# Hawking radiation for closely orbiting black holes

Suppose we have two black holes of radius $R_b$ orbiting at a distance $R_r$. I believe semi-classical approximations describe correctly the case where $R_r$ is much larger than the average black body radiation wavelength due to Hawking radiation.

Do we have approximations for Hawking radiation temperature where the distance $R_r$ is of the same order, or in the case where it is much shorter than the radiation average wavelength?

In the absence of a concrete analysis for either one, Do we have any physical insight to affirm if Hawking radiation will be either inhibited or increased in the above situations?

• I guess the radiation would be inhibited, since the black holes absorb radiation from each other, thus lose mass and hawking-radiate more slowly. – resgh Dec 8 '12 at 14:27
• remember the distance between the BHs is the same order or smaller than the wavelength of the radiation. There might be nontrivial boundary effects that qualitative change the Bogoliubov transformation – lurscher Dec 8 '12 at 14:40
• Just a guess. I'm not bothering to do any serious mathematics so I can't prove anything. – resgh Dec 8 '12 at 14:42
• What exactly do you mean by at a distance $R_r$? – MBN Dec 8 '12 at 14:55
• @MBN Read the first sentence of the question. – resgh Dec 8 '12 at 14:58

This problem would be very difficult. Obviously for it to be physically relevant this would have to pertain to two small black holes in a mutual orbit. Large astrophysical black holes are are at temperature $6.1\times 10^{-8}K(M/M_{sol})$ and Hawking radiation is insignificant,
Geroch looked at the issue of the shape of a black hole horizon due to external distributions of matter. This was way back in the 1970s. Two mutually interacting black holes will also distort their horizons. In a sense the more curvature there is to the horizon per unit area the more radiation there will be. Think analogously with a flat radiating surface and one with a rough surface, or why fins and the rest are used in units to dissipate heat. The gravity field near a horizon area that is curved will have a larger gradient to the gravity field. The surface gravity $g^2~=~\nabla^a\xi_b\nabla_a\xi^b$ will be larger if the Killing fields have a greater divergence.