Coarse Graining Holographic Black Holes

TL;DR

<3-5 sentence high-level summary>We formulate the outer entropy as a holographic coarse-grained entropy obtained by maximizing the boundary von Neumann entropy while keeping the exterior bulk data fixed. For surfaces that are minimar, we prove a sharp equality: $S^{( ext{outer})}[ ext{μ}] = rac{ ext{Area}[ ext{μ}]}{4G ext{ħ}}$, and we construct the interior that saturates this bound, together with the required junction conditions. The paper also defines a boundary dual, the simple entropy, and shows it matches the outer entropy to all orders in perturbation theory around equilibrium, thus linking bulk coarse-graining to a boundary thermodynamic-like law. It further analyzes the outer entropy for extremal and non-minimar surfaces, establishing bounds for untrapped and trapped cases, and provides a robust framework for understanding holographic second laws on screens foliated by minimar surfaces. Prospects for semiclassical extensions and broader applications to non-AdS settings are discussed.

Abstract

We expand our recent work on the outer entropy, a holographic coarse-grained entropy defined by maximizing the boundary entropy while fixing the classical bulk data outside some surface. When the surface is marginally trapped and satisfies certain "minimar" conditions, we prove that the outer entropy is exactly equal to a quarter the area (while for other classes of surfaces, the area gives an upper or lower bound). We explicitly construct the entropy-maximizing interior of a minimar surface, and show that it satisfies the appropriate junction conditions. This provides a statistical explanation for the area-increase law for spacelike holographic screens foliated by minimar surfaces. Our construction also provides an interpretation of the area for a class of non-minimal extremal surfaces. On the boundary side, we define an increasing simple entropy by maximizing the entropy subject to a set of "simple experiments" performed after some time. We show (to all orders in perturbation theory around equilibrium) that the simple entropy is the boundary dual to our bulk construction.

Figures (15)

• Figure 1: A conformal diagram of an asymptotically AdS black hole formed from collapse. The figure shows a compact, spacelike, codimension-2 surface $\sigma$ (purple dot), and the decomposition of the spacetime into the future of $\sigma$, $I^{+}[\sigma]$, the past of $\sigma$, $I^{-}[\sigma]$, the exterior of $\sigma$, $O_{W}[\sigma]$, and the interior of $\sigma$, $I_{W}[\sigma]$.
• Figure 2: A cartoon showing the different ways of denoting the orthogonal null vector fields and hypersurfaces generated from a surface $\sigma$. The left panel figure shows the vectors $\ell^{a}$ and $k^{a}$ at a point on $\sigma$ in $D=3$ dimensions. In the center panel, the null congruences $N_{\ell}$ and $N_{k}$ are shown with $(D-2)$ spacetime dimensions suppressed. The final panel figure shows $N_{\ell}$ (orange) and $N_{k}$ (purple) in $D=3$.
• Figure 3: A table summarizing the classification of surfaces by the expansion of null congruences fired from them. Our conventions are such that whenever one expansion vanishes, we take $k^{a}$ to be the corresponding generating null vector, and for untrapped surfaces $\theta_{(k)}>0>\theta_{(\ell)}$.
• Figure 4: A conformal diagram of maximally-extended Schwarzschild-AdS, which contains spherically-symmetric surfaces of all types under the classification of Table \ref{['fig:table']}. There are trapped surfaces in the black hole region, anti-trapped surfaces in the white hole region, and untrapped surfaces in each asymptotic region. As with all stationary black holes, the future event horizons are foliated by marginally trapped surfaces; the past event horizons are foliated by marginally anti-trapped surfaces. The bifurcation surface (black dot) is extremal.
• Figure 5: Decomposition of the outer and inner wedges of $\sigma$. $\Sigma$ is a Cauchy slice of the full spacetime, and In$_{\Sigma}[\sigma]$, Out$_{\Sigma}[\sigma]$ are the components of $\Sigma$ as split by $\sigma$.

Theorems & Definitions (3)

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