Entropy of "frozen stars"

[Lorentz Jr]

Has it ever occurred to anyone that the "frozen star" model of black holes provides a simple and obvious explanation for the fact that the entropy of a black hole is proportional to the BH's surface area? (the Bekenstein-Hawking formula) If the BH is frozen on the inside, then only a layer on the surface has any unrestrained degrees of freedom. There's no need for complicated hologram theories. (a) General relativity breaks down at the event horizon; (b) the vacuum inside the BH is incapacitated by some kind of stress associated with the gravitational potential, so nothing inside the event horizon can move*; and (c) the only source of entropy is the outer layer of matter at the event horizon.

 

* (In other words, the vacuum's state might be something vaguely analogous to whatever high-pressure/high-temperature solid allotrope of iron Earth's inner core is made of.)

Edited January 4, 2023 by Lorentz Jr

[MigL]

Black Holes were called 'frozen' stars by Soviet Physicists, like Y Zeldovich, simply because the maths describe time running slower, and finally stopping, or 'freezing', at the Event Horizon ( to a far-off observer ).
And also because the term 'black hole' has rude connotations in Russian.

[Lorentz Jr]

40 minutes ago, MigL said:

the maths describe time running slower, and finally stopping, or 'freezing', at the Event Horizon ( to a far-off observer ).

I don't see any reason not to take the math seriously.

I also don't understand why Georges Lemaître called the event horizon a nonphysical coordinate singularity. The θ=0 or π axis is a nonphysical singularity because the coordinate system is defined in terms of it, but the event horizon of a black hole is a calculated result. Personally, I suspect that Lemaître's singularity at r = 0 is nonphysical and the BH is full of matter suspended in the inactive vacuum all the way out to the event horizon.

Black holes as frozen stars

It looks like the Russian thing may be a myth.

Edited January 4, 2023 by Lorentz Jr

[Mordred]

11 hours ago, Lorentz Jr said:

I don't see any reason not to take the math seriously.

I also don't understand why Georges Lemaître called the event horizon a nonphysical coordinate singularity.

 

 

Its an apparent horizon where the radius can alter depending on observer. Another term commonly used is a coordinate horizon or coordinate singularity. Mathius Blau if I recall also discusses this in his lecture notes on General relativity on arxiv. It is more often described in literature as an "artifact of coordinate choice".

 If I recall the Kruskal-Szekeres coordinates will eliminate the R=0 singularity condition at EH.

Page 182 covers it in 

Lecture Notes on General Relativity Sean M. Carroll

https://arxiv.org/pdf/gr-qc/9712019.pdf

LMAO I guess one could say the frozen can be thawed with a different choice of coordinates

 

 

Edited January 5, 2023 by Mordred

[Lorentz Jr]

16 hours ago, Lorentz Jr said:

I also don't understand why Georges Lemaître called the event horizon a nonphysical coordinate singularity.

5 hours ago, Mordred said:

Another term commonly used is a ... coordinate singularity.

Thank you, Mordred. You're absolutely right about that. 🙂

5 hours ago, Mordred said:

Its an apparent horizon ...

I guess one could say the frozen can be thawed with a different choice of coordinate

I don't think so. This is the thing about relativity that reminds me of New Age mysticism. "Every individual's perspective is equally valid."

I think GR breaks down at the event horizon and that's the end of it. It's not a non-physical singularity, it's a physical singularity. No more time evolution for anything inside. I'm sure the math from the infalling observer's frame is very interesting and fun to work out, and it makes for entertaining fiction, with all those newly created universes and whatnot, but I don't think it really matters. The observer is completely time dilated in the coordinate frame, and I think that frame most closely approximates "ontological reality". This is the Lorentzian view of space and time.

This is also something that distinguishes black holes from neutron stars and ordinary matter. The effect that prevents neutrons and electrons from collapsing in a weaker gravitational potential is their own internal degeneracy pressure, but black holes are a whole 'nother story. What prevents them from collapsing any more than they do is the disabling of the vacuum itself. Like it turns to molasses because of the gravitational stress, and then keeps getting thicker and thicker*, until the infalling observer is in a state of ... "suspended animation", I guess you could call it.

It's hard to say for sure, of course, because we can't look inside the BH. But I think this picture is more "valid" than the relativistic one, because it's naturally consistent with Bekenstein-Hawking radiation.

 

* (meaning more and more "viscous", so to speak)

Edited January 5, 2023 by Lorentz Jr

[Markus Hanke]

1 hour ago, Lorentz Jr said:

I think GR breaks down at the event horizon and that's the end of it. It's not a non-physical singularity, it's a physical singularity.

On Earth, the standard lat/long coordinate chart we are all commonly using for navigation is singular at both poles. Does this mean the poles do not physically exist, or that there is a physical singularity located there? Does this mean that our models of aeronautical navigation “break down” there? Evidently it means no such thing.

There are simple, standardised ways to tell apart physical singularities from coordinate singularities on differentiable manifolds - these issues really have nothing to do with GR at all, they are mathematical questions that are considered in-depth within the discipline of differential geometry. It is trivial to show in a fully coordinate-independent way that the event horizon is not a physical singularity, in the sense that the manifold (which is entirely different from the coordinate chart) is completely smooth and regular there, and everywhere geodesically complete.

1 hour ago, Lorentz Jr said:

I'm sure the math from the infalling observer's frame is very interesting and fun to work out, and it makes for entertaining fiction, with all those newly created universes and whatnot, but I don't think it really matters.

You’ve got this exactly backwards, I’m afraid.

The length of a world line is a quantity that all observers agree on. If an in-falling observer finds his world line to be of finite length, then all other observers - including the distant stationary one - will also agree that it is in fact of finite length. This physically means that everyone agrees that the in-falling clock reaches (and crosses) the horizon in a finite amount of time as measured by itself, since the accumulated time on this clock is by definition identical to the length of the world line it traces out. On the other hand, what the distant stationary clock shows (divergence to infinity) is not the length of the in-fall world line, so it is entirely irrelevant to the physical outcome of the in-fall. You cannot use a distant clock to argue local physics, so it is really the distant clock that doesn’t matter, and not the other way around.

[Lorentz Jr]

1 hour ago, Markus Hanke said:

The length of a world line is a quantity that all observers agree on.

On the other hand, what the distant stationary clock shows (divergence to infinity) is not the length of the in-fall world line, so it is entirely irrelevant to the physical outcome of the in-fall. You cannot use a distant clock to argue local physics, so it is really the distant clock that doesn’t matter, and not the other way around.

I'm not convinced. The length of a world line is just the behavior of a local, comoving clock, and I don't trust a clock in a gravity well to measure anything except its own behavior, any more than I would trust a person who has been drugged with tranquilizers to count out seconds for me. If the laws of physics are the same everywhere, then there's no such thing as "local physics"; there's only physics.

1 hour ago, Markus Hanke said:

It is trivial to show in a fully coordinate-independent way that the event horizon is not a physical singularity

Actually, I wasn't entirely comfortable with the word "singularity". I'm not sure what would be the best word for that. My position is that the event horizon is physically meaningful because any "observer" inside it will be totally time dilated, i.e. in a sort of "suspended animation".

It's the same argument as in the twin paradox: You can talk all you want about "local time" or "local physics" or "local reality", but the astronaut will have a rude awakening when he lands back on Earth and discovers that he's missed out on several years of history during his journey to Alpha Centauri and back. The only difference is that we don't know how to get things back out of black holes.

Edited January 5, 2023 by Lorentz Jr

[Markus Hanke]

2 hours ago, Lorentz Jr said:

and I don't trust a clock in a gravity well to measure anything except its own behavior

Yes, and this is true for all clocks - including the distant stationary one! Now think about this - if the distant stationary clock measures purely its own time (“behaviour”), as you correctly say, then why do you expect it to be able to accurately measure any process that does not, in fact, take place within its own frame? Can you see the issue? 

The reverse situation is just as true - the in-falling clock cannot, based on its own readings, expect a distant stationary clock to tick out the same amount of time.

2 hours ago, Lorentz Jr said:

If the laws of physics are the same everywhere, then there's no such thing as "local physics"; there's only physics.

Exactly! This is just the basic principle in GR - the laws of physics are covariant, so all observers agree on them. In this case, all observers agree that the in-fall world line does in fact intersect the horizon, including the distant Schwarzschild observer. He just doesn’t physically measure this on his own clocks and rulers, because those are inextricably linked to his distant, stationary frame, and thus unable to measure anything about the in-falling particle. They can only measure things in their own local frame.

2 hours ago, Lorentz Jr said:

Actually, I wasn't entirely comfortable with the word "singularity". I'm not sure what would be the best word for that.

There are two entirely separate concepts to consider here - there is the manifold, which is spacetime itself, ie the set of all points in space at all instances of time (“events”); and then there is the coordinate chart that covers the manifold, which is simply a map that assigns a unique label to each event. The manifold is like the physical set of streets that makes up a city, whereas the chart is the choice of names we assign to those streets. 

It should be obvious straight away that the choice of street names is entirely arbitrary (so long as they are unique) - we can erase and re-write all street names, without affecting any of the physical layout and geometry of our city. You can also have different people employing different names for the same street; there’s potential for confusion when you do this, but so long as both sets of names are unique, there will be no problems or contradictions; you can map them into each other 1-on-1 in a unique way. The time it takes you to drive from one address within the city to another is not affected by the way the streets are named in any way. Essentially, the street names have no physical significance so far as the layout of the city is concerned.

And so it is with spacetime - you’ve got the spacetime manifold and its geometry, which is given by the distribution of gravitational sources (“the city). This is entirely separate from the coordinate chart which you choose to label each event on that manifold (“the street names”). How long it takes to inertially free-fall from one event to another as measured on a co-moving clock is likewise not affected in any way by what kind of coordinate chart you choose to use. Just as is the case for the city, the choice of coordinate chart has no physical significance whatever so far as the geometry of spacetime is concerned.

What happens at the event horizon is that the Schwarzschild coordinate chart becomes singular, in the same way as spherical coordinates become singular at the poles on Earth. So the question then becomes whether this is purely a coordinate singularity, where only the coordinate chart fails due to the way it is defined, but the manifold itself remains perfectly regular; or whether this is a curvature singularity, where both the coordinate chart fails and the manifold ceases to be smooth and regular. The former has no physical significance, it’s just an artefact of the way we choose to label our events; whereas the latter means that the manifold is geodesically incomplete, ie we cannot physically extend free-fall geodesics past that region. 

The simplest way to distinguish between them is to try and cover our spacetime with a different coordinate chart (remember that this choice is arbitrary and has no physical significance!), and see what happens at the horizon. Instead of Schwarzschild coordinates we can use (e.g.) Gullstrand-Painleve coordinates, Novikov coordinates, Eddington-Finkelstein coordinates, Kruszkal-Szekeres coordinates, or any other convenient choice. If we can find even only one coordinate chart that remains smooth and regular at the horizon, then we know that the original singularity was of the coordinate kind, and thus has no physical significance so far as the manifold is concerned. And that’s indeed the case here - in Schwarzschild spacetime, there are many coordinate choices that remain smooth and continuous even at the horizon.

To be absolutely sure, we can also check in a more direct way, by considering a covariant quantity that does not depend on coordinate choices, such as the curvature tensors. More specifically, one looks at the principle invariants of the Riemann tensor and the Weyl tensor, which indicates how the curvature of spacetime behaves at the region in question. There are altogether five of such invariants. When we calculate them at the horizon (using any coordinate chart of our choice), we find that they are all finite and well defined, indicating that spacetime is smooth and regular there.

This is in contrast to the central singularity - no matter what coordinate chart we choose, the central singularity is always singular; we cannot eliminate this by choosing different coordinates. Also, the curvature invariants all diverge there. This indicates that the central singularity is a physical one - a region of true geodesic incompleteness.

There are other ways to distinguish these singularity types, but you get the idea - the event horizon is a coordinate singularity (no physical significance), whereas the central one is physical.