# Is there a geometric reason why two merging black holes never “decay” into two separate black holes

All black hole mergers seem to result in one Kerr black hole after ring-down.

Could it be possible, that two initial black holes with two separate event horizons form an “intermediate compound state” with one event horizon, which - at a later time - “decays” in two final black holes?

I know that the answer to this question is “no” when applying Bekenstein-Hawking entropy where the decay would violate the second law which postulates $dA/dt > 0$.

And the answer seems to be “no” when relying on numerical relativity for black hole mergers.

My question is if there is a more elegant, geometrical proof following entirely from general relativity, which rules out such a decay.

Thanks

• I think there’s a theorem that says exactly that this can’t happen from pure classical GR considerations, hopefully somebody remembers more specifics! – knzhou May 21 '18 at 8:34
• – Qmechanic May 21 '18 at 8:48
• I would think that it would be related to the statement "never scatter off each other" – anna v May 21 '18 at 9:17
• Hawkibg's area theorem is purely geometrical/GR theorem that implies it cannot happen. – MBN May 21 '18 at 15:02

This geometric reason for that statement is that there is only one null geodesic lying on the horizon going through each point of the horizon. If the black hole have split into two, then there would be a point on the horizon from which two different null geodesics lying within the horizon would start, which is impossible. So black holes could form (starting with one point from which all of the horizon springs into existance) and merge, but not bifurcate.

This is not a consequence of the Hawking area theorem but was also formulated by Hawking:

• Hawking, S. W. (1972). Black holes in general relativity. Communications in Mathematical Physics, 25(2), 152-166, doi, OA text at ProjectEuclid. (section 2, p. 156).

• Hawking, S. W., & Ellis, G. F. R. (1973). The large scale structure of space-time. Cambridge University Press, Chapter 9.

As $\tau$ increases, black holes can merge together, and new black holes can form as the result of further bodies collapsing. However, the following result shows that black holes can never bifurcate.

The following result is a proposition 9.2.5 which simply formally states that.

Incidentally, it seems that this property has no generalization in higher dimensions where for example black strings (extended black holes with horizon topology of, say, $R_2\times S_2$ for 5D gravity) would fragment into many snaller black holes as a result of Gregory–Laflamme instability.

• Lehner, L., & Pretorius, F. (2011). Final state of Gregory-Laflamme instability. arXiv:1106.5184.
• Thanks to A.V.S. for the link to Hawking’s article on black holes. I have to understand some technical topics in more detail, but in essence this answers my question! – TomS May 22 '18 at 10:52

Here are links scattering of black holes studies:

We study systems consisting of several maximally charged, nonrotating black holes (‘‘Reissner-Nordstrom’’ black holes) interacting with one another. We present an effective action for the system in the slow-motion, fully strong-field regime. We give an exact calculation of black-hole–black-hole scattering and coalescence in the slow-motion (but strong-field) limit.

The classical and the quantum scattering of two maximally-charged dilaton black holes which have low velocities are studied. We find a critical value for the dilaton coupling, a2=1/3. For a2>1/3, two black holes are always scattered away and never coalesce together, regardless of the value of the impact parameter.

We describe the quantum mechanical scattering of slowly moving maximally charged black holes. Our technique is to develop a canonical quantization procedure on the parameter space of possible static classical solutions. With this, we compute the capture cross sections for the scattering of two black holes. Finally,we discuss how quantization on this parameter space relates to quantization of the degrees of freedom of the gravitational field

The low-energy dynamics of any system admitting a continuum of static configurations is approximated by slow motion in moduli (configuration) space. Here, following Ferrell and Eardley, this moduli space approximation is utilized to study collisions of two maximally charged Reissner--Nordström black holes of arbitrary masses, and to compute analytically the gravitational radiation generated by their scattering or coalescence. The motion remains slow even though the fields are strong, and the leading radiation is quadrupolar. A simple expression for the gravitational waveform is derived and compared at early and late times to expectations.

Italics mine.

So, I would think that a general geometrical proof of impossibility of scattering, would be against all these calculations, and as they are in peer reviewed literature, the probability of such a mistake is small.

Binary black hole:

binary black hole (BBH) is a system consisting of two black holes in close orbit around each other.

Italics mine.

Here it treats the shape of the horizon as a merger starts

One of the problems to solve is the shape or topology of the event horizon during a black-hole merger.

In numerical models, test geodesics are inserted to see if they encounter an event horizon. As two black holes approach each other, a ‘duckbill’ shape protrudes from each of the two event horizons towards the other one. This protrusion extends longer and narrower until it meets the protrusion from the other black hole. At this point in time the event horizon has a very narrow X-shape at the meeting point. The protrusions are drawn out into a thin thread. The meeting point expands to a roughly cylindrical connection called a bridge.

In a system where black holes scatter of each other , it will depend on the kinematics whether there will be a scatter or a merger, as with the binary black hole system. See figure two . Imo, once there is a coalescence the question of "decay" addresses one black hole, and by construction it cannot happen.

• Scattering is a good descriptor, but if the black-hole horizons never touch then I'm not sure it falls into OP's intent. – Emilio Pisanty May 21 '18 at 11:30
• @EmilioPisanty I expect there is a mathematical continuity between coalescence and scattering. In the two links I have access too the event horizon is not dealt separately. Seems to me if one event horizon exists, coalescence has happened, and decay has as much meaning as asking whether a black hole can decay ( independent of how it was formed) – anna v May 21 '18 at 12:47
• My question is not about scattering but about merger (coalescence) of two horizons into one with subsequent decay. From simple time reversal this should be allowed. So the question is whether there is a theorem which guarantees that after coalescence subsequent decay is forbidden – TomS May 21 '18 at 13:42
• @TomS are you sure that time reversal holds? Wouldn't this mean that gravitational waves have to be absorbed? – timm May 21 '18 at 16:21
• Regarding the area theorem: is says that for the area A the condition dA/dt > 0 holds under some reasonable assumptions. However, this does not say that a “decay” is forbidden in general, it rules out decays for some explicitly known geometries like Schwarzschild and Kerr. – TomS May 22 '18 at 6:48