The large D membrane paradigm for general four-derivative theory of gravity with a cosmological constant

Abstract

We find the membrane equations which describe the leading order in 1/D dynamics of black holes in the D → ∞ limit for the most general four-derivative theory of gravity in the presence of a cosmological constant. We work up to linear order in the parameter determining the strength of the four-derivative corrections to the gravity action and hence there are no ghost modes in the theory. We find that the effective membrane equations we obtain are the covariant version of the membrane equations in absence of the cosmological constant. We also find the world-volume stress tensor for the membrane whose conservation gives the membrane equations. We apply the membrane equations to predict the light quasi-normal mode spectrum of black holes and black branes in the theory of gravity under consideration.

General theory of large D membranes consistent with second law of thermodynamics

“…For static black holes of Einstein-Hilbert gravity, F (r) = r h r D−3 . For examples of blackening factor in other situations see [2](for Einstein-Maxwell theory), [6] (for Einstein-Gauss Bonnet theory) and [8]( for general four derivative theory of gravity). In general F (r) is obtained by solving an ordinary differential equation (ODE) along r for F (r) with the above boundary conditions .…”

“…The dynamics of this dual non-gravitational system is given by a set of effective equations on a finite number of variables in a 1/D expansion. In a formalism developed in [1][2][3][4][5][6][7][8][9], the effective non-gravitational system consists of a co-dimension one membrane moving in an ambient space-time that is equivalent to the asymptotic space-time of the dual black holes. 1 In this paper we will make some general observations about the effective equations of these dual membranes for a general class of gravity theories.…”

Large D membrane for higher derivative gravity and black hole second law

“…In this section we briefly review the large D membrane paradigm of [13][14][15][16][17] aligned to the discussion in the context of higher derivative theories [20,21]. We take the large D limit by restricting the dynamics to finite number of directions so that the spacetime preserves a SO(D − p − 3) isometry as D → ∞.…”

“…Hence, it is easy to 11 In practice, it is convenient to work in terms of a set of effective gravity equations in a reduced p + 3 dimensional spacetime involving the p + 3 dimensional metric and a dilaton field φ.The results obtained this way can then be recast in terms of quantities in the full D dimensional spacetime. The results obtained are then shown to be independent of the choice of p.The details of this procedure can be found in [13,14,20,21] see that at first subleading order in 1/D the EGB equations when evaluated in the patch coordinates on the ansatz metric, evaluate to a non-zero quantity of the form 12 Since, the part which remains unsolved at the subleading order is only a non-trivial function of the patch radial coordinate R 13 , the necessary corrections need only be functions of the R coordinates. i.e.…”

Black rings in large D membrane paradigm at the first order

“…While the other branch, known as the "large D membrane paradigm" [53][54][55][56][57][58][59][60][61][62][63][64][65][66][67][68][69] states that the dynamical black holes in large D have one-to-one correspondence to the dynamics of a codimension one hypersurface, called a "membrane", which doesn't backreact on its background. In this nongravitational dual, the membrane dynamics is governed by what are called the "membrane equations of motion", and the dynamical data associated with the membrane completely defines the corresponding black hole solution.…”