Low-frequency scalar absorption cross sections for stationary black holes

We discuss the absorption cross section for the minimally-coupled massless scalar field into a stationary and circularly symmetric black hole with nonzero angular velocity in four or higher dimensions. In particular, we show that it equals the horizon area in the zero-frequency limit provided that the solution of the scalar field equation with an incident monochromatic plane wave converges pointwise to a smooth time-independent solution outside the black hole and on the future horizon, with the error term being at most linear in the frequency. We also show that this equality holds for static black holes which are not necessarily spherically symmetric. The zero-frequency scattering cross section is found to vanish in both cases. It is shown in an Addendum that the equality holds for any stationary black hole with vanishing expansion if the limit solution is known to be a constant. Properties of black holes have been attracting much attention in connection with string theory since it was found that the Bekenstein-Hawking entropy is reproduced by string theory for some black holes [1]. The absorption cross section for the minimallycoupled massless scalar field was studied in this context [2, 3, 4, 5], and Das, Gibbons and Mathur showed that it is exactly the horizon area for a static and spherically symmetric black hole in any dimensions in the zero-frequency limit [6]. (See, e.g. [7, 8, 9, 10] for earlier calculations of absorption cross sections for black holes.) It is natural to