Holographic complexity bounds

Abstract

We study the action growth rate in the Wheeler-DeWitt (WDW) patch for a variety of D ≥ 4 black holes in Einstein gravity that are asymptotic to the anti-de Sitter spacetime, with spherical, toric and hyperbolic horizons, corresponding to the topological parameter k = 1, 0, −1 respectively. We find a lower bound inequality {left.frac{1}{T}frac{partial {overset{cdot }{I}}_{mathrm{WDW}}}{partial S}right|}_{Q,{P}_{mathrm{th}}}>C for k = 0, 1, where C is some order-one numerical constant. The lowest number in our examples is C = (D − 3)/(D − 2). We also find that the quantity left({overset{cdot }{I}}_{mathrm{WDW}}-2{P}_{mathrm{th}}Delta {V}_{mathrm{th}}right) is greater than, equal to, or less than zero, for k = 1, 0, −1 respectively. For black holes with two horizons, ∆Vth = {V}_{mathrm{th}}^{+} − {V}_{mathrm{th}}^{-} , i.e. the difference between the thermodynamical volumes of the outer and inner horizons. For black holes with only one horizon, we introduce a new concept of the volume {V}_{mathrm{th}}^0 of the black hole singularity, and define Delta {V}_{mathrm{th}}={V}_{mathrm{th}}^{+}-{V}_{mathrm{th}}^0 . The volume {V}_{mathrm{th}}^0 vanishes for the Schwarzschild black hole, but in general it can be positive, negative or even divergent. For black holes with single horizon, we find a relation between {overset{cdot }{I}}_{mathrm{WDW}} and {V}_{mathrm{th}}^0 , which implies that the holographic complexity preserves the Lloyd’s bound for positive or vanishing {V}_{mathrm{th}}^0 , but the bound is violated when {V}_{mathrm{th}}^0 becomes negative. We also find explicit black hole examples where {V}_{mathrm{th}}^0 and hence {overset{cdot }{I}}_{mathrm{WDW}} are divergent.