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Two black holes can collide and merge into one bigger black hole, but not split into two. Does this mean colliding black holes violate time reversal symmetry?

Related: Do black holes violate T-symmetry? Based on the answer to that question, time-reversing a black hole yields a white hole. However, that seems to imply that white holes are very unstable because they can spontaneously split into two, which would then split into four, ad infinitum, and the universe would be covered with tiny white holes all over.

Allure
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You missed something: the gravitational waves.

A black hole merger spacetime contains gravitational waves leaving the merger at the speed of light. Time reversal reverses time across the entire spacetime, and this converts those escaping gravitational waves into a converging gravitational wave front, as well as the black holes into white holes. These waves converge in on the central (now-)white hole and get so strong at that central point of convergence as to be able to "buck" it apart into two separate white holes.

Without those incoming gravitational waves, such a split would not occur.

EDIT: As A.V.S. points out in the comments, in fact, a better answer to this question would be that the future evolution of white holes is in general undetermined, or better unrestricted, in the sense that multiple future trajectories from identical phase-space points will satisfy the dynamical equations, though of course that means still that we must highlight that a crucial element of the answer here is that the time reversal turns the black hole into a white hole. (Indeed, this is part of why they're called "white" - technically that's understating it: they can literally spit out anything - even unicorns, no seriously, it'd be entirely [though unlikely] consistent with the equations for a 1-horned ungulate to pop out, as much as literally anything else.)

In a realistic black hole collision case, which is what I assumed in the answer above, then of course, yes, you will have the gravitational waves and so forth and you do have to take them into account in the reversal. But the situation is even more serious.

Since the future evolution of white hole is unrestricted, you can build scenarios with a totally causeless, spontaneous split of the white hole, and have it be consistent with the dynamical equations. As it is a consistent evolution, it doesn't violate time reversal symmetry. The reason that the Universe isn't covered with tiny white holes is that they are next to impossible to form in the first place - and likely, general relativity is not the final description of these things.

(I want to point out that there is actually an analogy for this within ordinary Newtonian mechanics called "Norton's dome". It is not physically achievable, but is still a system within the mathematical theory which has a similar property of its present state being equiconsistent with multiple future evolution trajectories.)

The_Sympathizer
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    Do you claim that arranging the gravitational front in specific ways you can make any object out of black hole? – Anixx Jan 12 '20 at 18:30
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    @Anixx, at most, this answer claims that by arranging the gravitational front in specific ways you could in principle make any object come out of a white hole. (It seems to me to be safe to assume that aren't any white holes in existence, and since presumably the thermodynamic implications make it impossible to construct one, I don't believe this has any practical implications. But I think it addresses the OPs question.) – Harry Johnston Jan 12 '20 at 20:22
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    @Harry Johnston : Indeed, I found something suggesting the white hole itself is unstable to a tiny perturbation - in a sense, if you "cut and pasted" one into a normal-time Universe, or even just introduced a little bit of "normal" time into it by putting a little normal mass nearby, the "normal" time would gobble it up and effectively instantly convert it to a black hole. Hence why white holes don't exist in our Universe any more than pencils balanced perfectly on their tip with no active stabilization or special surfaces. Literally. Everything. Has. To. Be. Right. – The_Sympathizer Jan 12 '20 at 22:50
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    Out of curiosity, do you know a book/paper where I could read up on that? I familiar with gravitational waves and the treatment of it in GR, but I have a hard time seeing how the your suggested scenario could work out... – Sito Jan 12 '20 at 22:59
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    @The_Sympathizer: while I have not seen the paper you're citing, I don't see how that argument could possibly be right, given that the Schwarzschild white hole solution comes from a continuous extension of the Schwarzschild black hole solution, and naïvely should have all of the same properties. – Zo the Relativist Jan 12 '20 at 23:19
  • @Jerry Schirmer : Part of this seems to relates to what Anixx was getting at with his post - think about reversing the egg-break scenario. The situation is entirely physically possible, but to actually occur requires a truly profound set of coincidences involving disturbances of just the right kinds - waves in the ground, the air, even the inner thermal vibrations within the shattered pieces themselves - all to conspire juuuuuuuuust so so as to cause the pieces - even the finest grains - to be bucked into the air, and then carried by precise air currents even, and – The_Sympathizer Jan 13 '20 at 04:54
  • more, in eeexaaaaaaactly the right way so as to reassemble into the egg. If you disturb this even phenomenally slightly, you'll end up with it simply ending as another high-likelihood configuration, i.e .just stirring the pieces around at best, with them remaining very much broken (at worst, it'll be indistinguishable from the thermal fluctuations going on all the time - i.e. macroscopically you notice nothing at all). The same goes with the white hole, by the same thermodynamics/statistics argument. – The_Sympathizer Jan 13 '20 at 04:54
  • Also, Schwarzschild solution is a vacuum. The moment you put any matter in there, you're no longer talking about the Schwarzschild solution, but a novel dynamical scenario. And while one of these is stable (i.e. an attractor, in effect), that doesn't mean the other is, too. – The_Sympathizer Jan 13 '20 at 04:54
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    Anyway, I found one reference titled "Death of White Holes in the Early Universe" by Eardley (1974). – The_Sympathizer Jan 13 '20 at 04:58
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    “-1” This is a wrong argument. Head on collision of two black holes of similar mass produces very small amount of gravitational radiation (less than 1% of either mass iirc). If we include converging gravitational waves into initial data on black hole merger it would be possible to pick them so that there is no outgoing gravitational waves at all. Time reversal of that would produce a white hole (without any converging radiation) splitting in two (and some outgoing gravitational waves). – A.V.S. Jan 13 '20 at 05:14
  • @A.V.S. Perhaps, but there's still going to be some sort of asymptotic decaying term, I'd think, even if on the order of exponential in time post-collision. It's not going to go mathematically perfectly constant in finite time. That still matters no matter how small it is when you consider the mathematical time reversal (just like it matters for, say, time-reversing a charging capacitor, which has a similar asymptotic behavior) even if it seems "too small to matter". When reversed, that exponentially tiny perturbation becomes exponentially growing and eventually disrupts the white hole. – The_Sympathizer Jan 13 '20 at 05:20
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    No. You are probably drawing your intuition from assumption of analyticity, that signal at finite time must have a precursor in the past, but it is possible to have solutions that are not infinitely differentiable and just start at some finite moment. White hole splitting solution can have the whole vicinity of $\scr I^{-}$ exactly coinciding with unperturbed Schwarzschild. Overall, you are concentrating on an interesting but ultimately extraneous phenomenon and claiming that it is all there is. – A.V.S. Jan 13 '20 at 16:01
  • @A.V.S. : Perhaps, but that would then mean that the time reversal symmetry implies the future evolution is not determined, either. Which, though, is what I've also heard regarding white holes - so perhaps that would be the better answer to give, or to add to this one. I might try that. – The_Sympathizer Jan 14 '20 at 03:06
  • @The_Sympathizer the white hole and the black hole are part of the same vacuum solution of the equation, though. if you introduce a time-reversible perturbation in the black hole region, it will be identical to producing the time-reverse of that perturbation in the white hole region. – Zo the Relativist Jan 14 '20 at 16:42
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Yes, to the same extent as falling egg violates time reversal symmetry (have you seen a broken egg spontaneously jump from the floor to your cup?). This is called "second law of thermodynamics" and it is not time-symmetric.

Anixx
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    I think this is incorrect? In principle a broken egg can spontaneously jump from the floor to your cup, because all the physics is time-reversible on the microscale; it's only at the macroscale that the second law of thermodynamics kicks in. – Allure Jan 13 '20 at 03:36
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    @Allure Colliding black holes sounds like a macroscale event to me. – Dmitry Grigoryev Jan 13 '20 at 09:42
  • @Allure it is the same with the black holes. There is no difference between merge of two black holes and a merge of two planets or stars. – Anixx Jan 13 '20 at 11:29
  • @Anixx I think in principle if you run time backwards you will see two planets or stars un-merge and split up. It violates the second law, but the second law is not a strict law; it's exceedingly unlikely but possible in principle. – Allure Jan 13 '20 at 11:49
  • @Allure the same is with the black holes. If you run the time backwards they will unmerge. Besides the second law there is no law that prohibits it. – Anixx Jan 13 '20 at 12:10
  • @Anixx are you sure black holes unmerge if you run time backwards? I attended a lecture yesterday where the speaker said it's not possible for black holes to break into two, which implies that it won't happen even if you run time backwards. – Allure Jan 13 '20 at 12:17
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    @Allure there are two things here. First, it takes infinite time for the black holes to entirely MERGE in GR, so given they cannot merge, they cannot unmerge either. What happens is that they look very similar to as if they merged. So they surely can roll back. Second thing to consider is the quantum effects. The black holes emit Hawking radiation, which includes photons and other particles. Rarely those particles can be atoms, even more rarely, molecules, and entirely rarely, tiny black holes. But if tiny black holes can be emitted via radiation, so can bigger ones, it is just very unlikely. – Anixx Jan 13 '20 at 12:41
  • @Allure Also, quantum fluctuations of gravitation field and metric always perturb the form of the horizon, and those perturbations can become so huge that the horizon would become like a dumbbell, and the black hole would separate in two. Why not? In any experiment you can trace where did the energy, momentum, charge go. So if you reverse time, you can easily determine where those things should come from in the reversed scenario. – Anixx Jan 13 '20 at 12:43
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When two black holes collide and merge into one, a lot of energy is sent out as gravitational waves. This energy spreading out without bound represents an increase in entropy, along with any increase due to the final BH surface area being greater than the sum of the original two.

If we had an exact solution to Einstein's field equation describing this, we could replace t with -t and have another valid solution. What this means in real life is that a black hole just sitting somewhere, when bombarded with converging gravitational waves arranged exactly the right way, could absorb those waves and split into black holes.

I don't know about ultra-advanced alien technology, but anything I can imagine like that isn't going to happen. It's just like the fragments and droplets of a dropped egg coming back together just right to make an unblemished whole egg.

DarenW
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Well assuming a black hole time reversed is still a black hole this does not violate cpt symmetry as hawking radiation is random and it is possible but very unlikely that a lot of radiation gets put into just the right spot with the right momentum to make an outgoing black hole

Not a scientist yet
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