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.
Highlights
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In this paper we further investigate the duality conjecture for stationary anti-de Sitter (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
We have investgated in this paper the original action growth rate (1.4)(1.5)(1.6) proposed in the recent papers [8, 19], which passed for various examples of stationary AdS black holes
Summary
where the cosmological constant Λ is related to the AdS radius L by Λ = −(D − 1)(D − 2)/2L2, h represents the determinant of induced metric on the boundary ∂M, K is the trace of the second fundamental form. The trace of the energy-momentum tensor of electromagnetic field T = (4 − D)F 2/16π is non-vanishing except for the case in four dimensions. r2 f (r) = 1 − (D − 2)ΩD−2 rD−3 + (D − 2)ΩD−2 r2(D−3) + L2 , where M and Q are the mass and charge of the black hole, respectively.
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