Well, there are actually several singularity theorems. The first one was by Penrose, it was for black holes only. The second one was by Hawking, it was for cosmology only. But then some generalizations have been made which are applicable to both black holes and cosmology.PeterDonis said:They're the same theorem.
As discussed in several posts above, formation of black holes is indeed irreversible, but it does not mean that gravity as such is irreversible. As in other deterministic physical theories, irreversibility can be traced back to the choice of initial conditions.martinbn said:This may be off topic, but another way in which gravity is irreversable is the formation of black holes.
Ok, show me initial conditions which lead to the formation of a white hole.Demystifier said:As discussed in several posts above, formation of black holes is indeed irreversible, but it does not mean that gravity as such is irreversible. As in other deterministic physical theories, irreversibility can be traced back to the choice of initial conditions.
Wald in the book General Relativity discuses it at page 155. Essentially, take a late state of the black hole and in this state replace all future oriented vectors (velocities etc.) by the opposite past oriented ones. If you take this as initial state, you get a white hole.martinbn said:Ok, show me initial conditions which lead to the formation of a white hole.
He discusses Schwartzschild, where nothing forms. If you have a formation of a black hole, and then reverse "time" you get a white hole, but it will violate some energy conditions.Demystifier said:Wald in the book General Relativity discuses it at page 155. Essentially, take a late state of the black hole and in this state replace all future oriented vectors (velocities etc.) by the opposite past oriented ones. If you take this as initial state, you get a white hole.
No, time reversal does not violate energy conditions. Take for example the energy condition that ##T_{\mu\nu}V^{\mu}V^{\nu}>0## for all time-like future oriented vectors ##V^{\mu}##. Now consider ##V'^{\mu}=-V^{\mu}##. Since ##V^{\mu}## is future oriented, it follows that ##V'^{\mu}## is past oriented. In other words, a transformation ## V^{\mu}\to V'^{\mu}## involves a time reversal. However, this does not violate the energy condition because we still have ##T_{\mu\nu}V'^{\mu}V'^{\nu}>0##. That's because the vector ##V^{\mu}## appears quadratically in the energy conditions, so the sign of ##V^{\mu}## doesn't matter.martinbn said:He discusses Schwartzschild, where nothing forms. If you have a formation of a black hole, and then reverse "time" you get a white hole, but it will violate some energy conditions.
I know this is the way the theorems are usually described, because, IIRC, that's the way Penrose and Hawking advertised them, so to speak. But mathematically, it's not correct. Both theorems are time reversible and neither one is restricted to a particular spacetime geometry; they only require that a trapped surface is present and that the spacetime satisfies an appropriate causality condition (in the original theorems I think that was global hyperbolicity, but I believe later extensions have relaxed that somewhat). So Penrose's theorem, by time reversing, could also be applied to an initial singularity such as the Big Bang, and Hawking's theorem, by time reversing, could also be applied to a final singularity such as a black hole.Demystifier said:The first one was by Penrose, it was for black holes only. The second one was by Hawking, it was for cosmology only.
This is not correct. For example, you can time reverse the Oppenheimer-Snyder 1939 idealized model for gravitational collapse to a black hole, and obtain a model where a white hole "expands" into an ordinary object. Since the original O-S model does not violate any energy conditions, the time reverse of it won't either.martinbn said:If you have a formation of a black hole, and then reverse "time" you get a white hole, but it will violate some energy conditions.
Demystifier said:No, time reversal does not violate energy conditions. Take for example the energy condition that ##T_{\mu\nu}V^{\mu}V^{\nu}>0## for all time-like future oriented vectors ##V^{\mu}##. Now consider ##V'^{\mu}=-V^{\mu}##. Since ##V^{\mu}## is future oriented, it follows that ##V'^{\mu}## is past oriented. In other words, a transformation ## V^{\mu}\to V'^{\mu}## involves a time reversal. However, this does not violate the energy condition because we still have ##T_{\mu\nu}V'^{\mu}V'^{\nu}>0##. That's because the vector ##V^{\mu}## appears quadratically in the energy conditions, so the sign of ##V^{\mu}## doesn't matter.
The energy in Einstein equation appears only as a second-rank tensor ##T_{\mu\nu}##, which is a deeper reason why the relevant energy conditions are always quadratic in the vectors, which is why the theory is invariant under time reversal.
Yes, I souldn't write without thinking.PeterDonis said:This is not correct. For example, you can time reverse the Oppenheimer-Snyder 1939 idealized model for gravitational collapse to a black hole, and obtain a model where a white hole "expands" into an ordinary object. Since the original O-S model does not violate any energy conditions, the time reverse of it won't either.
You could be right in a practical sense, that without violating an energy condition one cannot get a white hole with physically reasonable initial conditions expected to be possible in practice. But if you just explore the space of mathematical solutions of the Einstein equations, white hole solutions that do not violate energy conditions certainly exist.martinbn said:But i do believe, although now i should be cautious, that you cannot get a white whole formation without violating an energy condition.
Demystifier said:No, time reversal does not violate energy conditions. Take for example the energy condition that ##T_{\mu\nu}V^{\mu}V^{\nu}>0## for all time-like future oriented vectors ##V^{\mu}##. Now consider ##V'^{\mu}=-V^{\mu}##. Since ##V^{\mu}## is future oriented, it follows that ##V'^{\mu}## is past oriented. In other words, a transformation ## V^{\mu}\to V'^{\mu}## involves a time reversal. However, this does not violate the energy condition because we still have ##T_{\mu\nu}V'^{\mu}V'^{\nu}>0##. That's because the vector ##V^{\mu}## appears quadratically in the energy conditions, so the sign of ##V^{\mu}## doesn't matter.
The energy in Einstein equation appears only as a second-rank tensor ##T_{\mu\nu}##, which is a deeper reason why the relevant energy conditions are always quadratic in the vectors, which is why the theory is invariant under time reversal.
Yes, I souldn't write without thinking.PeterDonis said:This is not correct. For example, you can time reverse the Oppenheimer-Snyder 1939 idealized model for gravitational collapse to a black hole, and obtain a model where a white hole "expands" into an ordinary object. Since the original O-S model does not violate any energy conditions, the time reverse of it won't either.
Yes, but i mean a solution that does not have a white hole at an ealry stage, but forms later. For black holes you have that.Demystifier said:You could be right in a practical sense, that without violating an energy condition one cannot get a white hole with physically reasonable initial conditions expected to be possible in practice. But if you just explore the space of mathematical solutions of the Einstein equations, white hole solutions that do not violate energy conditions certainly exist.
In that sense you are right, provided that you work with purely classical gravity. For black holes, you don't have black hole solutions which are not black holes at late stages. By time inversion of that, it follows that you don't have white hole solutions which are not white holes at early stages.martinbn said:Yes, but i mean a solution that does not have a white hole at an ealry stage, but forms later. For black holes you have that.
What about my follow up question. What are the shortcomings of Penrose's idea? Other than it is not fully developed.Demystifier said:So I think now I can answer the initial question of @HomesliceMMA .
The idea is motivated by two assumptions, that:martinbn said:What about my follow up question. What are the shortcomings of Penrose's idea? Other than it is not fully developed.
These are alternatives, that may be more convinsing, but they are not shortcomings of his idea. I am interested if anyone has any objections based only on what Penrose proposes.Demystifier said:The idea is motivated by two assumptions, that:
(i) Hawking radiation is fundamentally irreversible,
(ii) wave function collapse is fundamentally irreversible.
From my (somewhat subjective) point of view neither of the assumptions is very convincing. About (i), most approaches to the black hole information paradox restore unitarity and reversibility at the fundamental level. About (ii), the collapse is widely believed to be somehow emergent from decoherence, and decoherence is based on assumption that fundamental dynamics is unitary and time reversible.
martinbn said:These are alternatives, that may be more convinsing, but they are not shortcomings of his idea. I am interested if anyone has any objections based only on what Penrose proposes.
Independent of the supposed physical mechanism (like decoherence) behind the collapse, the non-local collapse that Einstein called "spooky action at a distance" is an artifact of a "first person in the moment" description of what happens. The description is not wrong per se, but a very subjective description, very much in the spirit of the Bayesian interpretation of probability. This is also what I meant when I wrote thatDemystifier said:About (ii), the collapse is widely believed to be somehow emergent from decoherence, ...
gentzen said:QBism is indeed only a "more honest" version of the interpretation put forward by defenders of the old Cohenhagen orthodoxy
I agree, but Penrose and some other guys think that collapse is not merely an artifact of a first person description, but a real non-local physical process. In fact, in some works Penrose goes even further and argues that the first person experience itself is physical and related to fundamentally non-deterministic collapse. More precisely, from the fact that humans can see that the Godel sentence is true, while the Godel sentence is a self-referential statement saying that it cannot be proved by an algorithm, Penrose argues that human intelligence is fundamentally not algorithmic and hence requires a fundamentally non-deterministic process such as collapse.gentzen said:But the objection remains the same, namely that an artifact of a certain description should better not be promoted to a real physical process.
A white hole "formation" is not possible because a white hole contains an initial singularity--there can't be anything "before" it, by definition. If there is, it's not a white hole.martinbn said:i do believe, although now i should be cautious, that you cannot get a white whole formation without violating an energy condition.
It should be noted that many others have raised objections to this argument (and Penrose is not the only one to have made an argument along these lines). See, for example, Dennett's review of The Emperor's New Mind:Demystifier said:from the fact that humans can see that the Godel sentence is true, while the Godel sentence is a self-referential statement saying that it cannot be proved by an algorithm, Penrose argues that human intelligence is fundamentally not algorithmic
Not exactly. The definition of black or white hole involves the existence of future or past event horizon. The singularity is not a part of the definition of black or white hole. Instead, the singularity is a result of a theorem, which assumes an energy condition. So if energy condition is not assumed, it is possible to have a black or white hole without singularity.PeterDonis said:A white hole "formation" is not possible because a white hole contains an initial singularity--there can't be anything "before" it, by definition. If there is, it's not a white hole.
Yes, that's true; a better statement of the required initial condition for a white hole would be that a past horizon has to exist. But a past horizon also can't be brought into being (i.e., there can't be anything "to the past" of it--in more technical terms, it must go all the way back to past timelike infinity), so white hole formation is still ruled out.Demystifier said:The definition of black or white hole involves the existence of future or past event horizon.
Yes, but note that the theorem does not connect a singularity with an event horizon. It connects a singularity with a trapped surface. They're not the same in general, although they happen to coincide in the idealized case of maximally extended Schwarzschild spacetime.Demystifier said:the singularity is a result of a theorem, which assumes an energy condition.
No, if the energy condition is not assumed, it is possible to have a trapped surface without a singularity. But this tells us nothing about whether or not it is possible to have a black or white hole without a singularity, since those are defined in terms of event horizons, not trapped surfaces.Demystifier said:if energy condition is not assumed, it is possible to have a black or white hole without singularity.
Maybe the theorem itself does not imply it, but explicit black hole solutions (such as the Bardeen black hole) without singularity are known when energy condition is not required. The simplest way to construct such a solution is to write down by hand a regular black hole metric that you want, compute the Einstein tensor for that metric, a finally use the Einstein equation to find the needed energy-momentum tensor.PeterDonis said:No, if the energy condition is not assumed, it is possible to have a trapped surface without a singularity. But this tells us nothing about whether or not it is possible to have an event horizon without a singularity.
Do you mean the metric described in equation (1) of this paper?Demystifier said:the Bardeen black hole
Yes.PeterDonis said:
From what I can gather, the maximal extension of this looks like Reissner-Nordstrom, the only difference being that there is just a regular timelike line at ##r = 0## instead of a timelike singularity. However, ##r = 0## is still behind a Cauchy horizon, meaning that this idealized solution is probably not physically realizable for the same reason as Reissner-Nordstrom is not: the Cauchy horizon (inner horizon) is unstable against small perturbations and would probably be replaced by something spacelike, looking more like the ##r = 0## singularity in Schwarzschild. Ultimately all this probably won't get resolved unless and until we have a confirmed theory of quantum gravity.Demystifier said:the Bardeen black hole
To expand on this somewhat, based on some numerical investigations I have been making (and on what is said in some other papers I have found), there is a critical value of ##g## at which the horizon is degenerate (outer and inner horizons merge). For smaller values of ##g##, there are two horizons and what I said in the quote above applies. The degenerate case corresponds to the similar case with Reissner-Nordstrom. For larger values of ##g##, there is no horizon at all; this corresponds to the "naked singularity" case of Reissner-Nordstrom, but there is no singularity at ##r = 0## so this case corresponds to a static object that is supported against gravity by a sort of "magnetic repulsion" near the center.PeterDonis said:From what I can gather, the maximal extension of this looks like Reissner-Nordstrom, the only difference being that there is just a regular timelike line at ##r = 0## instead of a timelike singularity. However, ##r = 0## is still behind a Cauchy horizon
Since you seem to be very interested in this stuff, here are some more papers you might be interested in.PeterDonis said:To expand on this somewhat ...
Thanks! These look interesting.Demystifier said:Since you seem to be very interested in this stuff, here are some more papers you might be interested in.
https://arxiv.org/abs/gr-qc/0506126
https://arxiv.org/abs/1911.01020
The Penrose diagram in Fig. 5 of this paper is very interesting. What it is saying is, if you include evaporation in your model as well as formation, there is no true black hole any more. That is, there is no region of spacetime that cannot send light signals to future null infinity, and thus no event horizon (which would be the boundary of such a region).Demystifier said: