# Time-reversal of a black hole merger

General relativity is time reversal invariant. There are solutions though which are physically not reasonable. An example are white holes which are a valid solution of Einstein‘s field equations, but are not believed to exist considering the violation of the second law of thermodynamics. My explicite questions refer to the merger of black holes:

A time reversal black hole merger would „start“ with a spherical black hole and infalling gravitational waves arranged such that it begins to oscillate with increasing amplitude and finally splits in two. Is this possible in principle assuming technical problems are solvable? If not, why?

Does time reversal symmetry of a black hole merger imply that a splitting of one oscillating black hole in two forward in time is impossible in the absence of infalling gravitational waves?

First of all

General relativity is time reversal invariant.

That is not strictly true. Solution of Einstein field equations locally under time reversal transform into another local solution of EFE (provided there is no matter with explicit time arrow).

Is this possible in principle assuming technical problems are solvable?

First, under time reversal, black hole becomes a white hole and its event horizon becomes past horizon (antihorizon).

Second, the largest part of technical problems is that just converging gravitational waves are not enough. One also has to provide initial conditions inside the white hole interior. White holes have past singularity and for a white hole splitting into two (white or black) holes those has to be emitted by this singularity so that they would eventually emerge in the splitting, at the exact same moment that white hole's apparent horizon/antihorizon is deformed by converging 'ring-up' gravitational waves.

Third, even if one can arrange this, the resulting solution is unstable. This instability was discovered by Eardley a few decades ago:

Another more recent (and nonpaywalled) paper:

• Barceló, C., Carballo-Rubio, R., & Garay, L. J. (2016). Black holes turn white fast, otherwise stay black: no half measures. Journal of High Energy Physics, 2016(1), 157, doi, arXiv.

The key to understanding this is the observation that even a tiny amount of matter from the outside, if it gets close enough to a white hole's apparent horizon would form a new event horizon surrounding antihorizon and preventing the splitting from occuring. Instead of white hole we now have a black hole with a slightly larger (if the perturbation was small) horizon radius. The longer the time interval between perturbation and the 'scheduled split' of white hole the smaller could be the perturbation necessary to prevent the explosion. (It is somewhat ironic that instability of explosive solution results in a non-explosion).

And if the perturbation has converted white hole into black the converging gravitational waves simply fall onto a black hole and are partly scattered partly absorbed by its growing event horizon.

Does time reversal symmetry of a black hole merger imply that a splitting of one oscillating black hole in two forward in time is impossible in the absence of infalling gravitational waves?

Not really solution without infalling gravitational waves may be possible but like any other white hole solution it would has the same kind of instability.

Actually it may be more intuitive to ask the question without time reversal:

Is it possible by providing converging gravitational waves to produce a black hole merger without any ringdown radiation?

I believe the answer is yes (just a belief on my part that noise cancelling would work generally for nonlinear GR equations). And that formulation should give us a robust solution: by providing somewhat imprecise converging waves we would receive simply a merger with suppressed ringdown modes.

The variations of this instability are present in antihorizons for other solutions of EFE: inner horizon of Kerr metric, various non-traversable wormholes (such as Einstein-Rosen bridge).

• I may be wrong, but I think your answer and mine may be complementary in the following sense. Mine talks about flipping the global arrow of time, while yours seems to be about what goes wrong when you try to put a time-flipped object like a white hole inside a larger universe. I think we're both pointing our trivialities, but different ones. I'm pointing out that in a time-orientable spacetime, it's arbitrary which orientation you pick. You're pointing out that we get nonsense in a universe in which the arrow of time is different in different places. – Ben Crowell May 23 '18 at 21:24
• @BenCrowell: Agreed, but the second paper I listed also makes another point: if we restrict ourselves to a short timescales then a transient past horizon is not such a remarkable feature. So locally it is not nonsense, only globally so. – A.V.S. May 23 '18 at 21:43
• @A.V.S.: Thanks „That is not strictly true. Solution of Einstein field equations locally under time reversal transform into another local solution of EFE (provided there is no matter with explicit time arrow).“ Do you say that the question ‚is time reversal true‘ is in general answered by thermodynamics?- „First, under time reversal, black hole becomes a white hole and its event horizon becomes past horizon (antihorizon).“Yeah, I should have known this. – timm May 24 '18 at 8:17
• @timm: What I am saying is that 'general relativity' is more than just solution of EFE. There is causality 'on top' of manifold which gives us how to specify initial data, which Green function to choose etc. Example is cosmology: Big Bang and Big Crunch are both valid solutions, but development of fluctuations (both in metric and in matter) gives us time arrow independent of scale dynamics. – A.V.S. May 24 '18 at 10:21

General relativity is time reversal invariant.

Not true. For a given spacetime, there is not in general any way to define a time reversal operation on it. A spacetime does not have to be time-orientable in the first place. GR does not have discrete global symmetries such as parity and time-reversal. (It does locally have the full symmetry of the Poincare group, but that's not what you're describing in your discussion of global symmetries.)

Does time reversal symmetry of a black hole merger imply that a splitting of one oscillating black hole in two forward in time is impossible in the absence of infalling gravitational waves?

All you need in order to rule out this process is the first law of black hole thermodynamics, i.e., conservation of energy. In an asymptotically flat spacetime, we have conserved, global, scalar measures of energy (e.g., the ADM energy), which means that the black hole fission process is impossible without the energy input from the infalling gravitational waves.

A time reversal black hole merger would „start“ with a spherical black hole and infalling gravitational waves arranged such that it begins to oscillate with increasing amplitude and finally splits in two. Is this possible in principle assuming technical problems are solvable? If not, why?

Even with infalling gravitational waves, I think this is impossible in classical GR, as proved by the second law of black hole thermodynamics, i.e. the law that area always increases. As always with these arguments that look like they prove a time-asymmetric law based on laws of physics that don't have such a time asymmetry, what is really happening is that the asymmetry creeps in through the assumption of certain boundary conditions. The area-increase law is proved as proposition 9.2.7 in Hawking and Ellis (p. 318). The time-asymmetry is provided because they define a horizon as a surface from which you can't reach future null infinity.

• White holes do not have 'horizons' they have past horizons (aka antihorizons) which is a separate entity. So one could say that the laws of black hole thermodynamics do not really apply. – A.V.S. May 23 '18 at 17:20
• @A.V.S.: If you simply take the proof of the second law in Hawking and Ellis and change $\mathscr{I}^+$ to $\mathscr{I}^-$ everywhere, then the proof goes through as before, and you have a second law for white holes, which has exactly the same form, but time-reversed. That was the point of the final paragraph of my answer, although I may not have made it very clear. – Ben Crowell May 23 '18 at 19:20
• Thanks, „ Even with infalling gravitational waves, I think this is impossible in classical GR, as proved by the second law of black hole thermodynamics, i.e. the law that area always increases….The area-increase law is proved as proposition 9.2.7 in Hawking and Ellis (p. 318).“Is it proved to be impossible by thermodynamics and GR (Hawking and Ellis) independently? Is an arbitrary solution of EFE time symmetric until this is disproved? Could you please put the first paragraph of you anser in simpler terminology, if that is possible at all? – timm May 24 '18 at 8:16
• Sorry, I've forgotten to address my comment to @Ben Crowell – timm May 24 '18 at 10:00