount=60"

(reading back it was choron who first noted this issue)

There's an issues here isn't there?...

 

[JesseM]

I'm not interested in the matematical solution. As it is not the maths I'm worried about. I'm worried that FP is in a universe surrounded by singularity, BUT there is not enough information in that universe to describe the singularity, and thereby proscribe what happens next.

Why do you think there is "not enough information in that universe to describe the singularity"? Do you think there is enough information in the universe outside the singular surface to describe it? Even if it's true that worldlines end when they hit the singularity and no information can pass between the two regions of spacetime, in both cases you have a region of spacetime with a "border" defined by the singularity, there's no reason to treat the two cases differently. If you imagine making the surface larger and larger, in the limit as the radius approached infinity the surface would just look like a flat wall dividing 3D space into two regions, and obviously in the case of an infinite flat singular surface you wouldn't treat the two sides differently.

 

[Instine]

Why do you think there is "not enough information in that universe to describe the singularity"? Do you think there is enough information in the universe outside the singular surface to describe it? Even if it's true that worldlines end when they hit the singularity and no information can pass between the two regions of spacetime, in both cases you have a region of spacetime with a "border" defined by the singularity, there's no reason to treat the two cases differently.

I've had a crack at explaining this bit before, but I don't think folk had got there heads round the rest of the problem at that point. So here's anothjer try:

When 'outside' a black hole, the universe can 'feel' the foces of its gravitational field, you can observe lensing around it, etc...

Although there is no information passing from the singularity, to an onlooker, its mass and other properties can be induced by making 'related' observations.

When 'inside a perfectly sperical black hole' all of these 'tricks' are taken from you. Not only is the mass (and thereby the cause) of the singularity beyond your world, its effect on your world is not observable. No cause, or effect... No singularity?...

Any clearer?

 

[Instine]

OK

I've been pondering this, on and off for many years and only now does this strike me. There woould be no inner event horizon!

Causing a naked singularity!

Again thanks for the chats you fellas. It was only describing it, that made me see this.

 

[Instine]

This requires a new silly picture:

 

[JesseM]

I've had a crack at explaining this bit before, but I don't think folk had got there heads round the rest of the problem at that point. So here's anothjer try:

When 'outside' a black hole, the universe can 'feel' the foces of its gravitational field, you can observe lensing around it, etc...

Although there is no information passing from the singularity, to an onlooker, its mass and other properties can be induced by making 'related' observations.

When 'inside a perfectly sperical black hole' all of these 'tricks' are taken from you. Not only is the mass (and thereby the cause) of the singularity beyond your world, its effect on your world is not observable. No cause, or effect... No singularity?...

Any clearer?

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?

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? Check out the wikipedia entry on idea of a vacuum metastability disaster, where a bubble of lower energy vacuum would expand outward at close to the speed of light--other sources I've read, like this one and http://www.phys.utk.edu/rhip/Articles/RHICNews/Essay%20Will%20Brookhaven%20Destroy%20the%20Universe%20Probably%20Not.htm , say it would expand at the speed of light, I'm not sure who's correct--destroying everything in its path. If it actually moved at the speed of light, there's no way any measurements before it hit us would give us warning of its approach, but you wouldn't say that means it's "not part of our universe" and thus can never harm us, would you? Similarly, even if the guy in the center of the collapsing singular surface could have no warning of its approach, I don't see why that should make it plausible that he's in some alternate universe where the surface won't collapse in on him and crush him into a point singularity.