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, ∆V_th = ${V}_{\mathrm{th}}^{+}$ − ${V}_{\mathrm{th}}^{-}$, 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.