Temperatures of AdS$_2$ black holes and holography revisited

TL;DR

This paper revisits the temperature evolution of AdS$_2$ black holes during evaporation, incorporating quantum backreaction, and finds that the temperature starts at zero, peaks, and then vanishes, with the maximum temperature never exceeding the eternal black hole's Hawking temperature.

Contribution

It provides a detailed analysis of black hole temperature dynamics during evaporation with quantum effects, highlighting the initial zero temperature and maximum temperature constraints.

Findings

Black hole energy remains constant initially and then decreases to zero.

Temperature starts at zero, peaks at a critical time, then vanishes.

Maximum temperature never exceeds the Hawking temperature of the eternal black hole.

Abstract

In a dilaton gravity model, we revisit the calculation of the temperature of an evaporating black hole that is initially formed by a shock wave, taking into account the quantum backreaction. Based on the holographic principle, along with the assumption of a boundary equation of motion, we show that the black hole energy is maintained for a while during the early stage of evaporation. Gradually, it decreases as time goes on and eventually vanishes. Thus, the Stefan-Boltzmann law tells us that the black hole temperature, defined by the emission rate of the black hole energy, starts from zero temperature and reaches a maximum value at a critical time, and finally vanishes. It is also shown that the maximum temperature of the evaporating black hole never exceeds the Hawking temperature of the eternal AdS$_{2}$ black hole. We discuss physical implications of the initial zero temperature of the evaporating black hole.