General Relativity simplifies dramatically in the limit that the number of spacetime dimensions D is infinite: it reduces to a theory of non-interacting particles, of finite radius but vanishingly small cross sections, which do not emit nor absorb radiation of any finite frequency. Non-trivial black hole dynamics occurs at length scales that are 1/D times smaller than the horizon radius, and at frequencies D times larger than the inverse of this radius. This separation of scales at large D, which is due to the large gradient of the gravitational potential near the horizon, allows an effective theory of black hole dynamics. We develop to leading order in 1/D this effective description for massless scalar fields and compute analytically the scalar absorption probability. We solve to next-to-next-to-leading order the black brane instability, with very accurate results that improve on previous approximations with other methods. These examples demonstrate that problems that can be formulated in an arbitrary number of dimensions may be tractable in analytic form, and very efficiently so, in the large D expansion.Comment: 50 pages, 2 figures; v3: improved discussion of conceptual point

Abstract:The gravitational field of a black hole is strongly localized near its horizon when the number of dimensions D is very large. In this limit, we can effectively replace the black hole with a surface in a background geometry (e.g. Minkowski or Anti-deSitter space). The Einstein equations determine the effective equations that this 'black hole surface' (or membrane) must satisfy. We obtain them up to next-to-leading order in 1/D for static black holes of the Einstein-(A)dS theory. To leading order, and also to next order in Minkowski backgrounds, the equations of the effective theory are the same as soap-film equations, possibly up to a redshift factor. In particular, the Schwarzschild black hole is recovered as a spherical soap bubble. Less trivially, we find solutions for 'black droplets', i.e. black holes localized at the boundary of AdS, and for non-uniform black strings.

Abstract:The limit of large number of dimensions localizes the gravitational field of a black hole in a well-defined region near the horizon. The perturbative dynamics of the black hole can then be characterized in terms of states in the near-horizon geometry. We investigate this by computing the spectrum of quasinormal modes of the Schwarzschild black hole in the 1/D expansion, which we find splits into two classes. Most modes are non-decoupled modes: non-normalizable states of the near-horizon geometry that straddle between the near-horizon zone and the asymptotic zone. They have frequency of order D/r 0 (with r 0 the horizon radius), and are also present in a large class of other black holes. There also exist a much smaller number of decoupled modes: normalizable states of the near-horizon geometry that are strongly suppressed in the asymptotic region. They have frequency of order 1/r 0 , and are specific of each black hole. Our results for their frequencies are in excellent agreement with numerical calculations, in some cases even in D = 4.

We derive a simple set of non-linear, (1 + 1)-dimensional partial differential equations that describe the dynamical evolution of black strings and branes to leading order in the expansion in the inverse of the number of dimensions D. These equations are easily solved numerically. Their solution shows that thin enough black strings are unstable to developing inhomogeneities along their length, and at late times they asymptote to stable non-uniform black strings. This proves an earlier conjecture about the endpoint of the instability of black strings in a large enough number of dimensions. If the initial black string is very thin, the final configuration is highly non-uniform and resembles a periodic array of localized black holes joined by short necks. We also present the equations that describe the non-linear dynamics of Anti-deSitter black branes at large D.arXiv:1506.06772v2 [hep-th]

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 GregoryLaflamme 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 k 9 for the sound mode and to k 8 for the shear mode.