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

Abstract

In this paper, we study the von Neumann entropy of Hawking radiation $S_{\mathrm{R}}$ 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 $CF{T}_{d+2}$ 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 $t_{\mathrm{Page}}$. For the first case, $S_{\mathrm{R}}$ grows linearly with time before the Page time and saturates after this time. Moreover, $t_{\mathrm{Page}}$ is proportional to $\frac{2S_{\mathrm{th}}}{cT}$, where $S_{\mathrm{th}}$ 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 ${\theta }_m\ne 0$, the entropy of Hawking radiation grows exponentially before $t_{\mathrm{Page}}$ and saturates after this time. Furthermore, the Page time is proportional to $\log \left(\frac{1}{G_{\mathrm{N},\mathrm{r}}}\right)$, where $G_{\mathrm{N},\mathrm{r}}$ is the renormalized Newton's constant. It was also observed that for both cases, $t_{\mathrm{Page}}$ 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, $t_{\mathrm{Page}}$ is independent of $z_m$, and for ${\theta }_m\ne 0$, it is a decreasing function of ${\theta }_m$.

32 pages, 7 figures, V2: Published version in JHEP

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