Condition for Superradiance in Higher-dimensional Rotating Black Holes

It is shown that the superradiance modes always exist in the radiation by the (4 + n)-dimensional rotating black holes. Using a Bekenstein argument the condition for the superradiance modes is shown to be 0 < ω < mΩ for the scalar, electromagnetic and gravitational waves when the spacetime background has a single angular momentum parameter about an axis on the brane, where Ω is a rotational frequency of the black hole and m is an azimuthal quantum number of the radiated wave. Email:[email protected] Email:[email protected] Email:[email protected] 1 Recent brane-world scenarios such as large extra dimensions [1,2] or compactified extra dimensions with warped factor [3] predict a TeV-scale gravity. The emergence for the TeVscale gravity in the higher-dimensional theories opens the possibility to make the black hole factories in the future high-energy colliders [4–7]. In this context it is of interest to examine the various properties of the higher-dimensional black holes. The absorption and emission for the different particles by the (4 + n)-dimensional Schwarzschild background have been studied analytically [8] and numerically [9]. It was shown that the presence of the extra dimensions in general decreases the absorptivity and increases the emission rate on the brane. The decrease of the absorptivity may be due to the decrease of the effective radius [10] rc ≡ √ σ∞/π, where σ∞ is a high-energy limit of the total absorption cross section. Although it may explain why the absorptivity is suppressed in the high-energy regime, it does not seem to provide a satisfactory physical reason for the suppression of the absorptivity in the full range of the particle energy. The enhancement of the emission rate may be caused by the increase of the Hawking temperature in the presence of the extra dimensions. This means that the Planck factor is more crucial than the greybody factor in the Hawking radiation. For the case of the minimally coupled massless scalar the low-energy absorption cross section(LACS) always equals to the horizon area [11]. Thus, for the brane-localized scalar the LACS is always 4πr H while for the bulk scalar it is equals to Ωn+2r n+2 H where Ωn+2 = 2π /Γ[(n+3)/2] is the area of a unit (n+2)-sphere. The ratio of the LACS for the Dirac field to that for the scalar was shown to be 2 for the brane-localized case [8] and 2 for the bulk case [12]. Therefore, the ratio factor 1/8, which was obtained by Unruh long ago [13], was recovered when n = 0. The dependence on the dimensionality in these ratio factors may be used to prove the existence of the extra dimensions in the future black hole experiments. The relative bulk-to-brane energy emissivity was also calculated in Ref. [9] numerically, which confirmed the main result of Ref. [10], i.e. black holes radiate mainly on the brane. For the higher-dimensional charged black holes the full absorption and emission spectra have been computed numerically in Ref. [14]. It has been shown that contrary to the effect 2 of the extra dimension the presence of the nonzero inner horizon parameter r− generally enhances the absorptivity and suppresses the emission rate. It has been shown also that the relative bulk-to-brane emissivity decreases with increasing the inner horizon parameter r−. The LACS for the minimally coupled massless scalar always equals to the horizon area. For the Dirac fermion the LACS becomes [12]