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

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 $\theta_m$.