Does horizon entropy satisfy a Quantum Null Energy Conjecture?

TL;DR

The paper investigates whether the causal holographic information (CHI) entropy, a holographic coarse-grained entropy, obeys the quantum null energy condition (QNEC) in a 1+1 dimensional holographic CFT dual to AdS$_3$ with conical defects. By constructing and examining CHI on a specific null surface and using conformal frames that simplify the defect geometry, the authors show that CHI satisfies a coarse-grained QNEC, with a delta-function contribution arising from the caustic crossing of the bulk horizon. This result strengthens the interpretation of CHI as a physically meaningful coarse-grained entropy and suggests new bounds on the rate at which bulk horizon generators emerge from caustics. The work also includes a dilaton-gravity 2d generalization of a coarse-grained generalized second law, indicating broader implications for entropy bounds in holographic settings and pointing to future extensions to higher dimensions and non-equilibrium contexts.

Abstract

A modern version of the idea that the area of event horizons gives $4G$ times an entropy is the Hubeny-Rangamani Causal Holographic Information (CHI) proposal for holographic field theories. Given a region $R$ of a holographic QFTs, CHI computes $A/4G$ on a certain cut of an event horizon in the gravitational dual. The result is naturally interpreted as a coarse-grained entropy for the QFT. CHI is known to be finitely greater than the fine-grained Hubeny-Rangamani-Takayanagi (HRT) entropy when $\partial R$ lies on a Killing horizon of the QFT spacetime, and in this context satisfies other non-trivial properties expected of an entropy. Here we present evidence that it also satisfies the quantum null energy condition (QNEC), which bounds the second derivative of the entropy of a quantum field theory on one side of a non-expanding null surface by the flux of stress-energy across the surface. In particular, we show CHI to satisfy the QNEC in 1+1 holographic CFTs when evaluated in states dual to conical defects in AdS$_3$. This surprising result further supports the idea that CHI defines a useful notion of coarse-grained holographic entropy, and suggests unprecedented bounds on the rate at which bulk horizon generators emerge from a caustic. To supplement our motivation, we include an appendix deriving a corresponding coarse-grained generalized second law for 1+1 holographic CFTs perturbatively coupled to dilaton gravity.

Figures (6)

• Figure 1: A conformal diagram of our 1+1 Minkowski space showing $p$, $p^+$, $p^-$ and $i^0$ with the associated past (blue) and future (green) Rindler horizons and a partial Cauchy surface $\Sigma$ extending from $p$ to the right to $i^0$. The domain of dependence $D(\Sigma) = I^-(p^+) \cup I^+(p^-)$ (shaded) is the right Rindler wedge. The dashed vertical lines define a strip. Identifying them conformally maps our Minkowski space into $S^1 \times {\mathbb R}$.
• Figure 2: The conformal boundary in our Global minus Defect (GmD) conformal frame for the case of defect angle $\Gamma < \pi$. The unshaded region is that associated with the boundary in the Poincaré minus Defect (Pmd) frame. Since $\Gamma < \pi$, the points $p^+$, $p^-$, and $i^0$ cannot enter the defect. However, the point $p$ still reaches the defect at $t=- \pi/2 + \Gamma/2 \le 0$. We have placed $p^+$ at $\phi =0$ as this will be the case emphasized in section \ref{['sec:testing']}.
• Figure 3: Left: A projection of the GmD description onto the $(r,\phi)$ surface. Shown are $p^+$ and a general bulk point $q$ in its causal past having $\phi(q) \ge 0$. The causal curve linking them (magenta) may be chosen to avoid the defect (dashed lines). Right: The same projection showing generators of $H^+_{bulk}$; i.e., of the past light cone from $p^+$ with $t(p^+) = \pi \ell/2$, $\phi(p^+)=0$. In empty AdS$_3$, all generators would reach the vertical (dotted) reference line at the same global time $t=0$. So for $\Gamma < \pi$, when traced backward from $p^+$ the caustic forms at GmD time $t=0$ when the the central $\phi=0$ generator (red) reaches the defect. Decreasing $t$ further, the caustic moves out from the center. It finally hits the conformal boundary at $t = - \pi/2 + \Gamma/2$ when the point $p$ reaches the defect. In contrast, for $\Gamma \ge \pi$ (not shown), tracing the generators backwards from $p^+$ one finds the caustic to form first at the boundary as measured by the GmD time $t$.
• Figure 4: Since we are interested only in $\phi _-\in (-\pi /2,0]$, even with the largest interesting defect angle ($\Gamma = \pi$), an extra defect (green) centered opposite $p^-$ does not affect computations with $\phi \le 0$ so long as we also work outside the original defect (black) opposite $p^+$.
• Figure 5: $L_c$ as a function of $\lambda _{PmD}$ for $\phi _+=\pi /4$. From left to right, $\Gamma = \frac{\pi}{8}, \frac{\pi}{4}, \frac{\pi}{2}, \pi$. The right-most two have $\Gamma \ge \pi- 2\phi_+$ and thus transition suddenly at some $\lambda_c$ to finite values from $L_c=-\infty$. Regulated curves may then be defined by replacing $L_c$ with e.g. $\frac{1}{\ell }L_{c,reg} = \ln\left[\cos \left(\frac{\pi}{2}\frac{\lambda - \lambda_c}{\ell \epsilon} \right) \right]$ for $\frac{\lambda_c}{\ell } - \epsilon < \frac{\lambda }{\ell } < \frac{\lambda _c}{\ell }$. Such curves approach the solid lines as $\epsilon \rightarrow 0$ and give a regulated $Q_{PmD}$ equal to $\frac{\pi}{32\epsilon^2G_{bulk}}$ (i.e., satisfying \ref{['2dQNEC']}) for all $\frac{\lambda_c}{\ell } - \epsilon < \frac{\lambda }{\ell } < \frac{\lambda _c}{\ell }$. The dashed lines are sample such curves with $\epsilon=0.05$. Plots for other values of $\phi_+$ are similar.