Hollow spherical blackhole thought experiment

[JesseM]

Although the BH is pinched off in the sense that nothing can get out, the space-time continuum is continuous through the event horizon.

But you're still talking about a normal black hole with a pointlike singularity in the center, right? Instine is imagining something totally different which probably could never form naturally but may still be theoretically permitted in GR--namely, an observer enclosed within a 2D sheetlike "singularity" of infinite density, shaped like the surface of a sphere. Even if such a thing were possible, my guess is it would probably have to collapse to a pointlike singularity, but I'm not sure.

 

[Instine]

I'm kind of glad that its still standing the test of scrutiny 9 years on. No one who actually gets the question, has even attempted to answer it yet (beyond, "I don't know", or, "no one can say"). Which is the mark of a fun thought experiment in my book.

I think (or at least I thought) Garth had got it. If not here's my silly illustration again "[URL .[/URL]

Once more, as I did muck up explaining it a little.

an event of monstrous proportions culminates in the situation as shown in the image linked above. Not only is the FP surrounded by event horizon, but also by a point thick sphere of infinite density. this is a complete information barrier. Time, space, information and energy are all meaningless if beyond the shell. The rate at which, the shell collapses in on FP and whether or not it evaporates depend on information that no longer exists in FP's universe.

As it is spherical, hollow, and he's inside it, he can not measure the properties of the BH shell (such as its mass or velocity, or anything else about it), indirectly even, as would normally be the case - net gravitational effect is zeor, electric field is zero, lensing not observable and physical depth of the event horizon from the singularity, although arguably in his universe it is not observable (or rather its thickness - see depth of grey area between the two event horizons in image above).

Being the only ways in which to induce and calculate the time taken for the collapse or probability of evaporation of the event horizon surrounding the FP, we're left with a paradox. How fast does it collapse in on FP from FP's view point? And if this can't be predicted ...?

 

[pervect]

I did not realize this was a paradox, just a gedankenexperiment.

The scenario of being inside a shell black hole is not so unrealistic.

The Schwarzschild metric becomes singular at
$$\frac{2GM}{rc^2} = 1$$
so the radius required to make a black hole is proportional to the mass, and therefore the density required is proportional to 1/r².

A solar mass has to be compressed into a sphere of radius ~ 1.5 kms.

An extremely large 10¹²M_solar globular cluster closely packed into a sphere of radius 1.5 x 10¹² kms, (a compact giant elliptical really!), would pinch itself off from the rest of the universe into a black hole from which no light or anything could escape. This would have a density of about one solar mass star every 150 million kms - one solar mass star/AU, close but not impossibly so.

Tidal forces would not be too great either as an unfortunate astronaut passed unsuspecting through the event horizon...

Your question would now be: "What if this globular cluster were hollow?" A impractical situation I know, but nevertheless surpose there were an enormous massive shell around an observer. Would there be an inner horizon as well? I do not think anyone has solved this problem, if anybody knows differently please post links.

I think I can provide some insight into the problem.

If we do not assume spherical symmetry, the problem becomes very hard and would require a numerical simulation.

In an actual, physical collapse, we probably could not assume spherical symmetry, because slight departures from symmetry are magnified during the collapse process.

If we do assume spherical symmetry anyway, Birkhoff's theorem gives us the general form of the metric. It will basically be a variant of the Schwarzschild metric, because it is spherically symmetric.

The "globlular cluster" idea would be one of the easiest cases to analyze, and would correpond more or less to a pressureless collapse of dust. We are basically assuming that the gravitational attraction between stars is much larger than other forces, such as the radiation pressure. Hence - pressureless dust.

Thus we would look to the pressureless dust collapse as our model. The interior metric turns out to be a time-reversed FRW metric.

Most readers here are familiar with the FRW metric already (I know Garth is), but for those who may not be, this basically that the distance between the stars gets multiplied by a "shrinking function" a(t), as the black hole shrinks, just as the distance between galaxies in our universe gets multiplied by an "expansion function" a(t), as our universe expands.

https://www.physicsforums.com/archive/index.php/t-57568.html

is where this was discussed on PF earlier. The references that talk about this in more detail are (from this earlier thread, my post near the end):

MTW"s "Gravitation" on pg 851. "Collapse of a star with uniform density and zero pressure".

http://math.ucr.edu/home/baez/physics/Relativity/BlackHoles/universe.html

I'm reasonably confident that the a(t) for the FRW metric reaches zero in a finite amount of proper time, though I haven't looked this up.

I don't think the hollow space at the center would change this much - I'd expect a FRW metric with a bubble in the center, and the whole thing collapsing to a point in a finite amount of time.

 

[RandallB]

I'm kind of glad that its still standing the test of scrutiny 9 years on. No one who actually gets the question, has even attempted to answer it yet (beyond, "I don't know", or, "no one can say").

In focusing on how and when your temporary bubble will collapse, I'm not sure you get the point of your own hypothetical. But that happens in science.

I’m more curious about - what is the “IT” of 9 yrs ago that is still standing what “scrutiny”?

 

[Instine]

If you'll forgive me, you have to think inside the box. Modeling is fine as long as it accounts for completely closing a subsystem. I don't know enough about the model you're talking about to know, but I'm guessing this is a likely oversight as a possible scenario, during pervious examinations of the model's validity. Though this is entirely assumption.

Again, my focus is on the outcome for the 'interior', which is no longer connected (even historically) to the universe we intuitively see as 'containing' it. It is not and never has been within the black-hole, within the greater universe, in any meaningful way. Just because the black hole in the greater universe is collapsing, does not imply the lesser universe must also collapse. It was in the greater universe, 'then' (although, importantly, there is no such continuous timeline for any given observer!) within the black-hole shell. At no point do the two situation occur in the same universe.

 

[Instine]

I’m more curious about - what is the “IT” of 9 yrs ago that is still standing what “scrutiny”?

Exactly. :) The 'it' is 'what is the answer?' 'Is there an answer?'. No I don't fully comprehend my own question, which is why I keep asking it. Otherwise there'd be little point.

Do you know the answer? as seen by the FP, how fast, if at all does the event horizon collapse in? This is one way of narrowing it down to be specific enough for anyone thinking this is too vague.

But the real question is greater than this? And more vague. It involves questions like, how about the good old Schroedinger's Cat experiment? This is a true macroscale schroedinger box. What issues does this raise if the even horizon evaporates, if any. Can you exist in such a bubble universe?

But if you don't want to get too wishy washy, you can stick to the time to collapse question.

 

[RandallB]

The 'it' is 'what is the answer?' 'Is there an answer?'.

Sorry, though your were saying you drew your example from a specific point made maybe 9 years ago - that’s what I was curious about.

On the other points I’m more interested in defining FP measurements before the horizon moves in.

 

[Instine]

I simply meant it was 9 years ago hat I came up with the idea, and I've been asking lecturers, undergraduates, friends, philosophers, ever since. Nobody's sure what would happen, or how it would happen to FP.

Back then I called it The Transportable Universe, but never bothered trying to publish anything.

 

[Instine]

I think I can provide some insight into the problem.

If we do not assume spherical symmetry, the problem becomes very hard and would require a numerical simulation.

In an actual, physical collapse, we probably could not assume spherical symmetry, because slight departures from symmetry are magnified during the collapse process.

I forgot to reply properly to this. This is a good point again (tho the rest of this post suggests your not putting yourself in the box yet - the model you describe is dealing with the wrong universe [see my other posts re 'in the box'])

If not symmetrical my thought experiment looses water fast. There would then be a measurable (to understate it) gravitation force on FP, and the paradox vanishes. So yet another question would be, would a 'prefect' sphere just destabilise morphologically, as seen from the lesser 'inner' universe. Thus avoiding the paradox? In this case FP meets his maker very quickly, and the thought experiment ends. On quite a sad note... Thoughts?

 

[chronon]

The spherical shell and the physicist will both end up at the singularity i.e. they will be crushed to a point. The confusing point is 'when' this will happen - trying to match the times of things happening in a black hole with times of things happening outside it doesn't really make sense. Also, while those of us outside the black hole think of things inside falling towards the centre, for the things themselves this is more like the normal progress of time.

As for whether Hawking radiation can save the trapped physicist, well I happen to think that Hawking radiation might actually prevent the formation of black holes in the first place - see http://www.chronon.org/Articles/blackholes.html. However, the hollow shell example seems to point in the other direction. Part of my argument was that if we think of a black hole starting at the centre of a collapsing star then when it is very small it will have huge Hawking radiation which will balance the inward pressure. The hollow shell argument seems to say that a black hole need not start off as a point, rather it can 'suddenly' be formed at a much larger size if matter happens to be in the right configuration.

 

[NateTG]

Is it even possible to have black holes with negative surface (I mean event horizon surface) curvature?

 

[JesseM]

The spherical shell and the physicist will both end up at the singularity i.e. they will be crushed to a point.

But Instine isn't talking about an ordinary spherical shell of finite volume, he's talking about a 2D singularity of infinite density that has the shape of a sphere. I don't know if GR even allows such things, but it does allow "ring singularities" in the center of a rotating black hole, and apparently there was a 2D plane singularity in the unphysical model of the galaxy suggested by Cooperstock and Tieu which Garth mentioned in post #31.

 

[Instine]

Exactly jesseM. This thought experiment is designed to help investigate what can be allowable. Specifically, the interior of a hollow spherical singularity shell, where the observer (FP) is 'inside', but not yet consumed by event horizon. I'm pretty sure a negative curve on an EH is possible. But its an interesting point. Know any relevant work NateTG or anyone else?

Why do you doubt its geometric possibility Nate?

The illustration again, for late comers: "[URL

 

[pervect]

If you'll forgive me, you have to think inside the box. Modeling is fine as long as it accounts for completely closing a subsystem. I don't know enough about the model you're talking about to know, but I'm guessing this is a likely oversight as a possible scenario, during pervious examinations of the model's validity. Though this is entirely assumption.

Again, my focus is on the outcome for the 'interior', which is no longer connected (even historically) to the universe we intuitively see as 'containing' it. It is not and never has been within the black-hole, within the greater universe, in any meaningful way. Just because the black hole in the greater universe is collapsing, does not imply the lesser universe must also collapse. It was in the greater universe, 'then' (although, importantly, there is no such continuous timeline for any given observer!) within the black-hole shell. At no point do the two situation occur in the same universe.

I don't understand why you say the "interior" is not connected to the universe. I'm not even positive I understand what that is supposed to mean.

I would agree that signals cannot go out.
I would not agree that signals cannot come in.

Is this your own idea, or do you have some references to the literature?

On a similar same note - you mentioned "Teare's paradox". Google doesn't find any hits. Do you have any references for this?

As far as FRW dusts go, I found a few more references.

http://en.wikipedia.org/wiki/Dust_solution

In general relativity, a dust solution is an exact solution of the Einstein field equation in which the gravitational field is produced entirely by the mass, momentum, and stress density of a perfect fluid which has positive mass density but vanishing pressure. Dust solutions are by far the most important special case of fluid solutions in general relativity.

The pressureless perfect fluid in a dust solution can be interpreted as a model of a configuration of dust particles which interact with each other only gravitationally. For this reason, dust models are often employed in cosmology as models of a toy universe, in which the dust particles are considered as highly idealized models of galaxies, clusters, or superclusters. In astrophysics, dust solutions have been employed as models of gravitational collapse.

Here I am employing one of the standard dust solutions (the FRW dust) in the last manner mentioned - as a model of gravitational collapse.

It's probably about as good as one is going to be able to do analytically.

 

[Instine]

I don't understand why you say the "interior" is not connected to the universe. I'm not even positive I understand what that is supposed to mean.

I would agree that signals cannot go out.
I would not agree that signals cannot come in.

Note again the shell of singularity dividing the two universes. Nothing can pass through this without time or space to connect the two universes. How could information pass through the singularity when no information can escape from either side? So all the information that is, from the perspective of FP, is contained 'within' the new, lesser universe. And if it is impossible to detect the state of the singularity, from any point within the new lesser universe [see previous posts], can the singularity exist? I don't believe it can. Not enough information exists for it to persist.

So if it goes, what next?...



Again this is very philosophical, but again, GR and SR wouldn't be here if it weren't for a little lateral thinking, once in a while.

Is this your own idea, or do you have some references to the literature?

On a similar same note - you mentioned "Teare's paradox". Google doesn't find any hits. Do you have any references for this?

I'm Teare I have no references. I studied for a Msci in Physics With the Philosophy of Science at King's College London, but never completed sadly (not too sadly, I like doing what I do now). Did ok in logic though. Few 100%s. But I spend too much time thinking to read. Although the internet is changing that.

Many thanks to everyone contributing to this thread. Like I say, its been swilling round in my head for a while, and its nice to mull it with some folk who know what they're talking about. My maths is the bit lacking, so just giving me scale was very helpful (re Schwarzschild ) was helpful, as I'd never got round to calculating it, and perspective is always nice.

Cheeky I guess, but I think the idea is mine, obviously let me know if its not (I don't want credit for someone else's work). Though after several years of searching the Internet, I've not found anything. If anybody is interested in writing about it, I would love to co author something one day. For me it would simply be for the love of it, but others could use it as the basis for academic work. Apologies if this is not appropriate etiquette. Again, let me know if so.

Cheers

 

[Crothers]

The Moon could not become a black hole, hollow or otherwise, for the simple reason that black holes do not exist. Schwarzschild's solution precludes them, as can be verified by reading Schwarzschild (who NEVER breathed a single word about such an object) at

www.geocities.com/theometria/schwarzschild.pdf

and a general proof using general relativity itself can be had at

arXiv: gr-qc/0102055

Unfortunately, the black holers refuse to even read Schwarzschild. The black hole is obtained from a corruption of Schwarzschild's solution in violation of the intrinsic geometry of the line-element.

 

[Instine]

Cheers. Nice post. More new and relevant info.

I'm assuming this one has been raised before. What is the counter to this from the "black holers"?

However this desn't kill black holes for me. He's talking about the angular velocity not approching infinty as you shrink the radious of orbit around the point mass, according to interpretation of one set set of equations. It doesn't explain to me how the escape velocity will always be sub-c.

Again my maths is lacking, so I'll not take him on, on those grounds, but conceptually, I think he's a little off the mark on this one. Anyone? Whats the counter, to this?

 

[Crothers]

It is also interesting to note that the black holers routinely make claims for black hole binaries, black hole mergers and black hole collisions. One need only peruse any textbook on General Relativity and the relentless postings to arXiv to verify this.

Now I remark that the black hole is alleged to be a consequence of Einstein’s General Relativity. Assuming for the sake of argument that this is correct (but in fact, it isn’t), it is evident that the black hole is the result of a solution to Einstein’s field equations for the configuration of a single gravitating body interacting with a test particle in vacuum. It is not a solution for the interaction of two comparable masses, such as two black holes.

Before one can talk of black hole binaries or black hole collisions it must first be demonstrated that Einstein’s field equations admit of solutions for multi-body configurations of such gravitationally coupled spherically symmetric comparable masses. This can be done in two possible ways, in principle:

1) by deducing a particular solution, or

2) proving an existence theorem.

There are however, no known solutions to the field equations for the interaction of two or more such spherically symmetric comparable masses, so option 1) has never been met. In fact, it is not even known if Einstein’s field equations admit of such solutions, as no existence theorem has ever been adduced, so option 2) has never been met either.

Furthermore, one cannot simply assert by an analogy with Newton’s gravitation that a black hole can be a component of a binary system or that black holes can collide or merge.

Consequently, all talk of black hole binaries, black hole mergers, black hole collisions, etc. does not deal with well-defined problems at all. The naïve and endless claims by a great many investigators for black hole binaries, black hole collisions and black hole mergers are not meaningful, as they have no actual scientific justification. One cannot test General Relativity by means of a chimera.

 

[Instine]

One cannot test General Relativity by means of a chimera.

Why?

This is fair induction. Model x And Model Y leads to Model Z. Such is all creative thought.

You didn't answer my question by the way, which was what is the counter to this, that "Blackholers" would normally use (I'm asuming there is one)?

Schwortzchild has only gone part way to showing why angular velocity may never tend to infinity around a point mass by an 'orbitting' massless body. This is a long way from disproving the creation of event horizons. Nor does it discount singularities. Nor does it imply 2d singularities could not form. Nor does it explain what could or could not pass through a plane of point mass.

Though still interesteing.

 

[Crothers]

The black holers' standard response is stoney silence. Other than that it is irrational abuse, for reasons not divulged. A favourite response in abuse is that the messenger is a "crackpot", even though the messenger is merely pointing to the verifible facts. But "crackpot" is not scientific method.

One cannot extract a black hole from Schwarzschild's solution. His solution is regular on 0 < r < oo. See his equation (14) and the definition of his auxiliary parameter R to verify. Also, see his arguments leading to his eq. (6), where he defines his variable r. In his equation (14) it follows that alpha < R < oo for his auxiliary quantity. To force a black hole from Schwarzschild requires an arbitrary claim that his r can go down to -alpha, contrary to the very definition of r to which his solution was constructed. Alternatively, it requires the arbitrary claim that Schwarzschild's R can go down to zero, again in contradiction to the structure of his R. The black holers have effectively called Schwarzschild's R by r and taken his range 0 < r < oo and applied it to his R. That violates the geometrical structure of the line-element obtained by Schwarzschild. It is an arbitrary move that has no basis in geometry. Consequently, it is false.

One cannot argue for a black hole binary since a black hole is allegedly a solution for a test particle and a SINGLE gravitating source. Without a sound basis in General Relativity itself for a solution for two or more comparable masses such as two black holes, there is no theoretical substantiation for such a configuration in GR. The claims for black hole binaries are due to an inadmissible analogy with Newton's gravitation, in which the interaction of comparable masses are well defined. This is not the case in GR since it has never been proved by anyone, as pointed out specifically in my initial posting, that the required multi-body configurations are possible in GR. Without the required proof all talk of black hole binaries is meaningless.

Moreover, the concept of escape velocity for a black hole is misguided. An escape velocity means that a object having an initial velocity less than the escape velocity will leave the host, travel radially outward a finite distance, come momentarily to rest, then fall back radially. It does not escape. If the object has an initial velocity equal to the escape velocity it will leave the surface of the host and travel radially outward to infinity where it comes to rest. It escapes. Similarly, if the initial velocity is greater than the escape velocity then the object escapes. But in the case of the black hole, it is claimed that nothing can even leave the event horizon, not even light (that's why it is black). So the escape velocity of the black hole cannot be the speed of light, because if that was indeed the case, light could escape, and an observer could see the escaping light. According to the black holers, no observer, however close to the alleged event horizon, can see light even leave it. But an escape velocity does not mean that an object or light cannot leave the host, it only means that it cannot escape to infinity. Black hole escape velocity is a play on words, nothing more.

 

[Instine]

The black holers' standard response is stoney silence. Other than that it is irrational abuse, for reasons not divulged. A favourite response in abuse is that the messenger is a "crackpot", even though the messenger is merely pointing to the verifible facts. But "crackpot" is not scientific method.

I for one am not hurling such abuse (yet), so keep this a little less emotional if you can.

One cannot extract a black hole from Schwarzschild's solution.

Possibly not (anyone?...) but I for one can from SR. And with a struggle (again maths is my downfall) GR.

According to the black holers, no observer, however close to the alleged event horizon, can see light even leave it. But an escape velocity does not mean that an object or light cannot leave the host, it only means that it cannot escape to infinity. Black hole escape velocity is a play on words, nothing more.

But the thing about light is, is that it's quite an all or nothing phenomina. I don't just mean its quantized, but that comes into it. Light either escapes or it doesn't. A photon would not halt 100 meters beyond the event horizon and then fall back in, as once it is observable, it must travel at the speed of... light! No? Although, yet again, this is an interesting twist. Even if you don't accept this (which you should if you believe in SR, or GR) this does not explain away the possiblity of a body dense enough to cause c fast or sub-c particles to halt a fixed finit radious away from the point mass. i.e. it doesn't explain away event horizons.

Thanks for the continuing responses.

Again keep them comeing.

 

[pervect]

Cheers. Nice post. More new and relevant info.

I'm assuming this one has been raised before. What is the counter to this from the "black holers"?

However this desn't kill black holes for me. He's talking about the angular velocity not approching infinty as you shrink the radious of orbit around the point mass, according to interpretation of one set set of equations. It doesn't explain to me how the escape velocity will always be sub-c.

Again my maths is lacking, so I'll not take him on, on those grounds, but conceptually, I think he's a little off the mark on this one. Anyone? Whats the counter, to this?

It's a pretty long paper to wade through. (I'm talking about
http://lanl.arxiv.org/PS_cache/gr-qc/pdf/0102/0102055.pdf)

The short answer is this.

There is a known set of variable substitutions to get rid of the apparently infinite metric coefficeints at the event horizon. This has been known for a long time (45+ years) and was even known by Abrams, as he references it in his paper. This is the well-known result by Kruskal and Szerkes in the 1960's.

In other words, the apparently infinite metric coefficients at the event horoizon are the result of a coordinate singularity, not a physical singularity, much like the coordinate singularity in lattitude and longitude at the North pole. There is no such thing as "east" at the North pole, but the actual physical geometry of the Earth is well behaved there.

The mathematical details can be found in any standard GR textbook - for instance, Wald, "General Relativity", covers this quite well.

It is rather pardoxical, but Abrams first off makes a point of pointing out that the geometry at the event horizon of a black hole has an area

As shown by Kruskal [4] and Fronsdal [5], with r∗ = α having the character of a two-sphere in the t = constant hypersurfaces, SH is analytically extendible to r∗ > 0, and the so-extended space-time contains a black hole. It follows that the theoretical foundation
of spherical black holes is based on the 1916 error of Hilbert

The error of Hilbert was to assume that the space-time at r* = a was pointlike, rather than a two-sphere.

Abrams points this out, then makes the same error that he berates Hilbert for!

The purpose of Abrams function C(r) is essentially to "chop out" the interior solution of the black hole, by re-labelling the r-coordinate of the event horizon with a different number (a perfectly fair thing to do so far), but rather than change the domain of the re-labelled 'r' coordinate to include negative values of the re-labelled r coordinate he arbitrarily "cuts off" the solution when the relabelled r = 0.

Abrams is then left in the position of asserting that r=0, which should be a point (according to his philosophical interpretation of coordinates) correpsonds to a section of the manifold that has a finite area (the event horizon of a black hole). Furthermore, this surgery to excise the interior of the black hole was carried out at a location where there was no real physical problem with space-time.

Essentially Abrams cuts out a spherical shell around the black hole, topologically identifies all points of this spherical shell as being "the same point", and procliams that this "eliminates" the interior of the black hole.

 

[chronon]

But Instine isn't talking about an ordinary spherical shell of finite volume, he's talking about a 2D singularity of infinite density that has the shape of a sphere. I don't know if GR even allows such things, but it does allow "ring singularities" in the center of a rotating black hole, and apparently there was a 2D plane singularity in the unphysical model of the galaxy suggested by Cooperstock and Tieu which Garth mentioned in post #31.

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

I think that you still get one big black hole. Since the radius of a black hole is proportional to the mass, having enough mass to create a sphere of black holes will mean that that sphere is smaller than the event horizon of the total mass

 

[Instine]

Frankly I'm not sure what to make of that yet. More reading and thinking needed. This is where I think too much is lost in physics. Such advanced (abstracted) applied mathematics is loosing its way in so much conjecture, that it is no longer the anvil it ought to be.

But often it seems unavoidable that such mathematical trickery is needed, in order to progress.

It does seem like a hack though doesn't it.

But then who is to say one can't change the form of the model to suit the situation. Singularities and event horizons are exactly the kinds of places such models are likely to reach the limits of their cohesion. And is not Newtonian mechanics still of use in today's world? Now we're (well I am) definitely drifting towards the Philosophy, and I don't want the moderator to bump this thread over to philosophy of science, as it belongs here, I think.

In short, for now at least, I'm not persuaded that Relativity is dented in anyway by these, mathematical issues, beyond that which its is innately, simply through being so counter intuitive in nature, itself, as a construct. Though I will be reading more on the issues raised.

So enough philosophy of science for now, back to the apparent lack of physical logic in the lesser universe. Who thinks the lesser universe contains enough information to sustain the reality of the event horizon, and why? Where is the relevant information (i.e. the properties of the singularity causing the event horizon) contained, for it then to be communicated in the physical events that will follow, within that lesser universe?

 

[Instine]

Essentially Abrams cuts out a spherical shell around the black hole, topologically identifies all points of this spherical shell as being "the same point", and procliams that this "eliminates" the interior of the black hole.

This is surely the kind of forgiveable thing we must try. I mean forbidding this is akin to saying you must never change from polar coordinates to Cartesian mid calculation.

Its like to joining up lots of little black holes into a spherical shell.

definately one way of looking at it.

I think that you still get one big black hole. Since the radius of a black hole is proportional to the mass, having enough mass to create a sphere of black holes will mean that that sphere is smaller than the event horizon of the total mass

Yet more cunning thoughts. I nearly dismissed this one, but now its really got me thinking. I like it. And I'd not thought of this as an issue. Anyone want to do the maths, or do I have to get my calculator out.
At first guess, I'd assume you could have both scenarios. i.e. One where the Singularity Shell was 'thin crust' of low mass but had a large radius, and therefore had a possible Cosmic Censorship issue. And then the other kind 'deep pan' (I must be hungry) with which there is no such issue.

 

[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:

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

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

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

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

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

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

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.