The large D membrane paradigm for Einstein-Gauss-Bonnet gravity

Abstract

We find the equations of motion of membranes dual to the black holes in Einstein-Gauss-Bonnet (EGB) gravity to leading order in 1/D in the large D regime. We also find the metric solutions to the EGB equations to first subleading order in 1/D in terms of membrane variables. We propose a world volume stress tensor for the membrane whose conservation equations are equivalent to the leading order membrane equations. We work out the light quasi-normal mode spectrum of static black holes in EGB gravity from the linearised fluctuations of static, round membranes. Also, the effective equations for stationary black holes and the spectrum of linearised spectrum about black string configurations has been obtained using the membrane equation for EGB gravity. All our results are worked out to linear order in the Gauss-Bonnet parameter.

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

“…So, in effect we are working with the strongest possible GB coefficient β in the large D limit which satisfies this criteria. 9 This matches with the ansatz metric written in [20] as to leading order in 1/D, dψ · dψ = K 2 D 2 . Also, we require β K 2 D 2 ≥ − 1 2 for the horizon to be at ψ = 1.…”

“…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.…”

“…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 → ∞.…”

“…Since, we want to study dynamical processes involving black holes, we promote ψ and u to be functions of the spacetime points with the following restrictions The ansatz metric solves for the EGB equation to leading order in large D [15,16,20] provided…”

“…The Gauss-Bonnet (GB) parameter β is O(D 0 ) as mentioned earlier. The necessity of this has been explained in [20] to make sure that the large D solutions are smoothly connected to the solutions of Einstein-Hilbert theory in the β → 0 limit. The basic reason being that the typical curvature squared objects are O(D 2 ) times the monomials in curvature term in the large D limit.…”