Abstract
We use the inverse-dimensional expansion to compute analytically the frequencies of a set of quasinormal modes of static black holes of Einstein-(Anti-)de Sitter gravity, including the cases of spherical, planar or hyperbolic horizons. The modes we study are decoupled modes localized in the near-horizon region, which are the ones that capture physics peculiar to each black hole (such as their instabilities), and which in large black holes contain hydrodynamic behavior. Our results also give the unstable Gregory-Laflamme frequencies of Ricci-flat black branes to two orders higher in 1/D than previous calculations. We discuss the limits on the accuracy of these results due to the asymptotic but not convergent character of the 1/D expansion, which is due to the violation of the decoupling condition at finite D. Finally, we compare the frequencies for AdS black branes to calculations in the hydrodynamic expansion in powers of the momentum k. Our results extend up to k9 for the sound mode and to k8 for the shear mode.
Highlights
Have support only in the near-horizon region, where they are normalizable states to all orders in 1/D; they capture properties specific to each black hole, such as the instabilities of certain higher-dimensional black holes and black branes, and the hydrodynamic modes of black branes [5, 8]
The modes we study are decoupled modes localized in the near-horizon region, which are the ones that capture physics peculiar to each black hole, and which in large black holes contain hydrodynamic behavior
Our results extend up to k9 for the sound mode and to k8 for the shear mode
Summary
We consider (A)dS black holes with metric ds2 = −f (r)dt2 + f (r)−1dr2 + r2 dσK2 ,D−2, 1The decoupled spectrum at large D was first identified numerically in [9]. For the curvature of the metric dσK2 ,D−2 on the ‘orbital’ space KD−2: a unit sphere, a plane, or a hyperboloid, for K = +1, 0, −1 respectively. If we rescale (t, R) → (R0t, R0R) the metric becomes the same for all values of K and λ. In the hyperbolic case (K = −1, λ < 0), when n is finite there exist black hole solutions with negative mass parameter r0n < 0, with the lowest negative mass corresponding to an extremal black hole. The large n limit yields a good near-horizon geometry only when.
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