Holographic Complexity Bounds

Abstract

We study the action growth rate in the Wheeler-DeWitt (WDW) patch for a variety of $D\ge 4$ 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 $\frac{1}{T}\frac{\partial {\dot{I}}_{\mathrm{WDW}}}{\partial S}{|}_{Q,P_{\mathrm{th}}}>C$ 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 $({\dot{I}}_{\mathrm{WDW}}-2P_{\mathrm{th}}\Delta V_{\mathrm{th}})$ is greater than, equal to, or less than zero, for $k=1,0,-1$ respectively. For black holes with two horizons, $\Delta V_{\mathrm{th}}=V_{\mathrm{th}}^{+}-V_{\mathrm{th}}^{-}$, 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 ${\dot{I}}_{\mathrm{WDW}}$ 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 ${\dot{I}}_{\mathrm{WDW}}$ are divergent.

arXiv commentLatex, 49 pages, typos corrected and references added

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