5.6 Absorption of gravitational waves by a black hole

5.6 Absorption of gravitational waves by a black hole

In this section, we evaluate the energy absorption rate by a black hole. The energy flux formula is given by [62 ]

( ) sum integral 2 2 2 2 2 5 dEhole = dw 2Slm-128wk(k--+--4~e-)(k--+-16~e)(2M--r+)-|~ZHlmw|2, (194) dt d_O_ lm 2p |C |2

where $~e = k/(4r+)$ , and

( )( ) |C |2 = (c + 2)2 + 4awm - 4a2w2 c2 + 36awm - 36a2w2 2 2 2 2 2 +(2c + 3)(96a w - 48awm) + 144w (M - a ). (195)

In calculating $~ H Zlmw$ , we need to evaluate the upgoing solution $up R$ , and the asymptotic amplitude of ingoing and upgoing solutions, $inc B$ , $trans B$ , and $trans C$ in Equations (20 ) and (21 ). Evaluation of the incident amplitude $Btrans$ of the ingoing solution is essential in the calculation. Poisson and Sasaki [45 ] evaluated them, in the case of a circular orbit around the Schwarzschild black hole, up to $O(e)$ beyond the lowest order, and obtained the energy flux at the lowest order, using the method we have described in Section 3. Later, Tagoshi, Mano, and Takasugi [55 ] evaluated the energy absorption rate in the Kerr case to $O(v8)$ beyond the lowest order using the method in Section 4. Since the resulting formula is very long and complicated, we show it here only to $3 O(v )$ beyond the lowest order. The energy absorption rate is given by

(dE ) 32 ( m )2 [ 1 3 ( 33 ) --- = --- --- x10x5 - -q - --q3 + - q- --q3 x2 dt H 5 M 4( 4 16 ) ] 1- 13- 2 35- 2 1-4 1- 4 3 3 + 2qB2 + 2 + 2 kq + 6 q - 4q + 2 k + 3q k + 6q B2 x , (196)

where

[ ( ) ( )] -1 (0) --niq---- (0) --niq---- Bn = 2i y 3 + V~ 1---q2 - y 3- V~ 1---q2 , (197)

and $y(n)(z)$ is the polygamma function. We see that the absorption effect begins at $O(v5)$ beyond the quadrapole formula in the case $q /= 0$ . If we set $q = 0$ in the above formula, we have

(dE ) (dE ) ( ) --- = --- x8 + O(v10) , (198) dt H dt N

which was obtained by Poisson and Sasaki [45 ].

We note that the leading terms in $(dE/dt) H$ are negative for $q > 0$ , i.e., the black hole loses energy if the particle is co-rotating. This is because of the superradiance for modes with $k < 0$ .