Entropy of Hawking Radiation for Two-Sided Hyperscaling Violating Black Branes

TL;DR

This paper investigates the von Neumann entropy of Hawking radiation for hyperscaling violating black branes, analyzing how the Page curve and Page time depend on various parameters in different matter field scenarios.

Contribution

It provides a detailed calculation of the Page curve and Page time for HV black branes coupled to thermal baths, considering both conformal and hyperscaling violating matter fields, revealing new dependencies on geometric exponents.

Findings

Page time proportional to thermal entropy over temperature for CFT matter.

Hawking radiation entropy grows linearly or exponentially depending on hyperscaling violation.

Page time varies with dynamical and hyperscaling exponents, decreasing or increasing accordingly.

Abstract

In this paper, we study the von Neumann entropy of Hawking radiation $S_{\rm R}$ for a $d+2$-dimensional Hyperscaling Violating (HV) black brane which is coupled to two Minkowski spacetimes as the thermal baths. We consider two different situations for the matter fields: First, the matter fields are described by a $CFT_{d+2}$ whose central charge $c$ is very large. Second, they are described by a d+2 dimensional HV QFT which has a holographic gravitational theory that is a HV geometry at zero temperature. For both cases, we calculate the Page curve of the Hawking radiation as well as the Page time $t_{\rm Page}$. For the first case, $S_{\rm R}$ grows linearly with time before the Page time and saturates after this time. Moreover, $t_{\rm Page}$ is proportional to $\frac{2 S_{\rm th}}{c T}$, where $S_{\rm th}$ and $T$ are the thermal entropy and temperature of the black brane. For the second case, when the hyperscaling violation exponent $\theta_m$ of the matter fields is zero, the results are very similar to those for the first case. However, when $\theta_m \neq 0$, the entropy of Hawking radiation grows exponentially before $t_{\rm Page}$ and saturates after this time. Furthermore, the Page time is proportional to $\log \left( \frac{1}{G_{\rm N,r}} \right) $, where $G_{\rm N,r}$ is the renormalized Newton's constant. It was also observed that for both cases, $t_{\rm Page}$ is a decreasing and an increasing function of the dynamical exponent $z$ and hyperscaling violation exponent $\theta$ of the black brane geometry, respectively. Moreover, for the second case, $t_{\rm Page}$ is independent of $z_m$, and for $\theta_m \neq 0$, it is a decreasing function of $\theta_m$.