A membrane paradigm at large D

Abstract

We study SO(d + 1) invariant solutions of the classical vacuum Einstein equations in p + d + 3 dimensions. In the limit d → ∞ with p held fixed we construct a class of solutions labelled by the shape of a membrane (the event horizon), together with a 'velocity' field that lives on this membrane. We demonstrate that our metrics can be corrected to nonsingular solutions at first sub-leading order in 1 d if and only if the membrane shape and 'velocity' field obey equations of motion which we determine. These equations define a well posed initial value problem for the membrane shape and this 'velocity' and so completely determine the dynamics of the black hole. They may be viewed as governing the non-linear dynamics of the light quasi normal modes of Emparan, Suzuki and Tanabe.

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].…”

“…In order to have a physically acceptable solution, we must have the condition (4.10). Therefore, we may just assume thatâ 12) whereb is of O(1). In terms ofb, P (z) now becomes like 13) which is regular even in the extremal limitb = 0.…”

Static Gauss-Bonnet black holes at large D

“…In order to make the above expressions simpler, let us introduce 8) then the condition (4.5) becomes 9) JHEP05 (2017)025 and eq. (4.6) becomes…”

“…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.…”