# Do colliding black holes violate time reversal symmetry?

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.

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.)

• Do you claim that arranging the gravitational front in specific ways you can make any object out of black hole? Jan 12 '20 at 18:30
• @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.) Jan 12 '20 at 20:22
• @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. Jan 12 '20 at 22:50
• 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
• “-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). Jan 13 '20 at 5:14

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.

• 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. Jan 13 '20 at 3:36
• @Allure Colliding black holes sounds like a macroscale event to me. Jan 13 '20 at 9:42
• @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. 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. Jan 13 '20 at 11:49
• @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. Jan 13 '20 at 12:41

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.