Black holes in presence of cosmological constant: second order in $$ \frac{1}{D} $$

Abstract

We have extended the results of [1] upto second subleading order in an expansion around large dimension D. Unlike the previous case, there are non-trivial metric corrections at this order. Due to our 'background-covariant' formalism, the dependence on Ricci and the Riemann curvature tensor of the background is manifest here. The gravity system is dual to a dynamical membrane coupled with a velocity field. The dual membrane is embedded in some smooth background geometry that also satisfies the Einstein equation in presence of cosmological constant. We explicitly computed the corrections to the equation governing the membrane-dynamics. Our results match with earlier derivations in appropriate limits. We calculated the spectrum of QNM from our membrane equations and matched them against similar results derived from gravity.

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 → ∞.…”

“…We describe in brief the mechanism for this below. For the details of the procedure, one can look at the previous papers on the large D membrane paradigm [15][16][17] 11 .…”

“…In one the effective equations are obtained in terms of a dual object defined by its effective mass and momentum [8][9][10][11] 1 . The other formulation (which is the primary focus of this paper), recasts the black hole dynamics in terms of dynamics of a co-dimension one membrane propagating in the asymptotic spacetime of the black hole [13][14][15][16][17][18]. The membrane is characterised by its local shape and a worldvolume velocity field.…”

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

“…Hence, the above equations are satisfied at R = 0 only if the sources of the homogeneous parts vanish at R = 0. This is not a content less statement as has been observed in [13,14,27,34]. In fact this condition puts the following constraint on the membrane data U = 0.…”

The fate of instability of de Sitter black holes at large D

“…However, due to the limitation of the linear analysis, the final stage of the evolution of the unstable black holes has not been settled yet, though some non-linear efforts have been made from the viewpoint of gravitational collapse [9].The goal of this work is to better understand the "Λ instability" of the two kinds of black holes by using the large D expansion method [10]. Taking the large dimension limit, the non-trivial dynamics of black holes is localized at the near region of the horizon, such that a full non-linear effective theory can be formulated [11][12][13][14][15][16][17][18][19][20]. The large D studies of the RN-dS and GB-dS black holes were initiated in [21] and [22].…”