I've recently read about the last parsec problem, which states that supermassive black holes should not be able to merge within the current age of the universe. Once they reach a distance of about one parsec, there is no more interstellar matter to slow them down, yet relativistic effects (gravity waves) are still too weak to have a major impact.

For Newtonian bodies, such as moons in the solar system, another factor for orbital changes are tidal effects with their planets. These lead to either a decay or a rise in the moons orbit depending on whether it is below or above the geostationary orbit of its host body. These effects can be very pronounced for black holes, which can sometimes rip apart stars.

This brings me to my question: can black holes themselves be affected by tidal effects? I expect the shape of the event horizon to slightly change depending on the surrounding gravitational field, which could count as a deformation. But as this isn't actually shifting matter around, is this doing any mechanical work? Tidal forces also depend on the matter density of the affected object, as they wouldn't have any effects on point masses. Thus, if a black hole was affected by it, I suppose that would leak information about the matter distribution inside the event horizon to an outside observer.

On the other hand, looking at it from the perspective of some particles inside the black holes, they should be affected by the change in potential like any other object passing by a black hole.


Although the shape of the event horizon can be deformed slightly by a tidal gravitational field (see ArXiv:1407.6983 and ArXiv:1505.03809), there is generally no response from the gravitational field. More precisely, the tidal Love numbers that encode the gravitational response of an object from applying an external tidal field, are thought to be zero for all black holes (in general relativity). This has been proven explicitly for Schwarzschild black holes (ArXiv:0906.1366). In the Kerr case it has (as far as I know) only been proven in the slow rotation limit (although there may have been more recent progress; this is an active field of study).

At a more intuitive level, the vanishing (or at least smallness) of the black hole love numbers can be somewhat understood from the fact that black holes are extremely tiny compared to material objects (like moons). Since tidal effects grow with the size of the object this immidiately supresses any possible tidal effects. It doesn't take much to convince oneself, that even if there were an effect from black holes deforming each other, it would still be subdominant to the loss of energy through gravitational radiation.

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  • $\begingroup$ "black holes are extremely tiny compared to material objects" smbh are less "dense" than earth. Would this mean their mass is concentrated at the center? $\endgroup$ – DeinFreund Aug 19 '19 at 6:54
  • $\begingroup$ The paper showing that love numbers are zero also cited arxiv.org/pdf/gr-qc/0501032.pdf as a basis and implicit proof. There, it is shown how energy is transferred to the black hole through a tidal field "the rates at which the tidal interaction transfers mass and angular momentum to the black hole". What am I missing that allows these two statements to agree? $\endgroup$ – DeinFreund Aug 19 '19 at 7:47

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