[CJQS-2020-040] Holographic complexity bounds-量子交叉研究中心

JHEP 07, 090 (2020) arXiv link

Holographic complexity bounds

Hai-Shan Liu¹^,2, H. Lü¹, Liang Ma¹ & Wen-Di Tan¹

1 Center for Joint Quantum Studies and Department of Physics, School of Science, Tianjin University, Tianjin, 300350, China

2 Institute for Advanced Physics & Mathematics, Zhejiang University of Technology, Hangzhou, 310023, China

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 ${\frac{1}{T} \frac{\partial {\overset{\cdot}{I}}_{W D W}}{\partial S} |}_{Q, P_{t h}} > C$ 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 $({\overset{\cdot}{I}}_{W D W} - 2 P_{t h} Δ V_{t h})$ is greater than, equal to, or less than zero, for k = 1, 0, −1 respectively. For black holes with two horizons, ∆V_th = $V_{t h}^{+}$ − $V_{t h}^{-}$ , 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 $V_{t h}^{0}$ of the black hole singularity, and define $Δ V_{t h} = V_{t h}^{+} - V_{t h}^{0}$ . The volume $V_{t h}^{0}$ 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 ${\overset{\cdot}{I}}_{W D W}$ and $V_{t h}^{0}$ , which implies that the holographic complexity preserves the Lloyd’s bound for positive or vanishing $V_{t h}^{0}$ , but the bound is violated when $V_{t h}^{0}$ becomes negative. We also find explicit black hole examples where $V_{t h}^{0}$ and hence ${\overset{\cdot}{I}}_{W D W}$ are divergent.