Effective theory of black holes in the 1/D expansion

Abstract

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.

Charged black rings at large D

“…for the decoupled perturbations [7] the black hole can be effectively regarded as a thin membrane [11,12] with the thickness ∼ r 0 /D. The shape of the membrane that is the topology of the horizon is described by the embedding of the membrane into a background spacetime, and the non-linear dynamics of the membrane is determined by the effective equations obtained by integrating out the radial direction of the Einstein equations.…”

“…The shape of the membrane that is the topology of the horizon is described by the embedding of the membrane into a background spacetime, and the non-linear dynamics of the membrane is determined by the effective equations obtained by integrating out the radial direction of the Einstein equations. By solving the effective equations with proper embeddings of the membrane, one can construct different black hole solutions and furthermore study their dynamics perturbatively to find the quasinormal modes or determine numerically the end points of their evolutions under the unstable perturbations [11][12][13][14][15][16][17][18][19][20][21][22][23][24]. From a broader perspective, the large D effective theory of the black hole is similar to the effective theories in the fluid/gravity correspondence [25] and the blackfold approach [26,27].…”

“…The non-trivial conditions are just the effective equations for the slowly rotating large D black holes. The large D effective equations could be interpreted as the conservation equations of the charge and the large D effective stress tensor at R → ∞ [11,19]. 2 These equations are listed as follows…”

Static Gauss-Bonnet black holes at large D

“…It was interestingly noted in [6,7,13] that the stationary large D black holes are the solutions of an elastic theory. For example, for the neutral static black holes the effective membrane embedded in the background must satisfy the equation…”

“…The essence in the large D expansion is that when the spacetime dimension is sufficiently large D → ∞, the gravitational field of a black hole is strongly localized near its horizon due to the dominant radial gradient of the gravitational potential. As a result, for the decoupled quasinormal modes [5] the black hole can be effectively taken as a surface or membrane embedded in the background spacetime [6][7][8][9][10][11]. The membrane is described by the way it is embedded into the background spacetime, and its nonlinear dynamics is determined by the effective equations obtained by integrating the Einstein equations in the radial direction.…”