In late 2012, one of the largest international prizes was awarded for contributions to the theory of black holes. In connection with this, a professional social network hosted a discussion on gravitational collapse between me, the author of the frozar theory, and one of the world's leading theoretical physicists — a professor from Western Europe.
Here he is designated as Professor N. He is also a laureate of prestigious international prizes and, especially important in the context of this discussion, a major specialist in gravitation and the author of a book on general relativity.
The central question was: what does GR actually predict for a collapsing star — a black hole with a horizon, or a gravitationally frozen object, a frozar?
The historical exchanges below are based directly on the original English thread. Obvious spelling and grammatical slips have been lightly corrected; omissions are marked where applicable. Modern clarifications are explicitly separated as Author's Comment (2026).
1. “You are describing an ordinary black hole”
Reading the abstract of your paper, I can guess what this misconception is about: you are exactly describing a horizon there, while claiming that horizons do not exist, a mistake made more often by beginners.
According to the standard rules of GR, you should use coordinates where singularities due to coordinate artifacts are removed. Look up in the book what Kruskal coordinates are. Then you can see that the horizon is real, and that, using proper coordinates, an ingoing observer will continue his journey to reach the real singularity in a very short time.
Just compute the observer's eigen time, and find out that he is still young when the outside world reaches time = infinity. That's where he thinks he crosses the horizon. But his time goes on while the universe reached its end.
Your “frozar” is exactly what in normal jargon is called a black hole. I don't care about names. Frozar, black hole, great spaghetti monster, it's all the same thing.
More to the point: the standard theory of black holes is not produced by complete idiots as you seem to be claiming.
2. “Your own slow learning process”
Although I did not check all your math, I don't think there will be points of disagreement there. But what you describe as a “frozar” is exactly what in GR is called a black hole. My earlier comments not only apply to your abstract but also to your paper and its conclusion.
Your own Figure 2 is exactly what we call a black hole: a situation where infalling observers reach a finite proper time while outside observers reach infinite time. As you can also see in your own figure, the asymptotic state is approached exponentially, so, within seconds, all that is left outside is the Schwarzschild configuration.
What you failed to notice is that the ingoing observer experiences no singularity or other barrier as he or she goes through the point r = r_s and enters a new region of the universe. The singularity is reached a short time later, but this is never registered by the outside observer.
So, instead of shouting that you find something new and disagree with standard literature, you should, more modestly, confess that your paper describes your own slow learning process on what a black hole really is.
It is not of interest to the professionals who knew this already long ago.
From the Socratic dialogue
I'm sorry, Zahid, I'm not as stupid as either Salviati or Sagredo. They both fail to understand what a black hole is. They got the math right, but fail to observe that it's the singular relation between proper time and Schwarzschild time that defines a black hole.
Yes, everybody knows that a distant observer thinks ingoing matter stops right at the horizon. That's what makes this a horizon in the first place. Now force your Salviati and Sagredo to look at what happens according to the ingoing observer. He can get through the horizon. Just transform to the right coordinate frame to see that.
3. Eight questions and eight answers
| Question by Z. Zakir | Professor N.'s answer |
|---|---|
1 In general relativity (GR), the surface of a spherical object as an extended body is defined on hypersurfaces of simultaneity t = const, and the proper time on the object's surface at any moment is strongly related to a finite moment of world time t. Do you agree? | 1 Not quite. In GR you may use any coordinate set you like. Simultaneity is not well-defined. The Schwarzschild coordinate t is convenient, but that's all there is to it. |
2 At any finite t, the corresponding proper-time moment is less than the value at which the surface would cross the gravitational radius. Passing to Kruskal coordinates does not change the physical simultaneity of distant events, and at finite t one again obtains r > r_g. Do you agree? | 2 Fine. |
3 As the surface closely approaches the gravitational radius, proper times rapidly freeze in terms of t: first at the center, where the time delay is maximal, then in higher layers, and finally at the surface, still outside the gravitational radius. Do you know this feature of standard stellar models? | 3 It is well-known. This defines gravitational collapse, and describes what we know as a "black hole". |
4 The freezing of all processes throughout the whole volume also means the practical stopping of collapse. How can you ignore that? | 4 I don't ignore that. That's what a black hole is about. I don't think we disagree about the mathematics. But, as you know, this is true when you use the time coordinate t. The point about the black hole is that the collapse completes at finite proper time for the ingoing observer. |
5 Thus, in GR there is a universal factor preventing collapse: strong gravitational dilation of proper times, a fundamental physical phenomenon by which GR differs essentially from Newtonian theory. Do you agree with this physical fact? | 5 True and essential. |
6 Collapse in GR leads to the formation of frozars (frozen stars), whose surface and all layers throughout the volume are frozen; the particles' worldlines remain timelike and become almost parallel to the t-axis and to one another. Do you agree that such an object is not a black hole? | 6 I agree with the math but not with the last statement. This IS what we call a black hole. |
7 In my paper, frozar formation is shown for standard idealized collapse models: a thin dust shell, a dust ball, a constant-density star, and a star with ultrarelativistic matter. Did you read or repeat my calculations before rejecting them, and what is your conclusion? | 7 Yes. And you can also derive easily that, ignoring quantum mechanics, no radiation will be seen to come out of this thing—or more precisely, any radiation that comes out will be suppressed exponentially in time t. This suppression is so fast that, after only a short amount of time, nothing will be seen to come out anymore. So our object is black. That's why it is called a black hole. |
8 Thus, in GR horizons and singularities, as artifacts of Newtonian theory, do not arise and black holes are forbidden. What is your conclusion, taking the above argumentation into account? | 8 Wrong. The horizon is not a singularity, but you have to use the right coordinates to see that. |
No need to change my opinion. I hope that you will learn what a black hole is. You got the basic math right, but you didn't draw the right conclusions.
(Professor N.'s note: “This stupid editor does not allow me to begin sentences with numbers, it seems.”)
4. Simultaneity: “Landau–Lifshitz is ancient literature”
Landau Lifshits is ancient literature. Their books were well respected but were written at a time that only about 1% of the research on GR, black holes and the Big Bang had been performed. Since then, our understanding has improved enormously.
Your quotes on simultaneity are probably a misunderstanding of what they might have wanted to say. One can “define” simultaneity by referring to a special coordinate frame, such as the one where the collapsing star becomes stationary, but this does not mean you could or should not define it any other way. There's no “god given” definition.
Anyway, some arbitrary procedure to synchronize clocks does not affect the properties of a black hole and has little to do with the previous discussion. You can use any coordinate system you like and discuss how light rays transport information.
If information about an ingoing observer can only reach the outside world when it is emitted before some moment τ1 in the proper time of the ingoing observer, while information emitted at, or after, τ1 never reaches the outside world, then we have a black hole. What you call a “frozar” is a black hole.
5. “The collapse has not yet taken place”
“As usual, we use light; we send a light signal from A to B and back to A.”
Oh, or is it from B to A and then back to B? If they are moving with respect to one another, the outcome will be different! What you are doing here is choosing the reference frame of A — or perhaps B? — to define simultaneity. That's possible, but ambiguous. And irrelevant.
Indeed, the collapse has not yet taken place in your definition of time. The black hole is the metric at all spacetime points where the dust shell has already passed. If, inside the Schwarzschild radius, your definition of time were used, that's the infinite future, but that's your problem.
What makes this thing a black hole is that an ingoing observer has only a finite amount of proper time at his or her disposal to escape. As soon as the Schwarzschild radius has been reached, no return is possible; not even light signals can be sent. But your definition of time does not allow you to see or understand that. Please read my previous comments.
6. Head in Paris, body in Hollywood
7. The relation between proper time and world time
8. Achilles and the tortoise
The same mistake was made by the Greeks when they proved that Achilles would never beat the tortoise. Achilles froze when he was about to reach the tortoise. ...?
You write that the constrained character of proper times is your new contribution. I'm afraid not; this was noticed all along by anyone studying black holes. But most of them realized that, after passing rSchw, proper time continues until the singularity is reached at r=0, a few proper moments later.
9. The map of Antarctica
Zahid, I'm sorry that I could not get my point through to you. As I said, I agree with most of your math. Except when you treat the “frozar” as a black hole with a sharp boundary on the horizon, as if an ingoing observer cannot get through. That's incorrect, according to GR — not according to Newton.
The horizon is a coordinate singularity. Think of a map of Antarctica, focusing on the South Pole. Suppose it is obtained by projection from the center of the Earth. The equator would then be at infinity on that map. This is like the line r=r_{Schw} in your coordinates.
But someone in a ship crossing the equator does not notice anything stopping him from crossing the equator. Our ingoing observer would also not notice a thing. His proper time will continue after τ_1. If you don't see that, I have nothing else to say.
Oops ... “The evaporation, as described by enthusiasts of black holes, is absent in GR...” ??? I see, you haven't understood that either. Sorry about that, but I won't try to explain it to you; this is hopeless.
To all others on this blog: Hawking radiation is a beautiful feature of black holes. There are points that can be discussed further, but papers of people who say that “there is no Hawking radiation” miss some important points of locality and causality in QFT. And you must do better than our friend Zakir, I'm sorry to say.
10. When does the inner region arise?
You misinterpret what Hawking and Ellis are trying to say. The only part of the black-hole metric relevant for physical black holes is the part of the Penrose diagram that lies in the future of the collapsing matter. That includes a part where 0<r<r_0, but it is after the collapse took place.
What we call black holes does not use “the hypothesis about pre-existing Schwarzschild geometry inside the gravitational radius”. Hawking and Ellis use this only to introduce the mathematics.
You want to restrict the solution to outside r=r_0. The problem is that the infalling observer can't tell whether he passed r=r_0 or not. So why make that restriction? You failed to notice that, after the collapse, the region r<r_0 can be reached by an infalling observer.
11. Local and global
“Global simultaneity” does not affect the argument, because the laws of physics are assumed to be locally defined, not globally. You have to talk about light signals and such.
The crossing is legal for the local observer, but the distant observer cannot see what happens after the crossing. If a late observer cannot see how the local observer crosses a line, then that line is called a “horizon”, and the configuration is called a “black hole”.
You are not following standard GR, I am. Standard GR is unable to tell an ingoing observer that he should die when he crosses a line that he cannot see.
12. Hawking radiation and the energy balance
In the standard Hawking picture, the positive energy of the outgoing quantum is balanced by a negative-energy contribution that falls through the horizon and reduces the black-hole mass. For a frozar, both particles of the pair have positive energy: the infalling particle increases the mass, so the outgoing positive-energy particle has no compensating negative contribution.
Zahid, I'm sorry but you keep repeating things that are not correct. You did understand some of the main features of gravitational collapse, but not everything.
Imagine an observer sitting just outside the collapsing dust shell. Yes, you can compute the metric exactly. You should be able to see how this observer, just after the shell of matter, dives into the r<r_0 region. According to local physics, his immediate environment is close enough to the vacuum state.
Your claim that evaporation of the star violates energy conservation is wrong. Also, that it would be against GR is wrong. According to GR, I may exchange your definition of time t for mine — say, the observer's proper time τ.
This transformation is very special: at τ≥τ_0, there is no external time t. So this is a new region of the universe. Hawking radiation is what you get if you apply this to QFT.
From what you say I can tell that you never considered that calculation. The quantum states at τ≥τ_0 are invisible to the outside world, hence they have to be averaged over, as is normal routine in statistical physics when there's a part of a system that you cannot see.
If energy is defined with respect to your external time t — as it indeed should be — then the particles in the forbidden region turn out to have negative energies. Defining the vacuum state |vac⟩ as the state of lowest possible energy for the ingoing observer, one finds that after the transformation this vacuum does not act as a vacuum for the outside observer, but is a mixed state. You then calculate that it contains Hawking particles.
It's very straightforward, and please don't start arguing without explicitly having done the calculation.
13. Closing exchange
Zahid: good bye for now. Come back after you at least understood how the Hawking effect is calculated.
The discussion did not end in agreement, but it made the central disagreement quite precise. Frozar theory requires strict consistency between the two descriptions of each particle's motion. From r(t), r(τ), and the equality r(t)=r(τ), one obtains τ(t). Corresponding relations for the particles in different stellar layers describe the successive gravitational freezing of their proper times relative to the world time of the surrounding world.
The central question throughout the discussion was therefore: which mathematically admissible continuations of spacetime are actually physically realizable states of a collapsing star?
In another branch of the same discussion I formulated the relation more explicitly. The same motion of a particle can be written as r = r(t) and as r = r(τ). Since these are two descriptions of the same event on the same worldline, the spatial position must coincide: r(t)=r(τ). This determines τ=τ(t). In the frozar theory this relation is constructed and enforced for the particles of every layer of the star.