Holographic Complexity and Two Identities of Action Growth

TL;DR

This work provides a formal proof of the complexity–action conjecture for stationary black holes by showing that the rate of action growth in the Wheeler–DeWitt patch equals the difference of generalized enthalpies between the outer and inner horizons, ${d{\cal A}}{dt} = {\cal H}_+ - {\cal H}_-$. The key steps are establishing two identities: (i) the horizon boundary contribution satisfies $L^{\rm surf}_\pm = T_\pm S_\pm$, with $S$ given by the appropriate entropy (Bekenstein–Hawking in Einstein gravity, Wald entropy in higher-derivative theories), and (ii) the bulk contribution reduces to a generalized enthalpy through the quantum statistic relation, ${\cal H} = F + TS$. The results extend from Einstein gravity to general covariant higher-derivative gravities and discuss limitations in theories with derivative couplings, as well as implications for information storage bounds and the thermodynamic interpretation of black-hole volume and pressure in AdS spacetimes.

Abstract

The recently proposed complexity-action conjecture allows one to calculate how fast one can produce a quantum state from a reference state in terms of the on-shell action of the dual AdS black hole at the Wheeler-DeWitt patch. We show that the action growth rate is given by the difference of the generalized enthalpy between the two corresponding horizons. The proof relies on the second identity that the surface-term contribution on a horizon is given by the product of the associated temperature and entropy.