I'm not sure what you're not getting when I say analogous; it's simple: the vacuum observer and the Schwarzschild coordinate time are not only analogous, they are equal, basically by definition; additionally, at constant radius the rate of clocking on the observer's watch and coordinate time is proportional by a constant factor; lastly, the local watch's rate always appears to be constant. This is not true of Kruskal (or other) coordinate systems! A watch worn by an observer "clocking" at the Kruskal time coordinate would be odd, if not nonsensical; it would clock at different local rates depending upon the observer's movement, and it would also bear little relation to coordinate time. You seem to accept this coordinate switching very casually. Yes and no. I have a proposal which is not an extension of GR but rather a contraction, as it were, of what it predicts. There would not be 4 quadrants required to be analyzed or explained; all of existence remains on the outside of the EH. If you combine infinite coordinate time for EH creation along with a method of dispersion in finite time (i.e. Hawking radiation, etc) then spacetime is maximally extended and all problems associated with BH analysis go away. I'm aware that Hawking radiation is a QM phenomenon and outside of the realm of GR, but even then the Schwarzschild coordinate system is maximally extended if we use the "compromised definition" that has been adopted to handle the singularity anyway. (A bit off-topic, but I'm offended by the exception made for the singularity out of convenience. Shall we claim the hairy-ball theorem is false if we can reduce the nontangent vector field to a single point?)[/quote] Of course they are; why do you think this would be so? (The answer resides in the logic of my OP Please Register or Log in to view the hidden image!) The rest of your post was fascinating and I don't want to trivialize the amount of work you did, but here's why it falls apart: one cannot accelerate for eternity. This is not a technicality, it is the crux of the problem. It presupposes an infinite energy source, and if we have such a source we could simply use it all at once to break physics in all kinds of ways rather than waiting for an eternity to do it. One last thing: when I searched for the Kruskal wiki page I stumbled across this which I found interesting. Not you, so much, but others on this site will abandon their own internal voices of logic and reason...until something has been approved for publication in a peer-reviewed journal, which they then take as gospel. Please Register or Log in to view the hidden image!
Which definition? You seem to be missing the whole point of my objection: it's not what you say, but why anyone should believe what you say. Schwarzschild coordinates are not arbitrarily meaningful and should not be taken literally in all circumstances just because you say so. Your explanation is redundant. I never said that Schwarzschild coordinates were completely randomly defined, and you are merely giving specific examples of what I already told you: most of the utility of Schwarzschild coordinates comes from the fact that the metric is static in those coordinates -- i.e. independent of coordinate time. You are still making a gigantic leap of faith in going from "Schwarzschild coordinates can be argued to have some physical significance" to "they are gospel and spacetime itself must end where the Schwarzschild chart does". You have done nothing in this thread to bridge that gap and actually justify the conclusions you are drawing by GR's own internal logic. Not at all. Maybe I'm wrong, but I thought we were talking about GR black hole solutions and what general relativity has to say about them, and not some strawman of your own invention. If you ever tire to actually learn GR, you will find it is set in the context of (pseudo)-Riemannian geometry and is deliberately and carefully constructed in a coordinate-independent manner right from the ground up. That means that coordinate systems are not accorded any special significance by default in GR and the coordinates one uses shouldn't much matter, and any grand claim that seems to depend heavily on taking a particular coordinate system "literally" is immediately suspect. Logically you start from there and work up, and not the other way around. In other words, "no". I already understood that you were proposing cutting off black hole solutions at the event horizon, and that is not the question I was asking you. There aren't four quadrants in general anyway. Those only appear in the Schwarzschild eternal black hole solution. The white hole and "parallel universe" don't appear for a black hole that has formed by stellar collapse. Justification for this claim? While I'm not sure how much is really established concerning Hawking radiation, it does seem to be generally expected that even an evaporating black hole could or would have a singularity. Suppose we have a model that can incorporate Hawking evaporation and we find that a black hole in Kruskal-like coordinates looks something like the fourth figure here. Then what? Cut that off at the event horizon and the resulting spacetime is not geodesically complete. You are acting as if GR spacetime geometries were simply plucked out of the sky. Extending a spacetime geometry is not simply a matter of switching coordinates or drawing a bigger picture. The extended geometry must still be a valid solution to the Einstein field equation. That is a differential equation that isn't going to play with infinities very well, so if you propose an extension to a full black hole geometry beyond the singularity itself, you have the problem of explaining what it means for your geometry to still be satisfying the Einstein field equation across the singularity in a meaningful and well-defined way. If you can't explain that, or if you end up finding by some definition that you can glue fifty different and inequivalent extensions past the singularity that are all "allowed" by the Einstein field equation, then the extension is not well defined. Like I said, if you know a way of extending a black hole geometry past the gravitational singularity in a unique and well-defined way, I'm all ears. Because I'd personally be surprised if that were possible, and even if the mathematicians could come up with a way of letting you "evaluate" the Einstein field equation across a gravitational singularity, given your general disdain regarding infinities I doubt you would accept it anyway. Oh, I don't know, maybe any number of boring reasons, such as calculational convenience or even just someone's arbitrary whim, that wouldn't involve a conspiracy of some sort? Why don't you actually look up how collapsing starts are modelled and try to understand why it's done the way it is instead of just making up whatever story suits the conclusion you want to reach? Your rejection is inconsistent and invalid: One does not need an infinite amount of energy merely to define a coordinate system. Did nature break in all sorts of ways when I submitted my last post? You seemed perfectly happy to make similar idealisations concerning Schwarzschild geometry. A physical clock or observer also cannot hover indefinitely just outside a black hole event horizon. Even some orbits too close to the event horizon are unstable. To the extent that you have tried to give objective criteria for determining where spacetime ends, you have said nothing about the physical limits of observers, such as the finite amount of fuel they can carry, featuring among your criteria. In this regard, you are moving the goalposts. An accelerating reference frame attached in the way I described to an observer that doesn't accelerate forever still remains pathological if you push it too far. Specifically, some events in spacetime still appear multiply or get stretched out to entire lines in the accelerating frame, and you can still have the "time axis flipping" I described in point 3) in my previous post (hell, in general it can even start to zigzag back and forth). You really can't have much experience with the physics community if you believe that. Results are not necessarily generally accepted and certainly not taken as gospel solely on the basis that they have been published. An article appearing in a peer-reviewed journal means that it has been given the green light usually by between one and three referees (varies depending on the journal). Not the entire relevant research community. By the way, off the top of my head, the only person I can think of here who regularly cites papers and expects them (or at least his interpretation of them) to be treated as gospel is Farsight. As a rule of thumb I would recommend you don't try to cite papers to support a case you are trying to make unless you are either a) damn sure the results are actually generally accepted by the relevant research community, or b) you are prepared to defend it yourself.
Except that electrons have too much angular momentum and charge to have an event horizon, so they are not "black holes" but "naked singularities" if GR is being used to model them. Thus theorems about black holes do not apply.
The reason is that the OP is talking about existence which is more of a philosophical topic than a strictly mathematical one. Philosophically, the "naturalness" of the relation between the Schwarzschild time coordinate to what we mean when we say things like "exists" is crucial. It's OK that you are not convinced, but I want to make sure that you understand what I'm suggesting: I believe that GR's analysis puts the event horizon and the singularity at the same point. They are not disparate, and in reality neither actually exist. The existence of black holes is a presumption, and Hawking radiation was then applied to it. I'm saying that a dispersion process of any sort in finite time eliminates the singularity problem. I will. Do you have a recommendation on a gold standard college text for GR? What about "Gravitation" by Misner? I'll purchase it and see how my questions are explained. What you say about accelerating frames is interesting food for thought.
I don't see why you think saying this would change anything. There is such a thing as bad philosophy, you know. The "naturalness" of the Schwarzschild coordinates is limited at best and very weak grounds for saying anything definite about black holes. Incidentally, I hope you are not under some sort of impression that Schwarzschild coordinates are in any way necessary for deriving black holes. It is merely a matter of history that black hole solutions were first given in Schwarzschild coordinates. You could take the black hole metric expression in Kruskal coordinates and check that it is a vacuum solution to the Einstein field equation and never need to know about Schwarzschild coordinates for instance. Would you be saying all the same things you are here if historically it had been done that way? If you are talking about GR's account of things (which I really hope you are, otherwise I have no idea what this whole thread's been about), then the event horizon and gravitational singularity unambiguously do not coincide. Again, you don't seem to have noticed that the problem isn't that I have not understood what you are saying. The problem is that you have not justified what you are saying. You say "any" and "eliminates" as if your conclusion were true of any conceivable theory of gravity capable of modelling an evaporating black hole. Can you justify the generality of that claim? I would expect it's done in Misner, through from what I've seen of that text I don't have a terribly high opinion of it (too much fluff and irrelevant analogies). I can readily confirm that gravitational collapse is treated in Weinberg's Gravitation and Cosmology because I've got a copy of it right in front of me. That said, I don't know what you are expecting to get out of studying GR's account of gravitational collapse without first thoroughly understanding GR itself.
I understand this but if we were wearing "Kruskal watches" then the meaning of something existing today would differ. Our disagreement lies in the consequences of coordinate changes. You say that changing them should not affect the system under consideration...but then you make the exception with Schwarzschild coordinates because they are undefined at the EH? Aren't Kruskal coordinates undefined for a vacuum observer (i.e. at infinity)? I'm not convinced that we can make these coordinate changes and then continue to apply our intuitive meanings to them. Perhaps. I don't have time to do this right now but I have a lead and I'd like to dig out a response you gave me on the issue of whether or not an free-falling body could be rescued from the EH by an accelerated body leaving at a later date. I feel you're becoming a bit contrarian here. I'm showing my willingness to investigate things and attempt to back up my claims to your satisfaction, and it sounds like you're going to discard what I say, regardless.
The meaning of something existing "today" is frame-dependent in special relativity and even there it is only reasonably well defined in inertial reference frames. Like I hope I illustrated with accelerating frames, there is an arguably "natural" way of locally defining simultaneity for an accelerating observer, but you get nonsensical results if you try to push it too far (like the same event appearing twice in the accelerating frame at different times). That is just in SR. GR is not SR and it certainly isn't Newtonian physics. Take it for what it is, not for what you wish it were. If you want to make claims of "existence" in GR then the only meaningful way to do it is on GR's own terms, i.e. something "exists" in spacetime if it exists on a spacetime manifold predicted by the Einstein field equation. If you want more granularity than that then the only way to do it that'll get you a non-arbitrary conclusion is to distinguish between past and future lightcones and the causal present, as those are coordinate-independent concepts. That is not an exception. The very fact that a coordinate change doesn't change the physics is precisely what reveals that the coordinate singularity in the Schwarzschild metric is just that: a feature of only the coordinates. If the singularity were physical then it should be impossible to remove it with a coordinate change. Yet for the Schwarzschild metric, the coordinate singularity disappears even for something as simple as a locally inertial reference frame attached to an infalling observer approaching the event horizon. It also doesn't appear in coordinate-independent measures of the gravitational field such as scalar contractions of the curvature tensor. Various ways of analysing the physics in the vicinity of an event horizon consistently reveal that there's really nothing special going on there. Huh? Kruskal coordinates map everything Schwarzschild coordinates do and more. I think it's only you here who is struggling to apply intuitive meanings to coordinates that shouldn't necessarily be attributed such meaning. For the rest of us the metric is there to keep track of that sort of thing. Really? You could prove mathematically, for instance, that no matter what theory we end up with, we will never get something like, say, this? (Which on the face of it looks perfectly reasonable to me.) Without even being especially familiar with the mathematics of GR itself? Are you sure you've understood what it means to actually have a convincing proof? My issue is with how far that willingness actually goes. To make a convincing case that GR black hole solutions can be interpreted a certain way, the first thing you would need to be sure about is that you are actually talking about GR and not just your best guess at it. I mean, realistically you do realise that right? I'm not even talking about you convincing me here. These should be your own standards for convincing yourself of anything. It simply doesn't make sense to me that you insist on holding strong opinions about predictions made by a theory that you've so far only half learned at best.
Yup, I concur - they'd need to be exposed ring singularities. My former words merely relate to pop-science jazz of the Brain Greene variety.
...which is exactly what I did in the OP. If the BH creation and EH formation do not occur in the past light cone of outside observers would you concede that they do not "exist" for those outside observers? If, in the end, you're going to fall back on the idea that GR allows the BH to exist on paper and therefore it "exists" in Nature...then this is fruitless. I've said it before, I don't actually care who thinks what with the exception that I understand their motivations and logical framework for doing so. Convincing someone to see my POV is not important to me. If this thread boils down to a semantic difference (i.e. we cannot agree on what it means to exist) then you can kindly go on your way and I will maintain my version of the truth. That being said, I'm still investigating for myself some of your objections.
And I think you got a fairly definite answer there. And then, for reasons best known to yourself, you apparently abandoned that line of reasoning in favour of the so-called "frozen star" interpretation of black holes which is based on taking Schwarzschild coordinates 'literally'. No, because I think your definition of "exists" is overly narrow and arguably an unnecessary corruption of English grammar. That is a blatant strawman in more than one way. First, you're misstating what GR has to say about black holes: it doesn't merely "allow" them, it specifically predicts that gravitational collapse is inevitable in certain circumstances. Second, when did this become a discussion about whether black holes really exist in nature? Whether black holes meaningfully exist in GR by GR's own logic and whether they exist in reality are two different questions. We've been discussing the former, not the latter. I have been deliberately explicit that that is how I have been reading this thread, and if that wasn't what you wanted you have had every opportunity to object. You can't participate in a long discussion about whether black holes meaningfully exist in the context of GR and then declare the discussion "fruitless" just because you couldn't support the conclusion you wanted to reach. Finally, note the "meaningfully". For the purpose of this discussion we have not simply been presuming that black holes exist in GR just because that is what GR textbooks say. The "meaningfully" in theory gave you the breathing room to argue that maybe some reinterpretation or reformulation of GR was possible which would eliminate black holes as a prediction in a non-arbitrary way. You just haven't been successful in arguing that, particularly the "non arbitrary" part. Well you say it yourself here: you need to understand the logical framework. The problem is, that logical framework is general relativity -- a theory that would normally take an intelligent person at least a few months to fully get to grips with. I and others can assert answers to definite questions you ask about GR (within reason), but if you really want to fully understand the reasoning behind those answers, and I mean really from A to Z, then realistically I don't think that's possible unless you have a level of understanding of GR comparable to that of a graduate student who has passed (and understood) a university level course on the subject.
Yes, I thought I had as well when you said However, you proceeded to pick at various things, including my proposed definitions, etc, and the we turned to this: You said that there is a finite spacelike distance to regions within the EH. I claim there is an infinite spacelike distance to the singularity. The singularity's "existence" is a prerequisite of the area within the EH, agreed? It must "exist" within that area's causal past, yes? Ahh! So what you're saying is that, in your own words, the only way to get a non-arbitrary conclusion is to use a definition which is overly narrow and an unnecessary corruption of grammar?
You are conflating two meanings of "prerequisite". The singularity is a logical prerequisite for defining the event horizon around it. Within the event horizon, the singularity is in the causal future. That is the defining property of a black hole's event horizon and is what makes a black hole a black hole: if an observer is within the region bounded by the event horizon, then the singularity fills the whole of their future light cone and they have no way of escaping from it. I never said any such thing. In fact I explicitly gave you an alternative: say a black hole "exists" if it is in your causal present. If part of a black hole overlaps with your causal present then it is "there" in the intuitive sense that a 12 year old would think of: he could travel into it any time he wanted or use it to permanently dispose of his little sister's Barbie dolls.
You accept this logic? The singularity is in the causal future of every frame today, yet it must lie in the causal past of the area within the event horizon for that event horizon to form. This is the point I made in the OP.
If you had actually made a point in the OP this thread probably wouldn't have needed to go this far. :EDIT: And yes, the future lies in the past is great logic.
Oh dear, oh dear. Erm, what? Just look at the standard Penrose diagram for gravitational collapse of a spherical body to see that your claim is false [e.g. p.300 in Hawking & Ellis' book]*. That the existence of an event horizon in a spacetime implies the existence of a singularity in that spacetime is a consequence of the classical Penrose-Hawking singularity theorems I've also just read your bit about infinite proper length to the singularity from outside the event horizon and I died a little on the inside. I honestly think if you don't begin to understand how coordinates work in general relativity soon, God is going to start killing kittens. How about this: I give you a curve \(\gamma\) in the Schwarzschild spacetime and, with help if needs be, you can convince yourself that this curve is spacelike, starts outside the event horizon, finishes at the singularity and \( \int_\gamma \mathrm{d} s \) is finite. Fair? Or must the kittens die? * Unfortunately you wont understand what this diagram means (unless you simply accept what other people tell you it means). And herein lies the major problem with your approach: you aren't scientifically curious enough to look into the details. Or perhaps you're not intellectually capable of understanding the details. Either way, you can't digest any of the advice your given about general relativity because you can't process anything beyond a wordy pop-science explanation. The fact that you still don't understand the role of coordinates in general relativity is testament to this fact.
I said logical prerequisite. Not causal prerequisite. The event horizon is defined in terms of the singularity, so you first need to establish that there's going to be a singularity before you can identify the event horizon associated with it simply by definition. But the event horizon is just a lightcone. It's an imaginary line you draw tracing back from the singularity. It is not a physical object that actually "forms" and it does not "cause" the singularity. Where you say that the singularity must be in the causal past of an event horizon, you are simply contradicting GR's own account of things.
