Black hole matter accumulation

[PeterDonis]

What 'essence' or geometric 'object' can be entirely absent >= r_b, yet there strongly for r_b>r>r_a, so as to explain it? And what's more it has to be shown to be cumulative in effect, and not a mere 'blip' that leaves no trace on exit past r<r_a, so to speak.

Um, a non-zero stress-energy tensor, which means a non-zero Einstein tensor, which is the primary geometric object in the Einstein Field Equation? That is what does the work in the non-vacuum region, r_b>r>r_a. For r<r_a, the vacuum Einstein Field Equation is enough to ensure that the "potential" is constant at its value at r_a.

One final comment: in #222 you mentioned agreement between yourself and DrGreg's finding in https://www.physicsforums.com/showpost.php?p=3559845&postcount=10, but I read him there as saying interior length are as at infinity, once the metric is applied. A misunderstanding?

No, just a difference in terminology. What he means by "once the metric is applied" is "from the viewpoint of an observer in the interior vacuum region". Such an observer can't tell that he is not in the flat spacetime region at infinity by purely local measurements; locally the two regions look the same. Only by global observations can the two regions be distinguished.

 

[zonde]

In this context, infinity is a limit where there are no t coordinates beyond that limit. A lack of coordinates, in a particular choice of coordinates, does not imply non-existence. It just means those coordinates don't cover the whole of spacetime.

Infinity is imaginary (not real) limit where there are no real values beyond this limit no matter what choices you make.
Look if you say that function asymptotically approaches value a as it's argument approaches infinity then it means a is the limit no matter what you do with the argument.

Maybe there is some confusion with my argument that I can still clear up.
I can explain my argument in two steps rather than one:
1) in Schwarzschild metric interior of black hole is completely disconnected from exterior because there is no future beyond infinite future and there is no past before infinite past (where you could hope to connect interior with exterior).
2) there can be any number of spacetime patches that are completely disconnected from our spacetime. There can be even any number of universes that are completely disconnected from our universe. As they do not affect our reality in any way it can be stated that they are not real or alternatively they do not exist.

 

[DrGreg]

As they do not affect our reality in any way it can be stated that they are not real or alternatively they do not exist.

Even in flat spacetime, the "inside" of a Rindler horizon does not "affect the reality" of a Rindler observer (and it is "beyond" T=∞ in Rindler coordinates), so does the "inside" of a Rindler horizon exist?

(See post #75, the "inside" is the blue region, the observer is the black line, the red and green lines specify Rindler coordinates.)

 

[PeterDonis]

1) in Schwarzschild metric interior of black hole is completely disconnected from exterior because there is no future beyond infinite future and there is no past before infinite past (where you could hope to connect interior with exterior).

This is not correct. The *geometry* of the interior is not disconnected from the geometry of the exterior. They are connected, as can be easily seen by analyzing covariant or invariant objects like geodesics, curvature tensors, etc.

It is true that the interior Schwarzschild *coordinate patch* is disconnected from the exterior Schwarzschild coordinate patch; that is what is meant by statements about the "infinite future" and whether anything is "beyond" it. But that statement does not support your argument, because it only applies to a particular coordinate system; it is not a statement about the underlying geometry, which is what is important for the physics.

 

[Dale]

Infinity takes a long time, but you more or less get that there are means to eliminate 'difficult' factors if present. Really in my example it's no more than the before/after thing - measure f, D, before shell present, and then the same with shell.

Yes, I get the idea, but different measurement techniques will give different answers which is why a complete description of the measurement technique becomes important. Once you have done that then the result of that measurement technique is guaranteed to be coordinate independent.

But this is not really the issue.

OK, so I am not sure what you still think the issue really is. You now understand that the anisotropy was merely an artifact of the coordinates, and I thought that the dissapearance of that anisotropy was what was bothering you.

 

[zonde]

You can't have two different "infinite futures". What you can do is you can construct two different dimensions and claim that they both can be considered as going in direction of "future" under different viewpoints. But under single viewpoint there will be only one "infinite future".

I am not certain that I understand what you are saying here. Let me rephrase it the way I would say it: A spacetime with multiple dimensions may have different regions considered to be the infinite future, but the worldline of a single observer will only have one infinite future.

If this is what you meant then I agree, otherwise could you clarify your meaning?

Yes, this needs clarification.

So I am stating that the same "infinite future" applies to set of observers that have parallel time dimension i.e. it applies to global coordinate system as a whole.
I suppose that you can come up with some nasty example where I would have hard time defining "observers with parallel time dimension" but as we are talking about spherically symmetric coordinate systems centered on black hole I can always come up with Euclidean coordinate system after factoring out time dilation (and change in radial length unit if something like that shows up).

 

[zonde]

Even in flat spacetime, the "inside" of a Rindler horizon does not "affect the reality" of a Rindler observer (and it is "beyond" T=∞ in Rindler coordinates), so does the "inside" of a Rindler horizon exist?

(See post #75, the "inside" is the blue region, the observer is the black line, the red and green lines specify Rindler coordinates.)

"Inside" of a Rindler horizon does not exist for Rindler observer.

But I have one question about Rindler coordinates.
It seems to me that time dimension can not be arbitrarily extended for Rindler observer, is it right?
There is certain point ahead of Rindler observer (in flat coordinates) where Rindler observer reaches speed of light and time stops for him.

And because of this it's hard for me to associate real observers with Rindler observer.

 

[Q-reeus]

OK, so I am not sure what you still think the issue really is. You now understand that the anisotropy was merely an artifact of the coordinates, and I thought that the dissapearance of that anisotropy was what was bothering you.

Main issue is that which I posted in #240. Realizing SC and ISC are two sides of the same coin took much out of the need out of that other thread, although I still wanted to see whether my expectations of anisotropic spatial distortions for r>r_b were confirmed by others calcs. Issue now, following #241, is to nail down just what property/operation of ET actually yields tangential contraction. I recall now yuiop did an analysis getting the opposite - zero tangential contraction, and a radial component that jumped back to potential free value. Was based on some work by Gron I think.

 

[Q-reeus]

Um, a non-zero stress-energy tensor, which means a non-zero Einstein tensor, which is the primary geometric object in the Einstein Field Equation? That is what does the work in the non-vacuum region, r_b>r>r_a. For r<r_a, the vacuum Einstein Field Equation is enough to ensure that the "potential" is constant at its value at r_a.

Well it would be nice to expand on that a bit. Best I could make of Einstein tensor is that it is divergenceless - so much for surmising about divergence as conceivable factor. Given the shell spherical symmetry and static state, can the operation of said tensor within shell wall be expressed entirely in terms of potential and gradients thereof, preferably in polar form? I think all but the T₀₀ term is operative as source on rhs, yes? So there should when it's all broken down, only be potential term and derivatives at work? So the specialness of matter region re tangent contraction should be explicable just in those terms. I think.

No, just a difference in terminology. What he means by "once the metric is applied" is "from the viewpoint of an observer in the interior vacuum region". Such an observer can't tell that he is not in the flat spacetime region at infinity by purely local measurements; locally the two regions look the same. Only by global observations can the two regions be distinguished.

Thanks for clearing that up. :zzz:

 

[DrGreg]

"Inside" of a Rindler horizon does not exist for Rindler observer.

But do you think that "inside" of a Rindler horizon exists for a Minkowski observer? If yes, then your interpretation of "existence" is observer dependent?

But I have one question about Rindler coordinates.
It seems to me that time dimension can not be arbitrarily extended for Rindler observer, is it right?

The Rindler time coordinate approaches ∞ as you approach the Rindler horizon, and coordinates can't go beyond ∞, so you are right. (The Rindler time coordinate equals the Rindler observer's proper time along his own worldline, and locally represents Einstein-simultaneity for any other observer at rest relative to the Rindler observer.) But you can do the thing that happens with Schwarzschild coordinates -- you can set up a separate "interior Rindler" coordinate system that covers the inside of the horizon. But neither of the two separate exterior and interior Rindler coordinate systems include the horizon itself (where there is a coordinate singularity in each system).

There is certain point ahead of Rindler observer (in flat coordinates) where Rindler observer reaches speed of light and time stops for him.

Not true. The Rindler observer gets ever closer to the speed of light as measured by any inertial observer but never actually gets there. The Rindler observer always measures the local speed of light relative to himself to be c, so from his point of view he never gets any closer.

 

[Dale]

So I am stating that the same "infinite future" applies to set of observers that have parallel time dimension

OK, I think that the concept you are trying to describe here is a congruence, which is essentially a family of worldlines. For example, you could associate one worldline with each spatial location in Schwarzschild coordinates. This would give a family of timelike curves which could each represent a stationary observer relative to the central mass. Then this set of observers would all share the same "infinite future" region.

The existence of one congruence which share the same "infinite future" does not in any logical way forbid the existence of another set of congruences which share a different "infinite future". Your line of reasoning seems to be that there is a timelike congruence which ends up in the usual "infinite future" therefore all timelike congruences must end up in the same "infinite future". This is not sound logic.

i.e. it applies to global coordinate system as a whole.

Many spacetimes don't admit a single global coordinate system that covers the whole manifold. The Schwarzschild coordinates certainly don't cover the whole manifold.

 

[PeterDonis]

Well it would be nice to expand on that a bit. Best I could make of Einstein tensor is that it is divergenceless - so much for surmising about divergence as conceivable factor. Given the shell spherical symmetry and static state, can the operation of said tensor within shell wall be expressed entirely in terms of potential and gradients thereof, preferably in polar form? I think all but the T₀₀ term is operative as source on rhs, yes? So there should when it's all broken down, only be potential term and derivatives at work? So the specialness of matter region re tangent contraction should be explicable just in those terms. I think.

Some clarifications:

* The entire stress-energy tensor is the "source" on the RHS of the Einstein Field Equation (EFE). The LHS is the Einstein tensor, which is defined in terms of the Ricci tensor, which is a contraction of the Riemann curvature tensor.

* For the non-vacuum region in question, a reasonable stress-energy tensor would be a "perfect fluid" with non-zero pressure:

$$T_{00} = \rho$$

$$T_{11} = T_{22} = T_{33} = p$$

where $\rho$ is the energy density and $p$ is the pressure. The off-diagonal components are all zero. For the spacetime to be static and spherically symmetric, both $\rho$ and $p$ must be functions of the radial coordinate r only. Also, $\rho$ should be positive everywhere in the non-vacuum region, but $p$ will not be; a negative pressure is a tension, and in order for the non-vacuum region to be stable, there will have to be tension somewhere, to keep it from falling apart (a positive pressure everywhere would be possible only if the non-vacuum region went all the way down to the center, r = 0, without any hollow interior).

* The fact that the stress-energy tensor (and therefore the Einstein tensor) is divergenceless is one way of expressing local conservation of energy; the divergence of the stress-energy tensor, over some small piece of spacetime, is just the net energy going in or out of that piece of spacetime. The divergence being zero just means energy in equals energy out; i.e., energy is conserved. So you're right that this probably doesn't help much for this particular problem.

* You can still define a "potential" in the non-vacuum region, because you can do that for any spherically symmetric geometry. And you can still view the gradient of this potential as being the "acceleration due to gravity". I *think* that the effects on the metric, including its tangential part, in the non-vacuum region can be expressed in terms of the potential, but I'm not certain; the metric coefficients may be more complicated than that in the non-vacuum region. The only metric coefficient that I'm pretty well certain can be expressed entirely in terms of the potential is $g_{00}$, the timelike coefficient.

 

[Passionflower]

* For the non-vacuum region in question, a reasonable stress-energy tensor would be a "perfect fluid" with non-zero pressure:

$$T_{00} = \rho$$

$$T_{11} = T_{22} = T_{33} = p$$

where $\rho$ is the energy density and $p$ is the pressure.

Could you demonstrate that such a 'blob' could turn into a black hole?

 

[PAllen]

Could you demonstrate that such a 'blob' could turn into a black hole?

Q-reeus example has nothing to do with black holes. He is claiming Minkowski geometry inside a spherical shell and Schwarzschild outside somehow leads to a contradiction. This whole topic is really a hijacking of the original purpose of the thread - to discuss implications of the inability of a black hole interior (if it exists) to influence the outside.

 

[Passionflower]

Q-reeus example has nothing to do with black holes. He is claiming Minkowski geometry inside a spherical shell and Schwarzschild outside somehow leads to a contradiction. This whole topic is really a hijacking of the original purpose of the thread - to discuss implications of the inability of a black hole interior (if it exists) to influence the outside.

PAllen, could you please give me the chapter and page number of MTW where you claim it is stated the formation of a black hole is in finite time?

 

[PAllen]

PAllen, could you please give me the chapter and page number of MTW where you claim it is stated the formation of a black hole is in finite time?

Section 33.1 is the clearest such discussion.

 

[noblackhole]

The black hole does not exist. It is a figment of an unscientific imagination. The alleged signatures of the black hole are an event horizon and an infinitely dense point mass singularity since the singularity is alleged to have mass but no volume. But a mass is not a point and a point is not a mass. Infinitely dense point masses are mathematical artifices; they are centres of mass, and hence have no physical existence. One can buy a bag full of marbles but one cannot buy a bag full of centres of mass of the marbles. Also, nobody has ever found an event horizon and nobody has ever found an infinitely dense point mass singularity and so nobody has ever found a black hole despite the claims of the astrophysical scientists that they have found them all over the place. All claims for the discovery of black holes are wishful thinking. It is also claimed frequently in the literature that the escape velocity of a black hole is the speed of light in vacuum. If that were true then light (photons) can escape from the black hole so that it can be seen by all observers and that material bodies can leave the black hole but not escape, merely travel out radially to a finite distance then fall back, as a ball thrown into the air on the Earth. On the other hand it is also claimed by the very same astrophysical scientists that nothing can even leave the black hole, including light so that no observers can see the black hole or material bodies leave it. This is a contradiction: merely a play on the words “escape velocity”. Furthermore, escape velocity is an implicit two body concept – one body escapes from another body. However, all black hole “solutions” pertain to a universe that contains only one mass, that of the black hole itself. There are no known solutions to Einstein’s field equations for two or more masses and no existence theorem by which it can even be asserted that the field equations contain latent solutions for two or more masses. The Principle of Superposition does not apply in General Relativity and so one cannot simply pile up masses in any given spacetime. An implicit two-body relation such as escape velocity cannot rightly appear in what is by definition a one body problem. For these reasons it is meaningless to talk of black holes existing in multitudes, interacting with one another or with other matter. Now the quantity r appearing in the so-called “Schwarzschild solution”, which I point out is not even Schwarzschild’s solution, has never been correctly identified by astrophysics. It is variously called the radius, the radial coordinate, the coordinate radius, the radial space coordinate, the radius of a 2-sphere, the area radius, the circumferential radius, and even a gauge choice that defines what r is. That this r is nevertheless always treated as the radius is manifest in the fact that a particular value of it always called the “Schwarzschild radius” of a black hole, namely the radius of the event horizon of the black hole – the radial distance from the infinitely dense point mass singularity to the event horizon of the black hole. All of these notions of what r is in the so-called “Schwarzschild solution” are incorrect because this r is not even a distance let alone a radius in the Schwarzschild manifold. It is easily proven that this r is in fact the inverse square root of the Gaussian curvature of the spherically symmetric geodesic surface in the spatial section of the related manifold, and so it is not even a distance let alone a radius in the Schwarzschild manifold. In addition, according to Einstein and his followers, his Principle of Equivalence and his Special Relativity must manifest in his gravitational field. Now both the Principle of Equivalence and Special Relativity are defined in terms of the presence of multiple arbitrarily large finite masses and photons and so neither can manifest in the spacetime of a black hole bearing in mind that all black hole “solutions” pertain to a universe that contains only one mass, that of the black hole itself. Moreover, Special Relativity forbids infinite densities and so forbids infinitely dense point mass singularities in General Relativity. It does not matter how it is alleged infinitely dense point mass singularities are formed in General Relativity because they cannot be reconciled with Special Relativity. Finally, the real Schwarzschild solution forbids black holes. One can easily verify this merely by reading Schwarzschild’s actual paper on the subject. Most astrophysical scientists have evidently never even read Schwarzschild’s actual memoir, given all the false claims that they attribute to him. What goes by the name of “Schwarzschild’s solution” in the literature is in fact a corruption of Schwarzschild’s actual solution, committed by David Hilbert in late 1916. It is from Hilbert’s corruption that the black hole was originally and incorrectly conjured.

 

[Drakkith]

Forgive me if I don't take your wall of text seriously noblackhole.

 

[Passionflower]

Section 33.1 is the clearest such discussion.

This chapter talks about light getting fainter, but the fact is that a dust ball collapsing from a finite R value takes a finite time for an observer on the 'surface' to reach the EH but for a far away observer it takes an infinite amount of time. For closer observers it is faster very close to the horizon but at the horizon is infinite for any outside observer.

 

[PAllen]

This chapter talks about light getting fainter, but the fact is that a dust ball collapsing from a finite R value takes a finite time for an observer on the 'surface' to reach the EH but for a far away observer it takes an infinite amount of time. For closer observers it is faster very close to the horizon but at the horizon is infinite for any outside observer.

Consider the matter already (and always) inside the eventual event horizon. What do you think happens to it? Does it jump outside the horizon to be suspended there? Does it cease to exist? Consider the surface of last influence. You know beyond this point that anything falling through the horizon will find all the original matter already in the singularity. This surface is a well defined surface outside the horizon. MTW explicitly calls it the 'time of formation of the black hole'.

You can try read this in strained ways, but I think the clear intent here (and in later sections, and in earlier sections on black hole collapse, rather than eternal black holes) is that black holes form in finite time - the singularity inside and the horizon outside.

If I get a chance I'll try to track down the links George Jones gave in another thread that went into detail about how the actual event horizon forms from the inside out. That is not covered in MTW, so far as I know.

 

[Passionflower]

Consider the matter already (and always) inside the eventual event horizon.

How about one step at a time:

So you agree with me that for each outside observer the formation of a black hole happens in infinite time?

I am very interested by what you mean already inside the eventual event horizon as a dust ball is a ball not a shell.

More realistic would be a fluid ball, however who can do that?

 

[PAllen]

How about one step at a time:

So you agree with me that for each outside observer the formation of a black hole happens in infinite time?

No, I don't agree. I agree that an external observer never directly sees matter cross the event horizon. However, the outside observer knows the star is a ball of matter. When it goes dark, and has well defined event horizon radius, I find it nonsensical to conclude anything other than that most of the star's matter is inside the event horizon (it always was inside). Also, that when it goes dark, I call that an event horizon. Further, it seems clear that's what MTW authors mean.

I am very interested by what you mean already inside the eventual event horizon as a dust ball is a ball not a shell.

More realistic would be a fluid ball, however who can do that?

I don't understand how you don't understand. You have a ball of gas. No horizon, no catastrophic collapse yet. There is matter at its center. I collapses, its surface goes red, then dark. I call that the event horizon (as do everyone I have ever read on GR). Where is the matter that was always at the center of ball?

 

[Passionflower]

No, I don't agree. I agree that an external observer never directly sees matter cross the event horizon. However, the outside observer knows the star is a ball of matter. When it goes dark, and has well defined event horizon radius, I find it nonsensical to conclude anything other than that most of the star's matter is inside the event horizon (it always was inside). Also, that when it goes dark, I call that an event horizon. Further, it seems

I don't understand how you don't understand. You have a ball of gas. No horizon, no catastrophic collapse yet. There is matter at its center. I collapses, its surface goes red, then dark. I call that the event horizon (as do everyone I have ever read on GR). Where is the matter that was always at the center of ball?

What about that matter? I think you misunderstand.

It is the same with Earth there is matter everywhere.

We only have a black hole when all the matter is beyond the horizon!

We are talking about the formation of a black hole right?

 

[PAllen]

What about that matter? I think you misunderstand.

It is the same with Earth there is matter everywhere.

We only have a black hole when all the matter is beyond the horizon!

No, I disagree with that. Let's say the ball of gas is big enough so 1% in the center is enough to catastrophically collapse. The event horizon forms in the center, moves outward (it does not carry any matter with it), and the matter inside collapses to singularity, but not all at once. By the time the whole thing has gone dark, the singularity (or QG alternative) has existed for some time.

 

[Passionflower]

No, I disagree with that. Let's say the ball of gas is big enough so 1% in the center is enough to catastrophically collapse. The event horizon forms in the center, moves outward (it does not carry any matter with it), and the matter inside collapses to singularity, but not all at once. By the time the whole thing has gone dark, the singularity (or QG alternative) has existed for some time.

References?

So does that mean you are now agreeing that a collapsing dust ball from a given R value (or infinity) takes an infinite time to form an EH for all outside observers?

 

[PAllen]

Anyway, you asked where MTW said what I said (that the black hole forms in finite time). In 33.1 they say:

"one can think of the surface of last influence as the birthpoint of the black hole".

This surface exists far outside the event horizon. So go argue with them, I'm done for now.

 

[PAllen]

References?

So does that mean you are now agreeing that a collapsing dust ball from a given R value (or infinity) takes an infinite time to form an EH for all outside observers?

No I don't agree. I distinguish visually seeing matter cross an event horizon from existence of the event horizon. I believe my usage is standard. There is an ongoing astronomy project to 'image the event horizon' at the center of the milkyway. They expect success.

 

[Passionflower]

Anyway, you asked where MTW said what I said (that the black hole forms in finite time). In 33.1 they say:

"one can think of the surface of last influence as the birthpoint of the black hole".

This surface exists far outside the event horizon. So go argue with them, I'm done for now.

I can write down the integrals for the proper time on the surface of the dust ball from a given r0 value and the proper time of an observer an infinite distance away and also any stationary observer. The result is that the proper time for the outside observers for a black hole to fully form (the collapsing dust ball form) is infinite.

 

[PAllen]

I can write down the integrals for the proper time on the surface of the dust ball from a given r0 value and the proper time of an observer an infinite distance away and also any stationary observer. The result is that the proper time for the outside observers for a black hole to fully form (the collapsing dust ball form) is infinite.

The event horizon is what prevents you from seeing matter cross it. The fact that you see object redshifting infinite amount getting blacker than CMB background is because the event horizon already exists. And at the point you cross the surface of last influence, it is reasonable (and MTW do so) to conclude the singularity inside the event horizon has fully formed.

 

[Passionflower]

By the way MTW says he same as I do. On page 847:

Hence, to the distant astronomer, the collapsing star appears to slow down as it approaches its gravitational radius: light from the star becomes more and more red-shifted. Clocks on the star appear to run more and more slowly. It takes an infinite time for the star to reach its gravitational radius; and as seen by the distant astronomer the star never gets beyond there.