Ignorance is Cheap: From Black Hole Entropy To Energy-Minimizing States In QFT

TL;DR

The paper extends Engelhardt–Wall’s classical coarse-graining of black hole entropy to the semiclassical regime by introducing quantum marginally trapped/minimar surfaces and a generalized entropy framework. It formulates a coarse-graining that maximizes generalized entropy behind a quantum minimar surface while keeping the exterior state fixed, relying on the quantum focusing conjecture and a quantum maximin principle; in the nongravitational limit, it connects to Wall’s ant conjecture about the minimum energy of half-space completions, with Ceyhan–Faulkner providing explicit states that realize the required properties. The results establish a concrete link between gravitational coarse-graining and quantum energy minimization, supported by AdS/CFT intuition and an explicit QFT construction, and they discuss boundary duals and semiclassical stretched states as broader implications. Overall, the work offers a unified perspective on how ignorance (coarse-graining) can be energetically costly or beneficial, tying entropy maximization to minimal interior energy in a precise semiclassical and QFT context.

Abstract

Behind certain marginally trapped surfaces one can construct a geometry containing an extremal surface of equal, but not larger area. This construction underlies the Engelhardt-Wall proposal for explaining Bekenstein-Hawking entropy as a coarse-grained entropy. The construction can be proven to exist classically but fails if the Null Energy Condition is violated. Here we extend the coarse-graining construction to semiclassical gravity. Its validity is conjectural, but we are able to extract an interesting nongravitational limit. Our proposal implies Wall's ant conjecture on the minimum energy of a completion of a quantum field theory state on a half-space. It further constrains the properties of the minimum energy state; for example, the minimum completion energy must be localized as a shock at the cut. We verify that the predicted properties hold in a recent explicit construction of Ceyhan and Faulkner, which proves our conjecture in the nongravitational limit.

Figures (7)

• Figure 1: Penrose diagram of a black hole formed from collapse in Anti-de Sitter space, showing a minimar surface $\sigma$ and its outer wedge $\mathcal{O}_{W}[\sigma]$ with Cauchy surface $\Sigma$.
• Figure 2: Coarse-graining behind a Killing horizon. Any cut $V_0$ can be viewed as a quantum marginally trapped surface in the limit as $G\to 0$. The state $\rho_{>V_0}$ on the Cauchy surface $\Sigma$ of the outer wedge is held fixed. The coarse-grained geometry is the original geometry. The stationary null surface $N_k^-$ is the past of $V_0$ on the Killing horizon. The coarse-grained quantum state demanded by our proposal lives on $N_k^- \cup \sigma\cup\Sigma$. We identify the properties the state must have, and we show that the Ceyhan and Faulkner "ant states" satisfy these.
• Figure 3: The spacetime region associated to the interval $V < v < V_0$ on the null surface for which all observables in the algebra should register vacuum values in the coarse-graining state.
• Figure 4: We would like to fix the data on $N_{-l}(t_{i})$ (green thick line), while coarse-graining in the interior of the QMT surface. Simple data in the boundary region $t>t_{i}$ fixes the causal wedge $C[t_{i}]$ and thus fixes only a portion of $N_{-l}(t_i)$. In order to fix all of $N_{-l}(t_{i})$ one must allow for sources that remove the excitations (red arrows) that enter the black hole after $\sigma$; this can cause the causal wedge to grow to include $N_{-l}$. In the coarse-graining set $\mathcal{F}$, the simple data must agree for all allowed sources.
• Figure 5: The left stretch is a classical analogue of the CF flow that generalizes it to nontrivial geometries. Left: The null surface $N_k$ split by the marginally trapped surface $\sigma$. Middle: The affine parameter is rescaled on $N_k^-$ but held fixed on $N^+_k$. This is the same initial data in nonaffine parametrization. Right: The two pieces are glued back together, treating the new parameter as affine. This yields inequivalent initial data.