Action growth for AdS black holes

Abstract

Recently a Complexity-Action (CA) duality conjecture has been proposed, which relates the quantum complexity of a holographic boundary state to the action of a Wheeler-DeWitt (WDW) patch in the anti-de Sitter (AdS) bulk. In this paper we further investigate the duality conjecture for stationary AdS black holes and derive some exact results for the growth rate of action within the Wheeler-DeWitt (WDW) patch at late time approximation, which is supposed to be dual to the growth rate of quantum complexity of holographic state. Based on the results from the general D-dimensional Reissner-Nordström (RN)-AdS black hole, rotating/charged Bañados-Teitelboim-Zanelli (BTZ) black hole, Kerr-AdS black hole and charged Gauss-Bonnet-AdS black hole, we present a universal formula for the action growth expressed in terms of some thermodynamical quantities associated with the outer and inner horizons of the AdS black holes. And we leave the conjecture unchanged that the stationary AdS black hole in Einstein gravity is the fastest computer in nature.

Complexity growth rates for AdS black holes in massive gravity and f(R) gravity

“…[32]. In the limit a → 0, the KN black hole recovers to the charged black hole which can be seen from the metrics (6) and (64).…”

“…[56][57][58][59][60][61][62][63][64][65]. The authors of [32] conjectured that only the stationary black holes in gravity theories without higher derivative terms of curvature can saturate Eq. (4).…”

“…However, the authors of Ref. [32] found that the above bound will be violated for both small and large charged static AdS black holes, so they proposed a new bound…”

Complexity is simple!

“…In this case, the late time action growth of the Wheeler-deWitt patch for this class of solutions was computed in [1,2,48]. Their result is…”

“…10 Thus, we are fixing one overall constant, but the bound is non-trivially satisfied as other parameters are varied, e.g., mass, rotation, and dimension. 11 The violation for small black holes seems to have been neglected in [1] and pointed out subsequently in [48].…”