Dialogues on the Fate of Stars:
From Black Holes to Frozars

Prologue

Sagredo. If you remember, almost 40 years ago you and I had an argument about gravitational collapse, and at that time you persuaded me that when a sufficiently massive star collapses, a black hole forms. But my doubts did not disappear then. Since that time I have studied the problem and found new arguments. Are you ready to continue that dispute?

Salviati. Yes, I remember, but back then I proved to you that this is a strict consequence of Einstein's general theory of relativity (GR)! Many specialists had doubts at the time, but now this concept has won a complete and unconditional victory: black holes have been accepted by the entire scientific community, and thousands of them have been discovered by astronomers. The very term “black hole” has become part of universal human culture; it is used in many fields and even children know it. So what is there to argue about?

Sagredo. We are not talking about every possible sort of hole into which the benefits of civilization disappear, but only about the physical concept of black holes and its applications.

Salviati. Then I am ready to return not to an argument, but to explain to you once again this remarkable theory of the most astonishing objects in the Universe and to tell you about the latest successes of this science.

Sagredo. You did not pay attention to what I said: I already know everything you are going to tell me about them, because your favorite concept has not changed since then, and I have read almost everything you have. This time the dialogue will not be between an expert and a novice, but a genuine debate on equal terms.

1. GR Leads to Black Holes. It Has Been Proved, and There Is Nothing to Discuss!

Salviati. I did not expect this from you; frankly, I had a better opinion of you. Still, why not argue, especially since the result is obvious to me in advance. But the conditions are by no means equal: on my side are the entire scientific community and tens of thousands of publications, hundreds of conferences and, ultimately, the lives and careers of thousands of people who have spent almost their whole lives developing various aspects of this outstanding achievement of twentieth-century science. An enormous literature for readers of every level, many films, and thousands of television programs deal with black holes. On the Internet you will find millions of references to this term and hundreds of discussions in dozens of languages. So you had better study some of this even more carefully before starting a serious dispute. Otherwise we shall simply waste both your time and mine, and you will acquire the bad reputation of a person who, against all evidence, tries to refute a reliably confirmed scientific theory that has entered the golden treasury of humanity and is obvious even to any student.

Sagredo. As I said, I fulfilled your condition beforehand, and now it is not you who will persuade me, but I who will be able to persuade you of the opposite, despite the temporary victory of your favorite hypothesis and its promotion on an unprecedented scale. At that time I did not understand many of the subtleties of GR, while you worked on it closely, and I trusted you. Since then I have studied the best literature on this problem—not so much textbooks, where proofs contain “blank spots” and often resemble hand-waving before trusting students, but the works of the founders themselves, who were not nearly as categorical and confident as their followers—and I have also followed the later scientific literature. To my surprise I learned, and you know it well too, how ambiguously even the founders themselves applied GR: they hesitated among different versions and often settled precisely on variants that were formally correct but physically untenable. Meanwhile my doubts about the soundness of the black-hole hypothesis were confirmed, and now I know exactly that this hypothesis does not follow from GR, while the interpretation of observations is unfounded and observations of objects of an entirely different nature are being presented as observations of black holes.

Salviati. So that is how you put the question! If I did not know you personally, I would not even continue this conversation. When I receive messages claiming that black holes do not exist, I do not read any further; I make their authors persona non grata for my mailbox, give negative reports on their dubious papers with obvious errors when they are submitted to journals, and at conferences and seminars I do not recommend putting their talks on the program, so as not to waste the audience's time.

Sagredo. Our qualifications hardly differ, so if, knowing this, you turn your back on me as well, then so be it—it means you are engaged in anything you like, but not science.

Salviati. But you know perfectly well that there have been hundreds of objections to black holes, and for more than eighty years thousands of highly qualified professionals have tried to construct alternative models. Even Einstein himself tried, but could not give convincing arguments for factors that would keep a doomed star from collapsing inside its gravitational radius. Many tried to crack this hard nut—after all, the very author of the term “black hole” called gravitational collapse almost the greatest crisis of twentieth-century physics. But later it was proved that all the troubles are safely hidden beneath the horizon inside a black hole, and the crisis was overcome. So for me this question is closed once and for all, and I advise you to treat it the same way. I do not want to continue this pointless dispute, which is equivalent to an attempt to refute GR, whose predictions have been brilliantly confirmed for ninety years, and to cast doubt on astrophysical data about thousands of existing black holes.

Sagredo. I am asking you to continue the scientific dispute in which you supposedly convinced me because of my inexperience and naivety, while you are saying that the matter is already the final truth. If you are so sure of it, what would it cost you to refute the new arguments as well? Erroneous hypotheses that have become generally accepted paradigms are usually sustained mainly by trust in authorities and by the fact that many people are not accustomed to using their own heads for their intended purpose.

Salviati. Very well, I am ready to hear your arguments, if only out of a desire to bring you to your senses. So what makes you distrust this theory? You have almost accused it of being unscientific! Why?

Sagredo. For one thing, we have not even touched the theory yet, and already you exalt it to the skies and shield it from criticism. First, you make the controversial claim that black holes are the sole and rigorous consequence of GR, and then you dismiss every argument against this idea, even arguments made within GR itself, hiding behind the authority of the theory and the demagogic phrase that such criticism is an attempt to refute GR. Second, observations of objects whose nature is still unknown and for which there is no evidence at all about their internal structure are loudly and confidently proclaimed by you to be discoveries of precisely the black holes you desire, on the assertion that allegedly nothing else can exist. Third, without knowing my arguments against them and without even trying to learn them, let alone discuss them, you confidently draw conclusions in your own favor. Is that a scientific approach?

Salviati. I do not want to start this dispute precisely because I want to preserve respectful relations between us. You like to analyze things and you think very critically, but there must be a limit to everything.

2. GR Leads Only to Frozars? That's Nonsense!

Sagredo. Yes, the years have changed you, and it seems I came to the wrong address. It is a pity the debate will not take place. But before we part, each keeping his own opinion, I would like to clarify the impressive statistics you cited. First, if of the several hundred alternative approaches to collapse we retain only those that remain within standard GR, only a few dozen are left. Second, if from these we also exclude approaches that postulate as-yet-undiscovered forces, particles, or mysterious forms of matter or energy, the whole set of realistic alternatives narrows to three or four. Third, not many specialists studied even these few credible alternatives, and unfortunately, having begun to move in the right direction, they made hidden or explicit errors, thereby driving themselves into dead ends and, worse still, discrediting these nearly correct paths in the eyes of others.

Salviati. Does it not seem to you that three or four equivalent alternatives are too many for a correct theory? Unlike this small and confused scientific opposition, black-hole theory is unambiguous, beautiful, mathematically rigorous, and confirmed by observations. So let us part here: I really am in a hurry to attend a major international conference on the internal structure of black holes at one of the world's finest resorts, and when I return I shall gladly tell you about new facts concerning actually existing black holes. Although, if you have got this absurd idea into your head that black holes do not exist, then no observations will ever persuade you otherwise, and I deeply regret that your sense of reality has begun to fail you.

Sagredo. I never said that all these few alternatives were equivalent. They all contained one common correct element, after which the variations and speculations began. It was precisely these latter parts that were doubtful. But I have become convinced that recently only one consistent interpretation of relativistic collapse has remained, one that is unambiguous and follows rigorously from GR without any hypotheses at all. This is the theory of frozars—relativistic stars with frozen matter and structure. Do you know about it?

Salviati. No, and I have no intention of learning about it, because the interpretation of frozen stars is not new and is a stage long since passed. It does not look as though you have studied the literature.

Sagredo. We shall talk about that later. And do not forget that everything new is well-forgotten old; if anything, the new idea becomes more solid when people return to it after so many years of trial and error. As for your trip, I doubt you will learn anything new there apart from the pleasures of the resort, because the structure of your black holes is simple and was described long ago in every textbook, while observations made within that same hypothesis are completely useless for questions about their internal structure. The external manifestations of compact stars that, as a result of collapse, have entered the frozar phase are, in simple cases, practically the same as those of your black holes. Since only these external effects are accessible to observation, everything they tell you about the thousands of black holes already discovered I now attribute instead to the thousands of frozars already discovered, and I advise you to interpret it in exactly the same way. I want to help you begin looking at existing hypotheses more critically and not be quite so trusting. Our goal, after all, is the same regardless of personal preferences: both of us are interested in how these compact stars are actually structured, are we not?

Salviati. That is true enough, but frankly we have had a rather strange dialogue: each of us has insisted on his own point and neither has listened to the other. So far the only thing on which we have agreed is to accuse each other of being unscientific. And here, let me remind you, you are accusing almost the entire scientific community, whereas your own view, which you keep trying to impose on me, is shared—apart from its author—by no more than one and a half supporters. Does that not strike you as doubtful and even somewhat immodest?

Sagredo. As for numerical superiority at the moment, you are of course right. But how many supporters does any correct idea have when it first appears if it is incompatible with what is generally accepted? Recall the history of any important consequence of GR before it was confirmed—for example, the expansion of the Universe. Even Einstein himself categorically rejected that idea at first and accepted it only later, admitting that he had made a mistake by trusting intuition too much instead of the equations of his own theory. The situation with frozars is the same: they follow from the equations of GR, while black holes are dictated by Newtonian intuition. In such a situation, what matters and is decisive for science is not the number of supporters who trust those working in the field, but something else—the scientific integrity of those people. I know both versions well, the one with black holes and the one with frozars, whereas you know only one, your own version, and do not want to know the second, which essentially had not yet been worked out when the authors you trust wrote their papers and books. So it is better to count the chickens in the autumn. And modesty in science means being adequate to reality; what is immodest is to attribute to the enormous real world, and impose on all humanity, something that seems correct to many and on which people have merely agreed, but which is never actually realized. Evading criticism is always easier than answering it on the merits.

Salviati. I am not evading your criticism or that author of yours, whoever he may be. It is simply too expensive a luxury to spend time and effort on people who doubt a theory accepted on a world scale. You simply do not imagine or appreciate the scale of it: the black-hole concept is taught at more than ten thousand universities by tens of thousands of professors to millions of students; it has been set out in hundreds of millions of copies of books and journals and brought by television to billions of viewers! Many billions of dollars have already been spent testing its consequences—and all of this is simply a mistake?

Sagredo. The broad popularization of black holes has contributed a great deal to the development of astrophysics, and that is a fact. Many results connected with phenomena outside the gravitational radius remain valid—and that is almost eighty percent of all scientific publications and public demonstrations—and that too is a fact. But everything concerning fantasies about the internal structure of black holes is a grand misunderstanding and one of the curious legacies of such a contradictory and ideologized century as the twentieth, and we should rid ourselves of that legacy as soon as possible. You know perfectly well that mass acceptance of a hypothesis does not yet mean it is true; propaganda does not need scientific argument. Recall how many entertaining feature films there are about a time machine that carries you into the past and lets you change something there. All those jumps over millions of light-years through wormholes faster than light, seriously promoted by famous scientists, are no better.

Salviati. Well, if you accuse it of ideologization, I begin to feel somewhat uncomfortable—where do you see all this? I do not. The concept is simply so paradoxical and its consequences so captivating to the imagination that millions of people enjoy taking an interest in these objects, that is all. For the moment I agree that it does no harm to hear an opponent's arguments, even if they have already been refuted thousands of times, and it is better not to stoop to reproaches. I see in you the signs of an emerging megalomania, and you need to be cured of that: you are claiming that you know the one and only correct answer while the entire scientific community and this enormous worldwide audience are mistaken. Very well, then, I am prepared to examine these frozars of yours, or whatever you call them. Just do not be offended if I smash them to pieces.

Sagredo. I am glad for you and congratulate you: you have just completed exactly half the path at the end of which you will become a supporter of frozar theory. You have overcome the main barrier on that path—the psychological one—and, although still prejudiced, you are now prepared to examine a theory that rejects black holes. So we shall yet see which interpretation gets smashed to pieces.

Salviati. Well, well. I am beginning to envy your enthusiasm, although even that makes me suspect that you have simply become carried away by something and that time might sober you up. But I shall do it faster. For now, please begin with what this interpretation has in common with the old black-hole concept, and then tell me what the difference is, why it arises, and what follows from it. Agreed?

Sagredo. Entirely. In the black-hole concept the collapse process has two stages—before and after (including the very moment of) the object's surface crossing the gravitational radius, where the infall velocity reaches the speed of light. The theory of frozars consists of the standard applications of GR to the first stage of collapse in terms of world time (the time of distant observers), plus the assertion that the second stage of collapse, predicted by Newtonian theory, is absent in GR. As you can see, at the first stage of collapse the two interpretations are essentially identical, because the same equations of GR are used with the same standard solutions, whether explicit or numerical; only different aspects are emphasized. Most predictions of the old black-hole concept concerning phenomena outside the stellar surface remain valid with small refinements. That is all.

Salviati. Not much to it, though I am a little calmer now. As I understand it, in this interpretation almost all observable consequences of the old concept are retained, and the dispute will mainly concern different interpretations of the unobservable internal structure of collapsed objects.

Sagredo. Broadly speaking, yes. The internal structure of frozars is fundamentally different from that of black holes, and in principle it is observable, although much more difficult and much rarer to observe than the external effects.

Salviati. Then the discussion moves onto purely theoretical ground, and now I have you. There are famous, mathematically rigorous theorems proving the inevitability of singularities in GR, according to which all the matter of a star very rapidly, in proper time, falls to the center and is compressed to a point of infinite density.

Sagredo. The issue is not the mathematical rigor of those theorems, but the fact that they are proved under the condition that “if the surface of the object crosses the gravitational radius, then...” and so on. That “if” is natural in Newtonian physics, but, as frozar theory shows, it does not occur either in GR or in the real world. The surfaces of objects never cross the gravitational radius either in world time or in proper time.

Salviati. There is the error. In GR the surface fails to cross the gravitational radius only in world time, but crosses it without difficulty in proper time, and that fact lies at the foundation of the whole black-hole concept.

Sagredo. And for that reason world time is declared to be a bad time coordinate and proper time a good one, correct?

Salviati. Yes. The true time expressing the natural course of physical processes at each point is proper time, whose intervals are invariant in GR. The world lines of the particles on the surface therefore continue calmly inside the gravitational radius when expressed in proper time. World time does not represent this, so it really is an unfortunate choice of coordinates.

Sagredo. Now we have come to the difference between the two interpretations. Frozar theory argues the opposite: that proper time is the poor choice for a time coordinate, whereas world time expresses the true course of the simultaneous evolution of all parts of the star in conjunction with temporal evolution on the global scale.

Salviati. So in this new interpretation everything is reversed: proper time is a bad time coordinate and world time a good one. Is that what you mean?

Sagredo. Exactly. And there are substantial grounds for this within GR.

Salviati. Well, this point of view at least has a right to exist in principle, although I cannot imagine what you will then do with the second stage of collapse. Will you simply deny it because it is absent in your particular choice of coordinates? That is exactly what I suspected. Many opponents of black holes usually reduced their entire proof to some special choice of coordinates and then exaggerated the significance of their arbitrariness. In the black-hole concept, the choice is made in favor of a physically distinguished and invariant coordinate—proper time—and arbitrariness is excluded here by definition.

Sagredo. Yes, you are right with regard to many earlier alternative interpretations. But not with regard to frozar theory. In GR both time coordinates express two physically equally essential aspects of temporal evolution, and therefore both are physically distinguished. So, to remove suspicions of trivial arbitrariness, let us make the definitions precise.

Salviati. Exactly—give precise definitions, otherwise we shall wander off in the wrong direction.

Sagredo. The point is that time has two properties: the rate at which it passes and the simultaneity of events at different points, and both properties of time are important and fundamental. In Newtonian physics there is no need to distinguish them, but in GR there is. Therefore, during the first stage of collapse, events along the world lines of particles on the surface and inside the star are parameterized by two times—proper time and world time. The first determines the rate of local processes; the second determines the simultaneity of these events in different parts of the star and with distant events. In this sense, the parallel description in world time gives physical meaning to instants of proper time by tying them to definite epochs of the large real world in which the star as a whole is immersed. During the first stage of collapse, which of these two kinds of time you use is formally a matter of convenience and of the task you have set, because for every event they are uniquely related to one another. Do you agree with that?

Salviati. Yes, yes, all of that is well known.

Sagredo. The advantage of proper time, as you said, is that its intervals are invariant and have a direct physical meaning. But proper time itself is a poor time coordinate, because its rate, slowed by gravity and by the infall velocity, differs for different layers of the star and even for the same layer changes strongly as compression proceeds. As a result, for a given layer its intervals are completely non-equidistant with respect to world time, with respect to the proper times of other layers, and even at different moments for that same layer. The main drawback of describing evolution by proper time is that from the proper-time instants in different layers one cannot define the star as a whole—as an extended object whose parts coexist simultaneously.

Salviati. I agree that proper time has to be used with great care.

Sagredo. The physical simultaneity of events in the star, by definition, is specified by instants of world time. Only by grouping events with the same value of world time can we assign them to separate parts of the star that coexist simultaneously and therefore at that moment form such an extended object as the star.

Salviati. So far you are right only in saying that simultaneity of events is indeed not taken into account in the black-hole concept and that this aspect of stellar evolution is not given the exceptional importance you are trying to make me give it.

Sagredo. The real world, as we know, is a collection of simultaneously coexisting objects. In GR there are many complications with the simultaneity of distant events, but wherever global simultaneity can be defined, as in the case of spherical stars, you must describe the evolution of extended objects as a whole by their “snapshots” at definite instants of world time. Any other picture has no relation to reality and is merely a mathematical abstraction. Do you agree?

Salviati. All of this is true in special relativity, and in the static field of a spherical star things should be almost the same. Do not forget, however, that in moving reference frames the picture changes.

Sagredo. Not almost the same, but exactly the same, though with some precautions in synchronizing clocks because the speed of light is reduced in strong fields. The motion of the observer's reference frame does not alter the physical picture of the star's evolution, so it is sufficient to describe that evolution in the rest frame of the star's center, where there are no artificial kinematic complications. Therefore, in the black-hole concept only the first stage of collapse is described correctly, although very briefly and not clearly enough. The second stage of collapse, however, in which proper time is assumed to be disconnected from world time and forced to run on by itself, ceases to have any relation to the real world, where all objects and all their parts always coexist at a definite instant of world time. In reality, proper time is precisely the poor time coordinate that distorts the true course of collapse. World time remains, as it always was, a good time coordinate for as long as the star exists in our Universe.

Salviati. So a good clock is not one that counts the beats of my heart, but one that shows world time. I do not object.

Sagredo. When you freeze with terror or, forgive me, die, your heart-clock slows down or stops altogether, whereas your world clock keeps running. The collapse of a star is practically its death, so black-hole theory is based on describing processes at the rate of the heartbeat of the dead, which is absurd.

Salviati. Once you began giving detailed definitions, the differences became clearer and frighteningly graphic. So before we get lost in details, start from the beginning. What is this new theory of collapse based on, and where does it lead?

Sagredo. Yes, it is better now to specify the main propositions of frozar theory. First, we work within standard GR without any hypothetical additions, and therefore we use standard solutions of Einstein's equations both in the exterior region of the star and material solutions in the interior, with the usual model simplifications.

Salviati. It is good that you add nothing to GR. It will be easier to refute you. Most opponents protect their hypotheses against direct logical refutation by cleverly adding some detail that cannot be tested for the next few million years.

Sagredo. Calm down. This approach is simpler and clearer than all the previous ones and is not based on any clever “inventions.” The whole secret lies precisely in removing from the theory the erroneous complications of previous attempts to describe collapse.

Salviati. I find that hard to believe. Most likely it is your claim about the erroneousness of generally accepted truths that is erroneous. So what comes next?

Sagredo. Second, we consider the collapse of an ordinary star containing only ordinary matter and known forces, and therefore initially its surface lies outside the gravitational radius.

Salviati. You add nothing here either? A pity. This is exactly where you could allow yourself some undetectable force, some ultra-dark matter or energy that could never be discovered, and there you would have it—a ready-made alternative to black holes! And, most importantly, there would be no way to refute it!

Sagredo. Like you, I hope, I am interested not in invented worlds, of which there are infinitely many, but in the one real world, for whose description the simplest form of GR turns out to be sufficient. So I shall continue. Third, during the first stage of collapse let us suppose that the time of events at every point is recorded by two kinds of clocks: proper time by standard clocks and world time by coordinate clocks. The rate of proper time slows differently in different places and the stronger the gravity, the greater the slowing. That is why coordinate clocks are needed, sped up relative to standard clocks so that everywhere they run synchronously with the clock of a distant observer. Do you agree that all this is consistent with GR?

Salviati. Yes, of course, so far you are stating standard things. I see nothing new. You are still harping on the first stage of collapse, although in GR all of this is elementary and clear.

Sagredo. Excellent. At least we have begun to reach agreement on something. The third circumstance is crucial for frozar theory, and I deliberately specified the details needed later so that you would not think something new, and therefore dubious, was being introduced. Moreover, I am glad that all this is elementary and obvious to you; it will make it easier for you to understand why GR leads precisely to frozars rather than black holes. It is also clear that these common propositions, identical in both interpretations, lead under certain conditions to identical consequences in both interpretations.

Salviati. Then if there is no difference in the consequences either, you must go straight to black holes! So this is much ado about nothing?

Sagredo. Do not hurry and do not generalize. I said that the consequences coincide not in every case but only under certain conditions.

Salviati. Then specify those conditions and I shall show you where the error in your reasoning lies, because if you had done everything correctly you would have arrived at black holes and nothing else. That is what happened to everyone who seriously studied this problem, and it will happen to you too, unless you have botched something.

Sagredo. For all your self-confident irony, I shall hold you to the words “if you had done everything correctly.” Why are you so certain that in the black-hole concept “everything was done correctly”? To avoid the usual misunderstandings on this point, let me clarify what I mean by that expression. You do everything correctly if, starting from the general propositions just stated and adding nothing that contradicts them, you work in our real Universe with its world time that on average runs uniformly, at every instant of which all physical objects coexist simultaneously, with small corrections to the rate of time due to their motion. For our collapsing star, at each instant of world time and at the corresponding instants of proper time, you then follow the circumference of the stellar surface and of every internal layer. Do you object to that?

Salviati. You are still exaggerating the importance of simultaneity and of world time connected with it. In the black-hole concept, as in GR in general, the true time in which any object lives is regarded as its evolution in proper time. Every student knows that in terms of proper time collapse proceeds exactly as in Newtonian theory: the surface crosses the gravitational radius without difficulty and all the matter very rapidly falls toward the center, forming a singularity, an infinitely dense state, even though for a distant observer, in his world time, the surface of the object freezes forever outside the gravitational radius. So first you want to replace proper time by an arbitrarily introduced world time and present your arbitrariness as the only correct description? I shall tell you at once: if that is your whole secret, this trick will not work. It has nothing to do with GR, which allows the times of different observers to be introduced, all of them relative, while only the object's own proper time is invariant.

Sagredo. But you agreed that up to the moment when the surface could cross the gravitational radius, every instant of proper time on the surface is uniquely related to a definite instant of world time, and that descriptions of the motion of the surface in terms of the two times are equivalent. And you also know perfectly well that according to GR, at each instant of world time the corresponding instant of proper time on the surface of a spherical star is also uniquely related to the instants of proper time in the internal layers all the way to the center, do you not?

Salviati. Up to that moment, yes, of course, I agree. Although the two times are in principle completely equivalent for describing trajectories, everything is much simpler in proper time. But when the surface crosses the gravitational radius, whether you like it or not, you have to move to other, more suitable coordinates.

Sagredo. So you know perfectly well that outside the star and on its surface, instants of world time also have a direct physical meaning and mark events simultaneous with the corresponding instants of the proper time of distant observers, since they are synchronized with the clocks of those observers. The simultaneity of distant events is what physically distinguishes world time, and it is no less important or objective than the invariance of proper times. If you remember, the theory of relativity began with Einstein's statement that “all our judgments in which time plays any part are always judgments of simultaneous events.” And you have already agreed that in the rest frame of a spherical star the simultaneity of distant events is defined in exactly the same way as in special relativity.

Salviati. Yes, of course, but only up to the instant of proper time when the surface crosses the gravitational radius. After that there can be no question of simultaneity—the star has gone beneath its event horizon, from which no signal can emerge. You put far too much emphasis on simultaneity. In GR one may in general choose arbitrary coordinates at each point, and simultaneity of distant events is not the rule but an exception, and a very rare one.

Sagredo. But you know perfectly well that precisely this rare case occurs around and on the surface of a spherical star, and that global simultaneity of events exists here!

Salviati. How many times must I repeat that yes, and can we please close this subject before my patience bursts! It is simply impossible. Why are you wasting so much effort and time—both yours and other people's—on this nonsense, which students are given as an easy exercise?

Sagredo. When Jupiter was angry, everyone knew he was wrong. But that was long ago. Do not boil over; we are already close to the finish, and we have clarified almost everything we need.

Salviati. Really? I have not noticed your frozars appearing anywhere—where are they? Perhaps, like a magician, you are about to pull them out of a secret box?

Sagredo. For a civilized person the secret box is his computer, and from it he can pull out and show on the screen anything he likes. In any good detective story it seems that people are wasting time on boring details and dreadful trivia—some objects, stains, and so on. Archaeologists too spend a long time digging up fragments of something incomprehensible and only afterward begin building chains of reasoning, reconstructing a complete picture from pieces of a mosaic.

Salviati. Then when will you begin this “chain,” as you call it, of “reasoning”? So far, as I see it, you are digging rather shallowly, and the result will probably be the same.

Sagredo. Before moving on to this logical chain, let us once again clarify which of the details needed for it you have already accepted. Within ordinary standard GR we study an ordinary standard star—spherical and nonrotating—whose fuel has run out and which has begun gravitational contraction, and let the contraction proceed until the radius of its surface becomes close to the gravitational radius.

Salviati. If everything is completely standard, then for simplicity take a thin dust shell, where interactions among the particles do not get in the way and only gravity remains, so the problem can be solved exactly. Then move on to a dust ball, where everything is more complicated but exact solutions have also been known for a long time. It was precisely with these simple and clear examples, with exact solutions within GR, that the inevitability of black holes was rigorously proved. For real stars, once interactions are included, collapse with the formation of a black hole also proved inevitable when the stellar mass is three or four times the mass of the Sun. I remind you of these elementary truths of this long-established science only so that we are talking about the same thing. Otherwise I have a feeling that you are about to muddle everything up, seize on some imprecision in the definitions, build your own elaborate contraption on it, and pull the wool over my eyes.

Sagredo. Building elaborate contraptions and pulling the wool over people's eyes are classic attributes of erroneous paradigms that have temporarily triumphed. They are the ones that avoid precise definitions and boring logic and quickly move on to mysterious phantasmagoria that excite the imagination. But I shall not lull your vigilance with unnecessary refinements any longer. I shall make just one final clarification, and then we shall take the bull by the horns: you will prove to yourself that GR forbids black holes and that collapse leads only to the formation of frozars—ordinary stars, but with gravitationally frozen internal structure. Not only will you reach this conclusion; you will also wonder how anyone can fail to understand such a simple and absolutely obvious fact.

Salviati. And off we go—do you not notice that you are beginning to lose it? I have already prepared a grave in my mind for these frozars of yours and even thought up the inscription: “Here lies a frozar from the heart of a quasar, which died before it was born.” You have already mentioned the horns; I can wait until it kicks the bucket too. Then, I hope, we shall sing the farewell hymn. I shall endure your final clarification, but no more.

Sagredo. What you are carelessly preparing for frozars will be needed for your beloved black holes, although in principle they themselves would be an ideal grave for anything. Incidentally, black holes would indeed be inevitable if Newtonian physics remained valid in strong gravitational fields. But in strong fields only GR is valid, and, as you are about to see, this theory leads only to frozars. So I shall continue clarifying the definitions.

Salviati. Just do not drag it out. Remember, your time is already running out, and so is my patience.

Sagredo. In science, success is the result of patience in thought, as Newton said in words to that effect. Take a deeper breath. When we reach the frozars and you understand how they form, I am sure you will feel such relief that your patience will be rewarded.

Salviati. A little more and I shall be suffering not from lack of air but from bursting with rage.

Sagredo. Irritation in response to criticism is a symptom of error. Incidentally, that is a fairly accurate diagnosis, so concentrate and calm down. Now then, let us clarify our positions.

3. Does GR Really Lead Only to Frozars? This Is Interesting!

Sagredo. Do you agree that in the black-hole concept collapse inside the gravitational radius is unobservable to external observers because, as textbooks say, it lies beyond an infinite interval of world time, whereas before that everything proceeds simply and even prosaically? A dust shell is a very convenient example because everything is known exactly. As a thin dust shell falls in its own gravitational field and approaches its gravitational radius, proper time on the shell and the process of falling itself slow relative to world time so strongly that for external observers the shell freezes forever above the gravitational radius. Empty space remains inside the shell, where the proper times of test particles also freeze forever in terms of world time, and the test particles themselves remain frozen where they happened to be before the freezing. Do you agree with this clarification?

Salviati. Yes, in terms of world time the first stage of collapse is just like that, although in proper time the picture looks completely different. I also asked you to consider a dust ball. Sketch your last—pardon me, your final—clarification there as well, and then we shall see where, and on what banana peel, you slip.

Sagredo. Yes, of course; now I shall turn to the dust ball. When the ball contracts under its own gravity and the particles fall freely toward its center, then as the surface approaches the ball's gravitational radius, the slowing of proper times in terms of world time becomes so strong that the surface rapidly freezes above the gravitational radius. According to exact solutions of the GR equations [Tolman (1934, 1939), Oppenheimer and Snyder (1939)], the slowing is stronger in the inner layers than at the surface and strongest of all at the center, so the center and the other inner layers freeze earlier than the surface.

Salviati. I do not recall reading anywhere that, during collapse in terms of world time, the center freezes earlier than the surface. Are you sure? Have you checked it yourself using the known solutions? Intuitively I always imagined that during collapse the interior plunges toward the center faster than the surface!

Sagredo. Yes, of course I checked, and it is remarkable. Although this fact was contained in those well-known solutions from the beginning, no one paid attention to it. All of this is a consequence of GR when we model the process using the equations of GR and their solutions and, epoch by epoch in world time, follow the positions of the layers of the dust ball, starting from a time when gravity is not yet so strong. We record both the position of each layer at every instant of world time, each epoch, and the readings of standard clocks on that layer, which show an ever greater stretching of proper-time intervals and freezing near definite values of proper time as the ball contracts. This dull picture of the proper times of the ball's layers freezing in world time is a strict consequence of GR and also occurs in the black-hole concept during the first stage of collapse, when the surface of the ball, in its own proper time, is still outside the gravitational radius. Correct?

Salviati. Yes, this picture is indeed a very dull, unfortunate, and distorting representation of the dynamics of the first stage of stellar collapse. But it is only a prelude to the real collapse, which becomes dynamic and interesting as soon as you pass to better coordinates that reveal the true picture of contraction in proper time, especially in the second stage of collapse, where they become indispensable.

Sagredo. So you agree that so far everything has been done correctly for the first stage of collapse?

Salviati. So far, of course, yes. But you still have not touched the second stage of collapse and are avoiding those instants of proper time when the surface of the dust shell or dust ball crosses the gravitational radius and everything falls toward the center.

Sagredo. Thus we have reached complete agreement between the two interpretations in their descriptions of the first stage of collapse, that is, up to the supposed moment when the surface crosses the gravitational radius. Congratulations: you are now one step away from becoming a supporter of frozar theory.

Salviati. Frankly, I did not notice that, and I do not feel that my convictions have changed in any way. When you are one step away from turning into a camel, I shall warn you in exactly the same way, and you will tell me that you do not feel it!

Sagredo. Turning into a camel is your fantasy; the fact that even while strictly following GR I have to prove that I am not a camel is reality. But when reality is sad, it is better not to despair, but to remain optimistic and act to improve it; then reality will change for the better.

Salviati. Really? And can you show an example of such a transformation of reality by the efforts of your thought alone? Now I understand where these frozars of yours are going to lead us!

Sagredo. Take your own case. As I said, even now you believe yourself to be a supporter of the black-hole concept accepted throughout the world, and therefore with a clear conscience you feel comfortable within a scientific community where only this concept is identified with serious science. You consider yourself entitled to enjoy all the benefits of civilization in the status of an accomplished, and therefore respected and well-known, scientist recognized as such by other respected scientists. But believe me: when we part, you will no longer have all of that. You will be convinced that GR leads not to black holes but to frozars and will therefore immediately find yourself in opposition to the entire scientific community. Then you will not be able to take part in that international conference to which you have been invited to give a report on applications of a concept you will no longer consider correct; you will no longer publish or lecture on black holes; and no one knows what effect all this will have on your career and well-being. There is an example of the transformation of your personal reality. How do you like it?

Salviati. What I dislike is not what you are predicting for me, but your self-confidence in thinking that you can manipulate my views, which have not changed at all during our conversation and, I think, are not in danger of changing in the foreseeable future. So now it is you who have begun to fantasize. But I have to go; I told you your time had run out, and mine even more so. Goodbye. Your frozars can do without me.

Sagredo. I too should have left long ago, so goodbye. Especially since I have almost completed my task and you have already admitted that frozar theory is consistent with GR.

Salviati. Really? What are you saying? When did I admit that? I still have not understood what kind of theory this is or what sort of creature a frozar is. So do not get carried away and do not attribute to me what you are so eager to hear.

Sagredo. I have already given the definition of this theory, and you saw that it is very simple. The description of the first stage of collapse within GR in terms of world time is frozar theory. A star at a definite finite instant of world time, with gravitationally frozen layers, is a frozar. At every instant of world time, the corresponding instants of proper time for the surface and the inner layers of the star are practically frozen at limiting values that differ from layer to layer. The center freezes earlier than the surface, at a somewhat smaller value of proper time. That is all!

Salviati. Well, if you are merely giving a new name to the first stage of collapse, which has long been known and is perfectly obvious, then say so instead of elevating it to the rank of a new theory. But I still do not understand what you have done with the second stage, when in proper time the stellar surface crosses the gravitational radius and falls toward the center. The description of precisely this second stage is the essence and chief achievement of the black-hole concept.

Sagredo. And the essence of frozar theory—the reason for introducing a new name—is that for the first time it argues that in GR this very second stage of collapse does not exist, and that collapse is entirely and completely reduced to its first stage. At every instant of time in the real world, a thin dust shell and a dust ball whose surfaces lie very close to their gravitational radii are in a frozar state with frozen internal structure, and nothing else ever happens to them. The instant of proper time at which the second stage of collapse is supposedly to begin never arrives, because proper time on the surface freezes gravitationally relative to world time, that is, relative to the average rate of evolution of the rest of the matter in the Universe. Instants of proper time are uniquely related to instants of world time, while world time on average runs uniformly on the scale of the entire real Universe.

Salviati. So you are saying that the old black-hole concept should be split into two parts: the part describing the first stage of collapse should be called frozar theory and regarded as the only realistic application of GR, while the part describing the second stage should be discarded as a mathematical abstraction. Is that right?

Sagredo. Exactly. At last you have begun to think clearly and soberly.

Salviati. So far I am only formulating your claims more precisely, and I do not agree with the claim that there is no second stage. So what, in your view, stops the collapse—simply freezing in terms of world time? That was well known before.

Sagredo. Yes, exactly. A star is immersed in the real Universe, where cosmological time on average passes practically uniformly, while on the star itself, according to GR, proper time slows relative to this cosmological time as gravity grows during contraction. Collapse therefore stops naturally because all processes on the surface and inside the star gravitationally freeze relative to the uniformly flowing cosmological time. Thus the black holes predicted by Newtonian theory never form within GR. According to GR, in the real world, at every instant of world time at least the surface of any star coexists simultaneously with other objects. At every instant of this real history of the Universe, or at any given epoch, a dust ball whose surface has frozen near the gravitational radius is a frozar with a frozen structure throughout its volume.

Salviati. Then let me clarify. What you are describing is the external manifestation of collapse in world time, and this picture is limited because world time does not cover the entire world line of the falling particles, so this coordinate system is incomplete. In reality nothing stops further collapse in terms of proper time for the falling observers themselves on the surface! Their own time does not freeze for them! The fact that from their point of view the temporal evolution of the rest of the world is merely enormously accelerated does not imply that they themselves will not continue falling toward the center of the star, so the second stage of collapse is inevitable. I told you that you would reduce everything to exaggerating the role of some special coordinate choice and then claim that only this choice is true and that therefore there is no passage beneath the horizon.

Sagredo. Do not rush to hasty conclusions.

4. GR Leads Only to Frozars. It Really Is Obvious!

Sagredo. To dispel your latest—pardon me, your final—suspicion, let us look at the collapse from the point of view of falling observers on the stellar surface as well. You yourself said that from their point of view all their own processes proceed normally, while events throughout the rest of the world occur at a greatly accelerated rate. At the same time, both you and your observer, being sensible people not inclined to exaggerate your role in the fate of the rest of the world, understand perfectly well that throughout the enormous Universe the average rate of time, that is, cosmological world time, cannot accelerate more and more merely because some layer of one particular star is falling. Therefore you will clearly realize that what is actually happening is the gravitational slowing of the proper time of the falling observers themselves relative to uniformly flowing world time. Correct?

Salviati. Yes, that is so. The relative slowing is obvious, and a sufficiently competent falling observer would most likely interpret it as a slowing of his own proper time relative to the time of the rest of the world. But so what? The point is not the interpretation, but that his proper time continues to run no matter how rapidly the rest of the world evolves. So I am right and you are reducing your whole proof to someone's private opinion.

Sagredo. But you have overlooked one crucial fact that I emphasized earlier. World time and proper time on the stellar surface are uniquely related to one another. Within GR both times label the same events along the same world line of each particle. Every instant of one corresponds to a definite instant of the other, because the event itself is one and the same. Thus the instant of proper time on the surface at which, in the old theory, your observer would calmly cross the gravitational radius will in fact occur only after an infinitely remote future of the real world. Translated from mathematical language into physical language, that means that this instant will never occur in the real world, not even for the falling observers themselves. They would have to wait until the entire Universe had actually lived through that infinite world time, which by definition has no end and therefore never terminates at any definite instant. Do you understand that? And in general, do you agree that in GR every instant of the two kinds of time is uniquely related to the other?

Salviati. I understand that and know that the two times are related when they describe the same events. But you are right that the surfaces of real collapsing objects have not yet reached the gravitational radius. That is why they are called collapsing objects, since their horizon has not yet formed—or, more cautiously, astrophysical black holes. This is a well-known fact.

Sagredo. But we agreed to adhere to standard GR. Then say it as it is: in real collapsing objects the horizon has not merely failed to form yet; it will never form, because you cannot specify any particular instant of world time at which it forms. At every instant of world time marking an epoch of the real world, the surface of your object will lie outside the gravitational radius, while its internal structure will be frozen in the state it had immediately before the last photons were emitted from its surface. Such an object, which you yourself have finally begun to acknowledge, is a frozar. This is indeed known, and the most cautious astrophysicists take precisely this view. Yet whenever one of these real objects is discovered, hotheads announce to the whole world with delight that another black hole has been found. At the same time they display a Newtonian picture of collapse and present it as a strict consequence of GR.

Salviati. Now, I think, your position is completely clear to me. Frozar theory, as that part of the black-hole concept which is confined to the first stage of collapse, describes astrophysical black holes whose horizon has not yet formed and which used to be called collapsing objects, while you call them frozars for short. Could you not have stated these banalities at the very beginning? I would not even have argued about them.

Sagredo. So you admit that the thousands of collapsed objects already discovered and presented as black holes are not in fact black holes in the literal sense, but collapsing objects, or frozars, with ordinary particles frozen at different distances from the center—and, most importantly, that they have neither a horizon nor a singularity at the center?

Salviati. Well, broadly speaking, yes.

Sagredo. And you also admit that they will never turn into black holes?

Salviati. In world time—yes, they will not.

Sagredo. Then in what time do they turn into black holes—in proper time?

Salviati. According to the black-hole concept, precisely in proper time. But, as you said, according to GR these instants of proper time will never occur in our world because the proper times in each individual object really freeze. If one reasons this way, then yes, they will never turn into black holes.

Sagredo. Some popular textbooks say that the freezing of a star's surface is an optical illusion connected with the slowing of the speed of light near the star. You understand that this is an error? Do you agree that the freezing relative to world time is absolute, and that any observers using any coordinates can compare these two times expressed in their own coordinates and verify that the stellar surface has frozen relative to world time? The same thing is shown by coordinate clocks directly on the surface, running synchronously with the clock of a distant observer.

Salviati. Yes, of course. It is definitely not an optical illusion, and by using coordinate clocks that run synchronously with the clocks of distant observers one can indeed determine the simultaneity of events on the surface with events far away. I think I am beginning to understand the importance of defining the simultaneity of events on the surface and at a distance. It follows that once we accept this physical simultaneity of events, the world line of the surface really cannot cross the gravitational radius at any instant of time in the real Universe. Then all the particles inside the star remain ordinary particles and simply freeze at different places, and nothing else really happens to the star? Then there truly is no room for a second stage of collapse... There will be neither a horizon nor singularities? That is astonishing! For the moment it is rather shocking... I need to think about this a little.

Sagredo. Just do not get upset. If you still have any doubts or arguments, let us deal with them now so that you do not change your mind later and return to the beginning.

Salviati. Not get upset? I am, you might say, stunned! Could such a simple error really have been made? At the very foundation of such a grand concept as black holes? No, I cannot believe it... Above all, it cannot be true! All right, let us calm down. Explain it once more, more simply and, most importantly, rigorously: why is there no second stage of collapse in GR?

Sagredo. Gladly. To convince yourself finally that a dust ball freezes into a frozar and that there is no second stage of collapse, you can take explicit or numerical solutions of the GR equations and follow how the circumferences of the surface and of every layer change as world time passes. You will then see clearly that before the surface comes arbitrarily close to the gravitational radius, the rate of proper time at the center practically stops and everything there freezes; then the freezing rapidly reaches the surface. After that the world lines of the particles of the dust ball remain forever parallel to one another at the distances from the center where the freezing process caught them. Can you picture that?

Salviati. Yes, I can. They really will remain parallel, and that is all—what a pity! I had always imagined that the interior of the star rapidly plunged toward the center and that we simply received no information about it. But indeed, we can model everything using the GR equations in terms of world time, considering sets of simultaneous events far away, on the surface, and inside the ball! If all this gives only a transition into a frozar state with parallel world lines of particles throughout the volume and nothing more, then I understand. Incredible! I wonder what changes if pressure and temperature are included?

Sagredo. The contraction of a dust ball ends in a transition to the frozar state, and that is all. If, in more realistic models, other kinds of particle interactions are included, contraction causes pressure and temperature to rise, and everything depends on the equation of state of the stellar matter, which relates density, pressure, and temperature. The transition to the frozar phase may then fail to occur: the star may overheat and stop, begin to pulsate, or even expand up to an explosion at the final stage. Such realistic modeling on the basis of GR is what frozar theory is. Now you can say to yourself—notice, not to me but to yourself—do you accept frozar theory?

Salviati. Alas, for the moment it seems that I do. I say “for the moment” because I still have to rethink all of this.

Sagredo. I am glad you managed to overcome yourself and have at least swallowed this bitter pill for now. I wish you pleasant digestion and further sobering up.

Salviati. Still, why drag things out when there are only two possible answers—you are right or you are not. Suppose that after I have thought everything through myself and dug into the literature, I agree with this new interpretation. I repeat: suppose so for the moment. Then what follows from it? What changes would there be in GR and astrophysics, and what would remain of the old black-hole concept? Can you answer these questions briefly?

Sagredo. Of course you must rethink everything for yourself, and I am not going to put pressure on you. You specialize in this, you have devoted part of your life to it, and it will not be easy to watch some of your work lose its value before your eyes, if it does not turn into waste paper altogether. But science advances by trial and error, so do not worry too much; simply move along a new and more correct path. I too developed and discarded quite a few versions before spending the last several years becoming convinced that only this theory is correct, even though at first I did not much like it either.

Salviati. Do not get ahead of events; I am not completely convinced yet. Move on to the consequences and then we shall see. A correct theory always brings unexpected and radical improvements and new predictions, so those signs will show whether your theory is capable of playing that role. What, then, do we gain if we abandon black holes and move to your frozars?

Sagredo. Remember the words you quoted about the greatest crisis being hidden beneath the horizon? According to frozar theory there is neither a horizon nor singularities in GR, and there is no crisis either, hidden or otherwise. Nor is there a point source in GR—indeed there are no objects smaller than the gravitational radius at all. In astrophysics this resolves the problem of collapse both for compact stars and for galactic nuclei and quasars. One difference between a frozar and a black hole is that under certain conditions, namely a rapid rise of temperature and pressure at the center, contraction can be replaced by expansion, all the way to an explosion at the final stage with the release of far more energy than had previously been known. Thus for frozars collapse is, in principle, reversible.

Salviati. Really? How large would the mass of a frozar have to be for a rise of central temperature, in principle, to produce expansion with an explosion at the final stage?

Sagredo. Here one has to study not a dust ball but more realistic models. Astrophysics observes explosions on very different scales whose energy source is still unknown, so frozar theory now allows us to model processes opposite to collapse—anticollapse—as possible sources of such extremely powerful ejections of radiation and matter. Objects that, during collapse, did not have time to enter the frozar state and began to thaw are proposed in this approach to be called gloozars, from the English “glowsar,” that is, “glow star”—an incandescent, reddened star.

Salviati. Yes, a great deal really would become simpler! If GR also permits recollapse, then boundless prospects open up—I can hardly even imagine them now. So if a real collapsing star has not frozen, it passes into a gloozar state, as you call it, with the prospect of some sort of activity? But how is that proved?

Sagredo. With dust matter as an idealization, formation of a frozar is a strict consequence of GR. Whether an object can pass into a gloozar state, however, depends on the particular interactions, equations of state, and parameters of the star. GR merely permits the formation of gloozars, so it makes sense to study their properties and external manifestations while supposing, for now, that in some cases such states may be realized and observed.

Salviati. Well, at least observing these gloozars does not seem as hopeless as observing frozars, so that gives some reason for optimism. Are there any other interesting consequences of the new theory?

Sagredo. No less important, and perhaps even more important, is that the new theory of collapse solves the problem of fluctuations with infinitely large energy in particle physics. Quanta with Planck energy—frozons—appear there, and according to GR particles with greater energy do not exist. The gravitational redshift of frequencies in the field of a source of such a quantum, with a source of the same size, places an upper limit on frequencies: no source can emit particles of greater energy, so they are not created. Quantum gravity, which until now could not be constructed, then becomes straightforward because the theory is greatly simplified under this restriction. All of these are quite radical simplifications that make unnecessary many of the hypotheses proposed in elementary-particle physics during recent decades.

Epilogue

Salviati. If that is so, it is very tempting. We have not discussed any of this.

Sagredo. We shall discuss it another time. Do not forget, you have to get to your black-hole conference.

Salviati. Yes, indeed. I have a major report there, and afterward we shall receive a large grant.

Sagredo. Will you say anything about frozars in your report?

Salviati. Are you out of your mind! I cannot go to war with the whole world! But when leading scientists in this field begin supporting that author of yours, I shall certainly support him too.

Sagredo. Yes, of course, and thank you even for that. I have tired you badly; would you like me to treat you to an Eastern fable to lighten the mood?

Salviati. Excellent, go ahead!

Sagredo. Khoja Nasreddin's uncle in a distant city left him a great deal of money as an inheritance, but Nasreddin had no money to travel there and collect it. He went to a wealthy neighbor and asked to borrow ten gold coins so that he could make the journey and receive an inheritance of ten thousand gold coins. The neighbor said he was ready to support him, but on one condition: “You go there, collect your ten thousand gold coins, and when you come back I shall certainly lend you ten gold coins.”

Salviati. Yes, the East is a subtle matter; in ancient times the culture there was different!

Sagredo. The scientific community works much the same way now...

Salviati. Hmm. You know, give me that paper about frozars. I shall read it carefully, send it to ten well-known specialists who know me, and explain it to them myself at the conference. Those will be my ten gold coins. Agreed?

Sagredo. Of course. Now you are finally using your own head for its intended purpose!