[Paper Review] Entropy of Hawking Radiation for Two-Sided Hyperscaling Violating Black Branes

Farzad Omidi|arXiv (Cornell University)|Dec 11, 2021

Black Holes and Theoretical PhysicsPhysics and Astronomy122 references48 citations

TL;DR

This paper computes the von Neumann entropy of Hawking radiation for two-sided hyperscaling violating (HV) black branes coupled to two Minkowski thermal baths, using the island rule in AdS/CFT. It derives the Page curve and Page time for two matter scenarios: a large-c CFT and a d+2-dimensional HV QFT with a holographic dual. The key result is that for θm ≠ 0, the entropy grows exponentially before saturation, and the Page time scales logarithmically with the renormalized Newton constant, contrasting with linear growth in standard cases.

ABSTRACT

In this paper, we study the von Neumann entropy of Hawking radiation $S_{ m R}$ 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 $t_{ m Page}$. For the first case, $S_{ m R}$ grows linearly with time before the Page time and saturates after this time. Moreover, $t_{ m Page}$ is proportional to $\frac{2 S_{ m th}}{c T}$, where $S_{ m th}$ and $T$ are the thermal entropy and temperature of the black brane. For the second case, when the hyperscaling violation exponent $ heta_m$ of the matter fields is zero, the results are very similar to those for the first case. However, when $ heta_m eq 0$, the entropy of Hawking radiation grows exponentially before $t_{ m Page}$ and saturates after this time. Furthermore, the Page time is proportional to $\log \left( \frac{1}{G_{ m N,r}} ight) $, where $G_{ m N,r}$ is the renormalized Newton's constant. It was also observed that for both cases, $t_{ m Page}$ is a decreasing and an increasing function of the dynamical exponent $z$ and hyperscaling violation exponent $ heta$ of the black brane geometry, respectively. Moreover, for the second case, $t_{ m Page}$ is independent of $z_m$, and for $ heta_m eq 0$, it is a decreasing function of $ heta_m$.

Motivation & Objective

• To resolve the black hole information paradox in two-sided hyperscaling violating black branes by computing the fine-grained entropy of Hawking radiation.
• To investigate how the Page curve and Page time depend on the dynamical exponent z and hyperscaling violation exponent θ of the black brane geometry.
• To compare two matter field scenarios: a large-central-charge CFT and a d+2-dimensional HV QFT with a holographic dual.
• To determine whether the island mechanism enforces unitarity by saturating the entropy at twice the Bekenstein-Hawking value.

Proposed method

• Uses the island rule in AdS/CFT to compute the von Neumann entropy of Hawking radiation via extremization of generalized entropy Sgen = Area(∂I)/4GN + Smatter(R∪I).
• Applies the holographic entanglement entropy prescription from ref. [97] to compute Smatter for disjoint intervals [b−,a−] ∪ [a+,b+] with and without an island.
• Calculates the Page curve by comparing the generalized entropy with and without islands, identifying the transition at the Page time.
• For the HV QFT case, derives the entanglement entropy using the RT formula in a zero-temperature HV geometry dual.
• Considers two cases: θm = 0 (reducing to CFT-like behavior) and θm ≠ 0 (introducing exponential growth), with separate analysis of z, θ, zm, and θm dependence.
• Uses Kruskal coordinates and computes distances in the black brane geometry to analyze the island location and entropy evolution.

Experimental results

Research questions

• RQ1 How does the Page curve of Hawking radiation differ for a two-sided HV black brane when matter is described by a CFT versus a d+2-dimensional HV QFT with a holographic dual?
• RQ2 What is the functional dependence of the Page time on the dynamical exponent z and hyperscaling violation exponent θ of the black brane geometry?
• RQ3 How does the presence of a non-zero hyperscaling violation exponent θm in the matter QFT affect the growth rate of the radiation entropy before the Page time?
• RQ4 Is the island mechanism sufficient to enforce unitarity, as evidenced by saturation of the entropy at 2SBH, and how does this depend on the matter theory?
• RQ5 What is the scaling of the Page time with the renormalized Newton constant GN,r in the HV QFT case?

Key findings

• For matter described by a large-c CFT, the entropy of Hawking radiation grows linearly before the Page time and saturates at 2SBH, with tPage ∝ 2Sth / (cT).
• When the matter is a d+2-dimensional HV QFT with θm = 0, the Page curve and tPage are qualitatively identical to the CFT case, with tPage decreasing in z and increasing in θ.
• For θm ≠ 0, the entropy grows exponentially before the Page time, and the Page time scales as tPage ∝ log(1/GN,r), indicating a fundamentally different growth dynamics.
• The Page time is a decreasing function of the black brane's dynamical exponent z and an increasing function of its hyperscaling violation exponent θ.
• For θm ≠ 0, the Page time is independent of zm and decreases with increasing θm, highlighting the dominant role of θm in late-time entropy dynamics.
• The results for z=1 and θ=0 reduce to the planar AdS-Schwarzschild black hole case, and the findings are consistent with existing results in critical gravity when higher-derivative couplings vanish.