Island finder and entropy bound

Identifying an entanglement island requires exquisite control over the entropy of quantum fields, which is available only in toy models. Here we present a set of sufficient conditions that guarantee the existence of an island and place an upper bound on the entropy computed by the island rule. This is enough to derive the main features of the Page curve for an evaporating black hole in any spacetime dimension. Our argument makes use of Wall’s maximin formulation and the quantum focusing conjecture. As a corollary, we derive a novel entropy bound.

Figure 1

Left: evaporating black hole. Right: its Page curve. After the Page time, the semiclassical entropy $S(R)$ of the Hawking radiation in the asymptotic region $R$ exceeds the Bekenstein-Hawking entropy of the black hole, $A_h/4G\hslash$. The “Hawking partners” in the black hole interior purify $R$. (Dashed lines indicate entanglement.) Therefore, adjoining $\widehat{I}$ to $R$ decreases the generalized entropy $S_{\mathrm{gen}}$. However, islands must have stationary $S_{\mathrm{gen}}$. Solving for this condition exceeds present analytic control over the entropy. The island finder presented here sidesteps this obstruction.

Figure 2

Island finder. Suppose that $I^{\prime }\cup R$ is quantum normal (top) or antinormal (bottom). Then the generalized entropy of $I^{\prime }\cup R$ decreases along the dashed lines to ${\tilde{I}}^{\prime }\subset \mathrm{\Sigma }$. An island $I$ with even smaller generalized entropy $S_{\mathrm{gen}}(I\cup R)$ must exist on the maximin Cauchy slice $\mathrm{\Sigma }$. If $S_{\mathrm{gen}}(I^{\prime }\cup R)<S(R)$, the island cannot be empty.

Figure 3

Maximin restricted to the domain of dependence (“wedge”) $D(I_0)$ returns a region $\widehat{I}$ on a maximin slice $\mathrm{\Sigma }$. If $I_0\cup R$ is quantum normal, then $\partial \widehat{I}$ cannot intersect $\partial D(I_0)$ (dashed). Left: If $\widehat{I}\cap \partial I_0\ne \varnothing$, then $S_{\mathrm{gen}}(\widehat{I}(\epsilon )\cup R)<S_{\mathrm{gen}}(\widehat{I}\cup R)$, contradicting the min of maximin. Right: If $\widehat{I}\cap \partial D(I_0)-\partial I_0$ then $S_{\mathrm{gen}}(\widehat{I}(\epsilon )\cup R)>S_{\mathrm{gen}}(\widehat{I}\cup R)$ on the deformed slice $\mathrm{\Sigma }(\epsilon )$ violates the max of maximin.

Figure 4

Evaporating black hole after the Page time. Hawking radiation has accumulated in $R$. As shown on the left, the boundary of the past of $R$ denoted by $r_0(v)$ intersects the stretched horizon (shown in purple) at the sphere $A_s$, which together with the $A_h$ sphere on the event horizons reside at retarded time $v_s$. We consider candidate regions $I^{\prime }$ with boundary $\partial I^{\prime }$ on a causal horizon spacelike to $R$ (gray regions). The generalized second law implies that $S_{\mathrm{gen}}(I^{\prime }\cup R)$ increases under future outward deformations. For future inward deformations, quantum normalcy follows from the (trivial) classical normalcy in the dark gray subregion, which is chosen to keep quantum corrections to the expansion small. The $I^{\prime }$ that minimizes $S_{\mathrm{gen}}(I^{\prime }\cup R)$ subject to these restrictions is shown in pink. Its boundary is located at $\mathrm{\Delta }v=r_s\log c_1$ to the future of $A_h$, as shown on the right. We show that it provides an extremely tight upper bound on the true entropy $S(\mathbf{R})=S_{\mathrm{gen}}(I\cup R)\le S_{\mathrm{gen}}(I^{\prime }\cup R)=A_h/4G\hslash +O(1)$.

Figure 5

Spatially flat radiation-dominated universe with negative cosmological constant, purified by a reference universe (thermal Minkowski space, right). If we choose a large enough reference region $R$ at $t_{\text{Mink}}=0$, then the region $I^{\prime }$ at the turnaround time $t=0$ satisfies our sufficient conditions. Therefore an island $I$ must exist.

Figure 6

A time-symmetric Cauchy slice of a bag-of-gold geometry. The bag has large entropy (gray) compared to the area of its throat, and it purifies the reference system $R$. Then it is easy to find a (classically and quantum) antinormal region $I^{\prime }$ (pink) such that $S_{\mathrm{gen}}(I^{\prime }\cup R)<S(R)$. Hence, there must exist an island $I$. $I$ is expected to be approximately the interior of the classically minimal surface labeled $r_0$.

Figure 7

Collapse of a spherical star that is maximally entangled with a distant reservoir $R$. $I^{\prime }$ (pink) is the interior of a sphere surrounding the star at a time close to the singularity. Hence $\partial I^{\prime }$ has a small area, and condition (i) can be satisfied by an arbitrarily large margin. But $I^{\prime }$ is quantum trapped and so fails to satisfy condition (ii). Indeed, $R$ does not possess any island in this spacetime.

Figure 8

The entropy $S(\mathbf{R})$ of an external or distant reference system $R$ must be less than the generalized entropy of any region $I_c^{\prime }$ that is normal or antinormal (blue).

Figure 9

Not a Schwarzschild black hole. This could be a highly dynamical spacetime far from a stationary black hole solution. $I_c^{\prime }$ is quantum antinormal, so $S(\mathbf{R})\le S_{\mathrm{gen}}(I_c^{\prime })$. In the left example, $R$ is on the conformal boundary. Quantum antinormalcy follows from the GSL (future null congruence), and from the classical area theorem (past null congruence) if quantum corrections are small. In the right example, $R$ is inside the spacetime. Quantum antinormalcy follows by the area theorem and smallness of corrections if $\partial I_c^{\prime }$ stays far enough from the horizon.