ETH-monotonicity and the black hole singularity

Abstract

We study the enveloping function of the fluctuation term in eigenstate thermalization hypothesis (ETH) statement for holographic conformal field theories. We use this function to identify and examine black hole microstates. We set down a set of desirable criteria for this function called ETH-monotonicity. It reinforces the Kelvin statement of the second law of thermodynamics over and above the universal entropic contribution. We show that higher-dimensional holographic conformal field theories possess ETH-monotonicity. Stronger contribution from ETH-monotonicity to the second law of thermodynamics is observed in smaller black hole microstates. It dominates other quantum fluctuations. It also measures the curvature at the horizon of the small black holes. In the smallest size limit, the black hole curvature singularity is constructed of microstates for which ETH-monotonicity starts competing with the entropic contribution. We expect that ETH-monotonicity will persist even in the ultimate quantum theory of gravity. Because it is a property of many-body quantum chaotic systems which becomes more prominent with decreasing system size, unlike other physical properties which are usually more predictable with increasing system size. Two-dimensional holographic conformal field theory does not possess all features of ETH-monotonicity which is in agreement with the absence of curvature singularity in the BTZ black hole.