Black hole matter accumulation

[Passionflower]

Perhaps some of the confusion is due to the fact that people are talking about different things.

The Schwarzschild solution is a valid solution of the EFE. We can calculate all kinds of things with it, including things passed the event horizon (although preferably not with Schwarzschild coordinates).

However a completely different question is for instance how long it takes for black holes to fully form, or how black holes form at all etc, this is something the Schwarzschild solution does not tell us. There are other solutions (with rather simplified pressure assumptions) that can give us hints but for instance I do not think it is possible for any black hole not to rotate and as of now we do not have analytic interior solutions of rotating collapsing masses.

 

[PeterDonis]

However a completely different question is for instance how long it takes for black holes to fully form, or how black holes form at all etc, this is something the Schwarzschild solution does not tell us. There are other solutions (with rather simplified pressure assumptions) that can give us hints but for instance I do not think it is possible for any black hole not to rotate and as of now we do not have analytic interior solutions of rotating collapsing masses.

This is true, but it doesn't affect the main point, which is that in all such solutions, there is an event horizon and a region of spacetime inside it, which can be reached by observers traveling on timelike worldlines; and once an observer is inside that region, they can't escape back out to the exterior. If the spacetime is that of a collapsing star (or other massive object), then the event horizon forms at some finite time instead of existing infinitely far into the past, and if the Hawking evaporation of the hole is included, then the event horizon disappears at some finite time after it forms instead of existing infinitely far into the future. But there is still plenty of room, so to speak, for observers to fall into the black hole and be trapped during the period when it exists.

 

[Passionflower]

This is true, but it doesn't affect the main point, which is that in all such solutions, there is an event horizon and a region of spacetime inside it, which can be reached by observers traveling on timelike worldlines; and once an observer is inside that region, they can't escape back out to the exterior. If the spacetime is that of a collapsing star (or other massive object), then the event horizon forms at some finite time instead of existing infinitely far into the past, and if the Hawking evaporation of the hole is included, then the event horizon disappears at some finite time after it forms instead of existing infinitely far into the future. But there is still plenty of room, so to speak, for observers to fall into the black hole and be trapped during the period when it exists.

So assume we have a rotating star, on which solution do you base your conclusions?

I am not saying you are wrong, but I do not place theories with simplified models above experimental verification.

 

[PeterDonis]

So assume we have a rotating star, on which solution do you base your conclusions?

On the exterior Kerr (or Kerr-Newman, if the collapsing rotating star has a nonzero electric charge) solution. Once the collapsing rotating star is small enough in radius, and a horizon forms around it, the exterior solution is enough to show that there are timelike geodesics which cross that horizon in a finite proper time, and that the curvature there is finite so there's no singularity or anything similar that would result in the geodesics just stopping at the horizon. As far as I know, Kerr (or Kerr-Newman) works basically the same as Schwarzschild in that respect.

I agree that the exterior solution can't tell us the details of the interior, and the interior may well be a chaotic mess rather than anything close to the interior Kerr solution. (I believe numerical simulations have been done that suggest this, but I'll have to do some link digging to verify that, unless some of the experts here can point us in the right direction. I also believe the same would be true even in the case of a non-rotating collapsing star, since perfect spherical symmetry is unstable against small perturbations.) But the chaotic mess still has to be *somewhere*, and that somewhere has to be inside the horizon.

 

[Passionflower]

I think you are missing my point.

One can say whether something is true with respect to a particular solution as it is simply a case of mathematics.
One can only surmise that something is true based on the existence of a solution that sets proper boundaries for all real situations found in nature.

What I think one cannot do is to claim something is true based on models that do not set those boundaries. All one can do is speculate it is true.

In my opinion, as long as we do not have a solution for a collapsing rotating star (both analytic or numerical) that does not show clear boundary conditions such that a black hole must form in all real cases all we can do is speculate and make educated guesses and conjectures. Those guesses might make total sense and even be true, but it is a disrespect to the scientific method to call it more than that in my opinion.

 

[PeterDonis]

Passionflower, I don't disagree with your general point; we shouldn't be claiming more than what the actual models we have show. My understanding is that, as you say, we don't have an analytic solution for a collapsing rotating star (i.e., a solution analogous to the collapsing FRW solution in the non-rotating case), but that we do have numerical models of such collapses and they all show a horizon forming. I'm not aware of any known models of collapse that *don't* show a horizon forming. But I admit I am not very familiar with the literature in this area.

 

[zonde]

I not only claim that they are not contradictory, I proved it through a rigorous derivation in the previous thread*. If you disagree then either point out the specific error in my derivation or post a correct derivation of your own showing the contradiction. If you cannot do either then you have no grounds for claiming that there is a contradiction.

*https://www.physicsforums.com/showpost.php?p=3316839&postcount=375

You can prove that proper time of reaching EH for object falling into the Black Hole is finite.
But that does not prove that object will cross EH.
That is your error.

Obviously you can enumerate infinite dimension using hyperbolic units. But that does not prove that hyperbolic units are meaningful beyond the point where finite value in hyperbolic units correspond to infinite value in linear units.
In similar fashion you can use hyperbolic units to reach infinite value at finite value of linear units. That does not prove that linear units don't extend beyond that point.

So the question is if we can we distinguish what should be considered linear unit and what should be considered hyperbolic unit. And we can't do that within mathematical model. The only meaningful way how we can do that is by establishing some correspondence to physical world.

 

[PeterDonis]

You can prove that proper time of reaching EH for object falling into the Black Hole is finite.
But that does not prove that object will cross EH.

We can prove more than just that the proper time of reaching the EH is finite. We can prove that, at that point (when the EH has been reached), the spacetime manifold is still continuous, the curvature of spacetime is finite, and the outgoing side of the light cone at that point stays at the same radius, so any timelike worldline has to continue to decrease in radius. That means one of two things must be the case: either (1) the timelike worldline on which the object is traveling when it reaches the EH continues inside the EH (meaning there is a region of spacetime in there); or (2) a timelike worldline can abruptly just *stop*, and an object traveling on it can abruptly cease to exist, without any reasonable physical cause--no infinite curvature, no break in the continuity of spacetime, nothing. You are right that we are implicitly assuming that (1) is the case, not (2), but unless you're prepared to defend (2) as an actual alternative, I don't see what other choice there is.

 

[Dale]

Also, we can prove that the object will cross the EH in a finite time in any coordinate chart which covers any open ball containing the intersection of the object's worldline and the horizon. The Schwarzschild coordinates do not cover the EH itself nor the interior, but many other charts do.

 

[zonde]

We can prove more than just that the proper time of reaching the EH is finite. We can prove that, at that point (when the EH has been reached), the spacetime manifold is still continuous, the curvature of spacetime is finite, and the outgoing side of the light cone at that point stays at the same radius, so any timelike worldline has to continue to decrease in radius. That means one of two things must be the case: either (1) the timelike worldline on which the object is traveling when it reaches the EH continues inside the EH (meaning there is a region of spacetime in there); or (2) a timelike worldline can abruptly just *stop*, and an object traveling on it can abruptly cease to exist, without any reasonable physical cause--no infinite curvature, no break in the continuity of spacetime, nothing. You are right that we are implicitly assuming that (1) is the case, not (2), but unless you're prepared to defend (2) as an actual alternative, I don't see what other choice there is.

Of course (2) is an alternative. Object approaches EH asymptotically. Idea that it just cease to exist comes from bad choice of coordinate system.

And ... hmm why do you think that infinite curvature and break in the continuity of spacetime are physical things? I suppose that these are the things that you have at central singularity where objects abruptly cease to exist just the same.

 

[zonde]

Also, we can prove that the object will cross the EH in a finite time in any coordinate chart which covers any open ball containing the intersection of the object's worldline and the horizon. The Schwarzschild coordinates do not cover the EH itself nor the interior, but many other charts do.

Seems interesting. What coordinate chart would you prefer for discussion purposes as an alternative to Schwarzschild coordinates?
I have looked at Gullstrand–Painleve coordinates and I suppose that I understood more or less what they are doing but I don't know how popular they are.

 

[PeterDonis]

Of course (2) is an alternative. Object approaches EH asymptotically. Idea that it just cease to exist comes from bad choice of coordinate system.

No, the idea that the object only approaches the EH asymptotically comes from a bad choice of coordinate system, and from confusing coordinate time with proper time. The proper time for an object to fall to the horizon is finite, and that can be calculated using Schwarzschild coordinates. See below.

And ... hmm why do you think that infinite curvature and break in the continuity of spacetime are physical things? I suppose that these are the things that you have at central singularity where objects abruptly cease to exist just the same.

Yes, GR predicts infinite curvature at the central singularity. But the curvature is *not* infinite at the horizon. There is *no* "break" in spacetime there. And yet we have timelike geodesics that reach the horizon in a finite proper time and can't go back outward there because the outgoing side of the light cone there stays at the same radius. So to maintain anything except (1), that the timelike geodesics continue inward from the horizon (meaning that there has to be spacetime inside the horizon for them to continue into) would require you to maintain that timelike geodesics, and objects traveling on them, could abruptly cease to exist when there is *not* infinite curvature.

This is not a matter of choice of coordinates. As I noted above, you can calculate that the proper time along a timelike geodesic to the horizon is finite in Schwarzschild coordinates. Those coordinates are singular at the horizon, so you can't, technically, model the extension of a timelike infalling worldline across the horizon in those coordinates; but you *can* tell that there must be such an extension, because you can compute that the curvature is finite at the horizon, and that the outgoing side of the light cone stays at the same radius there, using a limiting process in those coordinates. Picking a better coordinate system makes all this easier to do, but it doesn't affect the physical result.

 

[PeterDonis]

Seems interesting. What coordinate chart would you prefer for discussion purposes as an alternative to Schwarzschild coordinates?
I have looked at Gullstrand–Painleve coordinates and I suppose that I understood more or less what they are doing but I don't know how popular they are.

Painleve would be fine, or ingoing Eddington-Finkelstein, or Kruskal. All of those are non-singular at the horizon and meet the requirement that DaleSpam stated.

 

[zonde]

No, the idea that the object only approaches the EH asymptotically comes from a bad choice of coordinate system, and from confusing coordinate time with proper time.

Can we agree that we disagree?
Our judgment what to consider good or bad coordinate system is different. The same about which observer should be chosen for global bookkeeping either distant observer that is not going to fall into particular black hole or the one that is falling towards particular black hole.

The proper time for an object to fall to the horizon is finite, and that can be calculated using Schwarzschild coordinates. See below.

Did I claim something different?

Yes, GR predicts infinite curvature at the central singularity. But the curvature is *not* infinite at the horizon. There is *no* "break" in spacetime there. And yet we have timelike geodesics that reach the horizon in a finite proper time and can't go back outward there because the outgoing side of the light cone there stays at the same radius.

But ingoing side of light cone continues inwards? Basically you are saying that light can cross EH and therefore massive object can do that as well.
Then the question is - why do you think that light can cross EH?

So to maintain anything except (1), that the timelike geodesics continue inward from the horizon (meaning that there has to be spacetime inside the horizon for them to continue into) would require you to maintain that timelike geodesics, and objects traveling on them, could abruptly cease to exist when there is *not* infinite curvature.

This is not a matter of choice of coordinates. As I noted above, you can calculate that the proper time along a timelike geodesic to the horizon is finite in Schwarzschild coordinates. Those coordinates are singular at the horizon, so you can't, technically, model the extension of a timelike infalling worldline across the horizon in those coordinates; but you *can* tell that there must be such an extension, because you can compute that the curvature is finite at the horizon

You can compute that the curvature is finite at the horizon in Schwarzschild coordinates?
How is that?

and that the outgoing side of the light cone stays at the same radius there, using a limiting process in those coordinates. Picking a better coordinate system makes all this easier to do, but it doesn't affect the physical result.

And still you haven't answered my question - why do you think that infinite curvature and break in the continuity of spacetime are physical things?

 

[?????]

(e.g. until the hole has completely evaporated due to Hawking radiation).

Could you clarify how long it would take for this to happen? In other words, I think I am reading that all the posters on this thread agree that to the outside universe, time goes very slowly (maybe even stops) at the EH. This applies to this falling towards the BH, so I assume it applies to things going out again. So the rest of the universe is sitting on our hovering spaceships and watching the Hawking radiation coming out past the EH. But how does it get out? The rest of the universe is long gone by the time this radiation manages the trip.

It's the same argument as things falling into the BH. Sure, the proper time on the falling object proceeds at the normal rate (whatever that is). But the rest of the universe outside the EH has lived out it's lifetime and no longer exists by the time the falling observer crosses the EH. All of the Hawking radiation will be frozen at the EH going the other way, just as all of the falling objects will be frozen at the EH. The BH will never lose any mass, as view by those of us who are in the rest of the universe, it will only exponentially gain mass until it consumes everything.

 

[DaveC426913]

Could you clarify how long it would take for this to happen? In other words, I think I am reading that all the posters on this thread agree that to the outside universe, time goes very slowly (maybe even stops) at the EH. This applies to this falling towards the BH, so I assume it applies to things going out again. So the rest of the universe is sitting on our hovering spaceships and watching the Hawking radiation coming out past the EH. But how does it get out? The rest of the universe is long gone by the time this radiation manages the trip.

It's the same argument as things falling into the BH. Sure, the proper time on the falling object proceeds at the normal rate (whatever that is). But the rest of the universe outside the EH has lived out it's lifetime and no longer exists by the time the falling observer crosses the EH. All of the Hawking radiation will be frozen at the EH going the other way, just as all of the falling objects will be frozen at the EH. The BH will never lose any mass, as view by those of us who are in the rest of the universe, it will only exponentially gain mass until it consumes everything.

I see your point. Virtual particle production occurs microscopically close to the EH. Events there should be stretched out over the age of the universe as observed by a distant observer.

It's part right and part wrong.

EM travels at the speed of light, so it will always climb out of the gravity well at a speed of c, so no weirdness there. But what the time dilation does do though is red-shift the EM. Essentially Hawking radiation will be red-shifted to virtually zero.

 

[PeterDonis]

Our judgment what to consider good or bad coordinate system is different. The same about which observer should be chosen for global bookkeeping either distant observer that is not going to fall into particular black hole or the one that is falling towards particular black hole.

It's not a question of whether particular coordinates are "good" or "bad". It's a question of whether or not a particular coordinate chart covers the whole spacetime, or only a part of it. That's not a matter of opinion, it's a matter of math. The exterior Schwarzschild chart only covers the region of spacetime outside the horizon. The other charts I mentioned all cover both that region *and* the region inside the horizon.

Obviously, if you're talking about a region of spacetime covered by multiple coordinate charts, you can pick whichever chart you want to to describe the physics. So as long as you're talking about events that happen outside the horizon, yes, you can choose the exterior Schwarzschild chart for "global bookkeeping". But as soon as you try to talk about what happens at or inside the horizon, you can no longer make that choice, because the chart simply doesn't cover that region.

If you want to argue that the region inside the horizon does not exist, you can try, but you can't do it using things that are particular to one coordinate system. You have to look at covariant objects (like vectors or tensors) and invariant objects (like scalars). Schwarzschild coordinate time is not such an object. The proper time along a particular timelike worldline between two particular events is. So is the curvature tensor; so are the null vectors that define the light cone at a particular event, such as the event where a particular timelike observer crosses the horizon. And so is that observer's 4-velocity vector at that same event. So when I say that there are timelike worldlines that reach the horizon and must still be going inward there, I'm talking entirely in terms of covariant and invariant objects.

But ingoing side of light cone continues inwards? Basically you are saying that light can cross EH and therefore massive object can do that as well.
Then the question is - why do you think that light can cross EH?

For the same reason I think a timelike object can cross the horizon: there are ingoing null worldlines that reach the horizon and are still going inward there, and the curvature there is finite so those worldlines can't just stop and cease to exist.

You can compute that the curvature is finite at the horizon in Schwarzschild coordinates?
How is that?

It's a standard computation that's given in all the major GR textbooks I'm aware of. I can't seem to find a good online link showing it, but it's straightforward if tedious; I can post an outline of it if you're really interested, but that will take some time. The only technical point about using exterior Schwarzschild coordinates is that you have to take a limit as r -> 2M, since the metric is singular at r = 2M. But the formulas for the Riemann tensor components are all non-singular at r = 2M, so the bit about taking a limit, as far as I can see, is only really necessary to satisfy mathematical purists.

Of course, you could also use any of the other coordinate systems I mentioned, which are not singular at r = 2M, and then you wouldn't even have to take a limit. Since the curvature is a covariant tensor, if it is finite in one coordinate system is is finite in any coordinate system. (You can compute scalar curvature invariants as well, which are the same in all coordinate systems, and are finite at the horizon.) So even if the computation could not be done in Schwarzschild coordinates, it wouldn't matter; it could still be done.

And still you haven't answered my question - why do you think that infinite curvature and break in the continuity of spacetime are physical things?

I haven't answered it because it's irrelevant to the question of whether objects can reach and cross the horizon. Curvature only becomes infinite at the singularity at r = 0.

If you insist on an answer even though it's irrelevant to the EH question, I'm not sure that infinite curvature is an actual physical thing. It might well be an indication that classical GR cannot properly describe what happens at r = 0 inside a black hole. We may need a theory of quantum gravity to do that. But, as I said, that's irrelevant to the question of whether objects can reach and cross the horizon, since the curvature is finite there (and can be arbitrarily small if the hole's mass is large enough).

 

[Passionflower]

The Schwarzschild r coordinate is actually a measure of curvature, just take 1/r to get the curvature. Clearly at the EH it is finite while at r->0 it goes to infinity.

Does the region passed the event horizon exist?
It certainly does not exist for outside observers, only observers who pass the EH will be able to experience this region.

 

[PeterDonis]

The Schwarzschild r coordinate is actually a measure of curvature, just take 1/r to get the curvature.

Actually, IIRC, the components of the Riemann tensor in Schwarzschild spacetime go as 1/r^3. According to Wikipedia (and their formula matches what I remember from MTW, anyway), at least one curvature invariant, the Kretschmann invariant, goes as 1/r^6.

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

 

[PeterDonis]

Does the region passed the event horizon exist?
It certainly does not exist for outside observers, only observers who pass the EH will be able to experience this region.

By "exist" I meant "is part of the spacetime as a whole", which is true. I don't like using words like "exist" to refer to things which are observer-dependent, which is why I don't like the sense of "exist" in which the region inside the horizon can "exist" for infalling observers but not for observers far away. But that's just my preference. If necessary, we can describe everything that goes on without using the word "exist" at all.

 

[?????]

I wonder about gravity itself being time dependent. For example, the notion that gravity waves are spreading through the universe at the speed of light is the basis for the LIGO experiment having been underway for many years now and costing a great deal of money. This idea means that gravity is "traveling" , so to speak, outward away from the gravitating bodies (e.g. two massive planets caught in each others orbit and spinning as a pair).

So, how does gravity get past this frozen time shell surrounding a black hole? If light is red shifted to zero at the EH, then what happens to gravity going outward on the same trip. Shouldn't the apparent gravitational pull of the black hole mass appear to be zero to the rest of the universe outside the EH?

 

[ZapperZ]

I wonder about gravity itself being time dependent. For example, the notion that gravity waves are spreading through the universe at the speed of light is the basis for the LIGO experiment having been underway for many years now and costing a great deal of money. This idea means that gravity is "traveling" , so to speak, outward away from the gravitating bodies (e.g. two massive planets caught in each others orbit and spinning as a pair).

Why is this odd? Take a charge particle under electrostatic situation. It has a static E-field everywhere. Now jiggle it up and down. How fast do you think the disturbance in the E-field propagate?

So why is this any different than gravity?

Zz.

 

[Drakkith]

I wonder about gravity itself being time dependent. For example, the notion that gravity waves are spreading through the universe at the speed of light is the basis for the LIGO experiment having been underway for many years now and costing a great deal of money. This idea means that gravity is "traveling" , so to speak, outward away from the gravitating bodies (e.g. two massive planets caught in each others orbit and spinning as a pair).

So, how does gravity get past this frozen time shell surrounding a black hole? If light is red shifted to zero at the EH, then what happens to gravity going outward on the same trip. Shouldn't the apparent gravitational pull of the black hole mass appear to be zero to the rest of the universe outside the EH?

If you view gravity as a curvature of spacetime then there is no need for it to "escape" the black hole. Imagine the bowling ball on a trampoline example. If gravity is simply that curved surface then it is immediately obvious why gravity can escape, as it is curvature of spacetime.

 

[PeterDonis]

So, how does gravity get past this frozen time shell surrounding a black hole?

For a good short answer to this, see the Usenet Physics FAQ entry on "how does gravity get out of a black hole?" at:

http://www.desy.de/user/projects/Physics/Relativity/BlackHoles/black_gravity.html

Key quote:

If a star collapses into a black hole, the gravitational field outside the black hole may be calculated entirely from the properties of the star and its external gravitational field before it becomes a black hole.

In other words, the gravity you feel as a "pull" towards the black hole is actually "propagating" from the past, from the object that originally formed the hole.

It's also important to draw a distinction between the gravity waves propagating from a time-dependent system, such as a binary pulsar, and the "gravity" of a static system such as a black hole. The latter type of gravity doesn't "propagate" in quite the same way as the former.

 

[?????]

For a good short answer to this, see the Usenet Physics FAQ entry on "how does gravity get out of a black hole?" at:

http://www.desy.de/user/projects/Physics/Relativity/BlackHoles/black_gravity.html

I looked at this link and do not find its reasoning to be compelling.

I cannot think of a single other instance where a scientific enterprise is so critically dependent on a mathematical model with featuring a singularity (the Schwarzschild radius). In my view, forming a theory around this undefinable mathematical region is beyond belief. The singularity issue trumps all other mathematical reasoning.

I feel reinforced in this belief by reading the very discussions going on in this thread. The posters here are obviously very informed in the technology, very dedicated and very intelligent - yet the large disparities in viewpoints seems beyond reconciliation, despite the fact that all involved can refer to the same Schwarzschild mathematical model to back up their viewpoint - and have done so repeatedly in this thread with conviction. That is the problem with singularities - since you can't define with certainty what is happening, anything can be happening. And the theories presented in this thread about what happens after the EH cover the gamut of possibilities, all with supposed mathematical justification. When a theory is so complex that nobody can truly understand it, then anyone can come up with a theory to explain it.

The singularity at the Schwarzschild radius is the starting point in the confusion by keepit (who began this thread) and by others, especially in the more adept posters I have referred to. The Schwarzschild Metric is the key mathematical starting point for all this confusion. This defines, by default, that it cannot possibly be correct.

 

[DaveC426913]

The singularity at the Schwarzschild radius is the starting point in the confusion by keepit (who began this thread) and by others, especially in the more adept posters I have referred to. The Schwarzschild Metric is the key mathematical starting point for all this confusion. This defines, by default, that it cannot possibly be correct.

Because we here on PF do not agree, this is enough for you to conclude that it cannot possibly be correct?

 

[PeterDonis]

I looked at this link and do not find its reasoning to be compelling.

Why not? The rest of your post talks about coordinate singularities, but there's nothing at the page I linked to about coordinate singularities; it doesn't even talk about coordinate systems at all. The discussion is entirely in terms of coordinate-free (covariant or invariant) concepts.

That said, let's talk about the "singularity" itself:

I cannot think of a single other instance where a scientific enterprise is so critically dependent on a mathematical model with featuring a singularity (the Schwarzschild radius). In my view, forming a theory around this undefinable mathematical region is beyond belief. The singularity issue trumps all other mathematical reasoning.

The theory is not "formed around" the coordinate singularity at the Schwarzschild radius. Nor does the mathematical model depend on it. See below for more comments on that.

But even if we restrict attention to Schwarzschild coordinates and the coordinate singularity they have at r = 2M, you're making an awfully big deal about something which is not even a physical concept at all, just a mathematical artifact of a particular coordinate system. Coordinate singularities are well-understood and not at all mysterious. Our standard system of locating points on Earth by latitude and longitude has a coordinate singularity--actually two of them, at the North and South poles. See, for example, the Wiki page here:

http://en.wikipedia.org/wiki/Mathematical_singularity#Coordinate_singularities

Does that mean the North and South poles are somehow mysterious, or that there is some problem with "forming a theory of Earthbound locations" using latitude and longitude?

I feel reinforced in this belief by reading the very discussions going on in this thread. The posters here are obviously very informed in the technology, very dedicated and very intelligent - yet the large disparities in viewpoints seems beyond reconciliation, despite the fact that all involved can refer to the same Schwarzschild mathematical model to back up their viewpoint - and have done so repeatedly in this thread with conviction.

You are mistaken. Some people have indeed referred to Schwarzschild coordinates to back up their views; others, such as I, have said that if you want to talk about the actual physics, you have to look at things that are covariant or invariant--i.e., that *don't* depend on a particular coordinate system being used. They *do* depend on the *geometry*, which is a mathematical object in its own right, independent of whatever coordinate chart or charts we use to describe it. But claims about a particular feature specific to Schwarzschild coordinates, whether it's a singularity at r = 2M or anything else, can't be used to make claims about the geometry.

So the different "viewpoints", as you call them, are due to the fact that people are talking about different things. Some are talking about a specific coordinate chart; others are talking about a geometry, independent of any particular coordinate chart, because it's the geometry that affects the physics. See below.

That is the problem with singularities - since you can't define with certainty what is happening, anything can be happening.

Really? So because the North and South poles don't have a well-defined longitude, anything can happen there?

This is a prime example of confusing coordinates with physics. The geometry of the Earth's surface is perfectly well-defined at the poles. It's just that latitude and longitude coordinates don't do a good job of describing it there. So we use other coordinates; for example, there are various "polar projections" that are used, as described here:

http://www.geowebguru.com/articles/242-polar-maps-and-projections-part-1-overview

Similarly, the geometry of the spacetime surrounding a black hole is perfectly well-defined at the event horizon; we can tell that by computing covariant and invariant quantities like the curvature tensor and showing that they are finite and well-behaved. It's just that Schwarzschild coordinates don't do a good job of describing the geometry at r = 2M. So if we want to describe things in terms of coordinates at r = 2M, we use other coordinates. There are a number of choices, which have been mentioned in this thread.

And the theories presented in this thread about what happens after the EH cover the gamut of possibilities, all with supposed mathematical justification.

As far as the "theories" presented in this thread, I don't see a gamut of possibilities; I see only two:

(1) Objects can reach and go inside the event horizon; i.e., there is a region of spacetime inside the horizon;

(2) Objects cannot reach or go inside the horizon; i.e., spacetime "ends" at the horizon, there is no region of spacetime inside it.

Theory #1 is based on looking at covariant and invariant quantities like the curvature tensor; theory #2 is based on looking at the coordinate singularity in the Schwarzschild exterior chart. So the different "theories" are because people are talking about different things.

When a theory is so complex that nobody can truly understand it, then anyone can come up with a theory to explain it.

What evidence do you have that nobody can understand GR's description of a black hole spacetime at and around the event horizon? You may have evidence that *some* people don't understand it (and you hardly need this thread to show that; there are plenty of others, not to mention plenty of other websites). But it's a long, long haul from that to the claim that *nobody* understands. If you really want to defend the latter claim, you'll need some really impressive evidence.

The singularity at the Schwarzschild radius is the starting point in the confusion by keepit (who began this thread)...The Schwarzschild Metric is the key mathematical starting point for all this confusion. This defines, by default, that it cannot possibly be correct.

No, it means that it can't describe the geometry at r = 2M. But it does just fine at describing it for r values much larger than 2M, and it even does a passable job describing it wherever r > 2M if you remember to allow for the "distortion" it introduces close to the horizon.

 

[zonde]

It's not a question of whether particular coordinates are "good" or "bad". It's a question of whether or not a particular coordinate chart covers the whole spacetime, or only a part of it. That's not a matter of opinion, it's a matter of math. The exterior Schwarzschild chart only covers the region of spacetime outside the horizon. The other charts I mentioned all cover both that region *and* the region inside the horizon.

Fine, you explained your position about what are good and bad coordinate charts.
Let me explain my position.
Good coordinate chart does not contain empirically meaningless regions of spacetime i.e. it does not make untestable pseidoscientific implications.
If coordinate chart breaks down at some point we simply say that it's domain of applicability ends there. Beyond that point live dragons.

Obviously, if you're talking about a region of spacetime covered by multiple coordinate charts, you can pick whichever chart you want to to describe the physics. So as long as you're talking about events that happen outside the horizon, yes, you can choose the exterior Schwarzschild chart for "global bookkeeping". But as soon as you try to talk about what happens at or inside the horizon, you can no longer make that choice, because the chart simply doesn't cover that region.

If you want to argue that the region inside the horizon does not exist, you can try, but you can't do it using things that are particular to one coordinate system. You have to look at covariant objects (like vectors or tensors) and invariant objects (like scalars). Schwarzschild coordinate time is not such an object. The proper time along a particular timelike worldline between two particular events is. So is the curvature tensor; so are the null vectors that define the light cone at a particular event, such as the event where a particular timelike observer crosses the horizon. And so is that observer's 4-velocity vector at that same event. So when I say that there are timelike worldlines that reach the horizon and must still be going inward there, I'm talking entirely in terms of covariant and invariant objects.

So let's assume that I take Schwarzschild coordinate chart and use the same mathematical manipulations that are used in construction of Gullstrand–Painleve coordinates. But instead of coming up with coordinates that describe interior of black hole I get coordinates that correspond to white hole interior and both interior descriptions are contradictory.
Could I argue in this case that region inside the horizon does not exist?

For the same reason I think a timelike object can cross the horizon: there are ingoing null worldlines that reach the horizon and are still going inward there, and the curvature there is finite so those worldlines can't just stop and cease to exist.

Outgoing null worldlines can stay there without moving anywhere. And they don't cease to exist because of that.
In case of massive object you have it's proper time. You are assuming that proper time should extend to infinity. If proper time does not extend further beyond certain value it means that object ceased to exist.
Well you can't apply the same reasoning to light. Light does not have proper time.

Besides we can say that at EH massive particles become photon like and that's the reason why they proper time does not increase.

It's a standard computation that's given in all the major GR textbooks I'm aware of. I can't seem to find a good online link showing it, but it's straightforward if tedious; I can post an outline of it if you're really interested, but that will take some time. The only technical point about using exterior Schwarzschild coordinates is that you have to take a limit as r -> 2M, since the metric is singular at r = 2M. But the formulas for the Riemann tensor components are all non-singular at r = 2M, so the bit about taking a limit, as far as I can see, is only really necessary to satisfy mathematical purists.

Can't say that I am really interested in that but if I can't check it I am not buying it.
Maybe we can try to leave out this point from our discussion for now? I will try to look for some online resources.

 

[zonde]

The singularity at the Schwarzschild radius is the starting point in the confusion by keepit (who began this thread) and by others, especially in the more adept posters I have referred to. The Schwarzschild Metric is the key mathematical starting point for all this confusion. This defines, by default, that it cannot possibly be correct.

From empirical point of view there are no correct theories. That does not mean that they are useless.
We just use simple elegant theoretical constructs as a baseline for interpretation of observed facts and to analyze possible success of our future actions.
For well established theories we just know their domain of applicability and do not try to apply them where they give poor results. Lengthy discussions around Schwarzschild Metric might indicate that there is no consensus yet about it's domain of applicability.

 

[?????]

I can agree with much of what is said above, including some criticism of the view that I expressed.

But consider the implications of the link http://www.desy.de/user/projects/Phy...k_gravity.html that I commented about. Someone felt it was necessary to formulate this link and pay for its continued presence on the web. That means that others have asked the same question about gravity escaping from black holes that I posed. Not only that, but these must have been credible persons and this questioning must have come up repeatedly over the years. If my question was not a legitimate issue, then no link would be necessary.

And as for my statement that the Schwarzschild Metric is not correct - I can think of two people who agree with me: Einstein and Schwarzschild. Unfortunately, this disagreement has been going on for ninety years and shows no signs of being resolved any time soon. I think that it is appropriate to mention, at least in passing, that theories that are often thought of as confirmed science are not so universally agreed upon.