Complexity Growth for AdS Black Holes

TL;DR

This work scrutinizes the Complexity-Action duality by computing exact late-time action growth rates for a variety of stationary AdS black holes (RN-AdS, BTZ, Kerr-AdS, GB-AdS). It reveals that the original bound on action growth is violated for charged AdS black holes and proposes a universal bound expressed in terms of horizon-based thermodynamic quantities: dA/dt ≤ (M-ΩJ-μQ)_+ − (M-ΩJ-μQ)_−, with equality realized for stationary Einstein gravity and certain GB cases. The results unify the growth rates across different spacetimes as simple horizon-difference formulas, connect BTZ cases to left/right-moving CFT data, and show stringy (Gauss-Bonnet) corrections generally slow the rate, reinforcing the notion that Einstein gravity AdS black holes act as the fastest computers. The analysis also highlights subtleties related to singularities behind horizons and the role of inner horizons in regulating WDW patch contributions. Overall, the paper strengthens the CA duality framework and clarifies how higher-derivative corrections influence holographic complexity growth.

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.

Figures (2)

• Figure 1: The Penrose diagram of the charged Gauss-Bonnet-AdS black hole. The singularities $r=0$ or $r=\widetilde{r}_-$ are presented with wiggly lines. The growth rate of WDW patch at late time approximation comes from the spacetime region that lies outside the inner horizon and inside the outer horizon.
• Figure 2: The Penrose diagram of the neutral Gauss-Bonnet-AdS black hole. The WDW patch can be ended either on both singularities $r=0$ or $r=\widetilde{r}_-$.