Page Curves, Islands, and Modern Lessons

The main idea

The previous page explained why AdS/CFT makes the black-hole information problem sharper: the exact boundary theory is unitary, but the semiclassical Hawking calculation seems to predict that the entropy of the radiation keeps increasing. The modern island story gives a remarkably concrete answer to the entropy question.

For a radiation region $R$ coupled to a gravitating system, the fine-grained entropy is not computed only by the ordinary entropy of quantum fields in the exterior. Instead, in semiclassical gravity one must extremize a generalized entropy over possible interior regions $I$:

$$\boxed{ S(R) = \min_{I}\;\operatorname*{ext}_{I} \left[ \frac{\operatorname{Area}(\partial I)}{4G_N} + S_{\mathrm{bulk}}(R\cup I) \right]. }$$

The region $I$ is called an island. Its boundary $\partial I$ is a quantum extremal surface. At early times, the dominant saddle is usually the empty island, $I=\varnothing$, and the entropy agrees with the Hawking result. At late times, a nonempty island near or behind the horizon can dominate. Then the radiation entropy stops following the naive Hawking growth and instead follows the Page curve required by unitarity.

The deep lesson is not that Hawking radiation becomes nonthermal in an obvious local way. The lesson is subtler:

$$\boxed{ \text{after the Page time, part of the black-hole interior is counted in the entanglement wedge of the radiation.} }$$

In other words, the late-time interior is not an independent Hilbert-space factor that can be freely traced over while also treating the radiation as a complete external subsystem. This is exactly the kind of statement one expects from the quantum-error-correcting structure of holography.

The island mechanism. The fine-grained radiation entropy is the lower envelope of competing quantum extremal surface saddles. Before the Page time, $I=\varnothing$ dominates and the entropy follows Hawking’s semiclassical growth. After the Page time, a nonempty island $I$ dominates, and the interior region $I$ belongs to the entanglement wedge of the radiation region $R$. Replica wormholes give a gravitational path-integral derivation of the island saddle.

This page explains the island formula, its derivation from replica wormholes, and the precise lessons one should take away from the modern Page-curve calculations.

Page’s question

Consider a black hole formed from collapse in a pure state and evaporating into radiation. Let $R(t)$ denote the radiation emitted up to time $t$, and let $B(t)$ denote the remaining black-hole system. If the total system is closed and unitary, then the state on $R(t)\cup B(t)$ remains pure:

$$S(R)=S(B).$$

At early times, the radiation Hilbert space is much smaller than the black-hole Hilbert space. A typical pure state on $R\cup B$ has radiation entropy approximately equal to the maximum possible entropy of $R$:

$$S(R)\approx \log \dim \mathcal H_R.$$

At late times, the remaining black hole is the smaller subsystem, so the same typicality logic gives

$$S(R)\approx \log \dim \mathcal H_B.$$

Thus a unitary evaporation process predicts the schematic Page curve

$$S(R)\sim \min\left(S_{\mathrm{rad}}^{\mathrm{coarse}}(t),\, S_{\mathrm{BH}}(t)\right).$$

The Page time is the time at which the coarse-grained entropy of the emitted radiation becomes comparable to the Bekenstein-Hawking entropy of the remaining black hole:

$$S_{\mathrm{rad}}^{\mathrm{coarse}}(t_{\mathrm{Page}}) \sim S_{\mathrm{BH}}(t_{\mathrm{Page}}).$$

Hawking’s leading semiclassical calculation instead gives an entropy that keeps growing as more thermal radiation is emitted. In the simplest cartoon,

$$S_{\mathrm{Hawking}}(R) \sim \int_0^t dt'\, \frac{dS_{\mathrm{rad}}^{\mathrm{coarse}}}{dt'}.$$

That result is natural if one traces over interior partners of Hawking quanta and assumes that those partners remain independent of the early radiation. The island formula changes the entropy calculation precisely when that assumption becomes inconsistent with the finite black-hole entropy.

The setup: an evaporating AdS black hole with a bath

A technical nuisance is that a large AdS black hole with reflecting boundary conditions does not evaporate away in the same way as an asymptotically flat black hole. Radiation reaches the AdS boundary and returns. To model evaporation while retaining holographic control, one often couples the boundary CFT to a nongravitating auxiliary bath.

Schematically,

$$\mathcal H_{\mathrm{total}} = \mathcal H_{\mathrm{CFT}}\otimes \mathcal H_{\mathrm{bath}},$$

with an interaction that allows energy to leak from the CFT into the bath. In the bulk, this corresponds to an AdS gravitational region connected to an exterior nongravitating reservoir. The radiation subsystem $R$ is chosen as a region in the bath.

The exact statement of unitarity is for the combined CFT-plus-bath system:

$$\rho_{\mathrm{total}}(t) = U(t)\rho_{\mathrm{total}}(0)U^\dagger(t).$$

The entropy $S(R)$ of the bath radiation is not conserved, because $R$ is only a subsystem. It can grow and later decrease, while the total system remains pure.

This setup is not a mere trick. It separates three notions that are often blurred:

Quantity	Meaning	Holographic description
$S_{\mathrm{BH}}$	coarse-grained black-hole entropy	horizon area over $4G_N$
$S_{\mathrm{bulk}}$	semiclassical matter entropy	QFT entropy on a fixed or slowly varying geometry
$S(R)$	fine-grained entropy of radiation	quantum extremal surface prescription
$S_{\mathrm{rad}}^{\mathrm{coarse}}$	thermodynamic entropy carried by outgoing radiation	Hawking flux entropy

The information paradox arises when one uses the second line to compute the third line beyond its regime of validity.

Generalized entropy and quantum extremal surfaces

The classical RT/HRT formula says that, for a boundary region $A$ in a holographic CFT,

$$S(A) = \frac{\operatorname{Area}(\gamma_A)}{4G_N}$$

at leading order in large $N$. Quantum corrections replace the area alone by the generalized entropy

$$S_{\mathrm{gen}}(\Sigma) = \frac{\operatorname{Area}(\Sigma)}{4G_N} +S_{\mathrm{bulk}}(\mathcal R_\Sigma) +S_{\mathrm{ct}}(\Sigma),$$

where $\mathcal R_\Sigma$ is the bulk region bounded by the boundary subsystem and the candidate surface $\Sigma$. The counterterm contribution $S_{\mathrm{ct}}$ is often suppressed in notation, but conceptually it is important: the area term and the bulk entanglement entropy are separately UV divergent, while the properly renormalized generalized entropy is finite.

A quantum extremal surface is a codimension-two surface satisfying

$$\delta S_{\mathrm{gen}}=0.$$

The quantum-corrected entropy prescription is then

$$S(A) = \min_{\Sigma}\;\operatorname*{ext}_{\Sigma}\,S_{\mathrm{gen}}(\Sigma).$$

For ordinary boundary regions in AdS/CFT, this is the quantum RT/HRT prescription. For Hawking radiation in a nongravitating bath, the new possibility is that the bulk region whose entropy is computed may include a disconnected gravitational region $I$:

$$S(R) = \min_I\;\operatorname*{ext}_I\,S_{\mathrm{gen}}(R\cup I).$$

The notation $R\cup I$ is crucial. The matter entropy is not $S_{\mathrm{bulk}}(R)$ but $S_{\mathrm{bulk}}(R\cup I)$. The island degrees of freedom are counted as part of the radiation subsystem for the purpose of computing the fine-grained entropy.

This sounds bizarre only if one expects a fixed microscopic tensor factorization between the black-hole interior and the radiation. Holography teaches that such a factorization is not fundamental in quantum gravity.

Empty island versus nonempty island

The island formula is most transparent when viewed as a competition between saddles.

The empty-island saddle

For $I=\varnothing$,

$$S_{\mathrm{gen}}(R) = S_{\mathrm{bulk}}(R).$$

This is the Hawking calculation of the entropy of the outgoing radiation. It grows as more Hawking quanta are emitted. In a simple approximation,

$$S_{\mathrm{empty}}(t) \approx S_{\mathrm{rad}}^{\mathrm{coarse}}(t).$$

At early times this is the correct dominant answer. The radiation is a small subsystem of a much larger remaining black hole, and its entropy should grow.

The island saddle

At late times, there can be a nonempty island $I$ whose boundary lies near the horizon. Its generalized entropy has the schematic form

$$S_{\mathrm{island}}(t) \approx \frac{\operatorname{Area}(\partial I)}{4G_N} + S_{\mathrm{bulk}}(R\cup I).$$

Because $I$ includes the interior partners of much of the outgoing Hawking radiation, the bulk entropy $S_{\mathrm{bulk}}(R\cup I)$ does not keep growing in the same way as $S_{\mathrm{bulk}}(R)$. The dominant time dependence is often approximately the decreasing black-hole area:

$$S_{\mathrm{island}}(t) \sim S_{\mathrm{BH}}(t)+O(N^0).$$

The physical entropy is the smaller of the competing extrema:

$$S(R) = \min\left(S_{\mathrm{empty}},S_{\mathrm{island}},\ldots\right).$$

Thus

$$S(R) \approx \begin{cases} S_{\mathrm{rad}}^{\mathrm{coarse}}(t), & t<t_{\mathrm{Page}},\\ S_{\mathrm{BH}}(t), & t>t_{\mathrm{Page}}, \end{cases}$$

up to model-dependent constants and subleading corrections. This is the Page curve.

A useful toy model of the saddle competition

A deliberately simple model captures the logic. Suppose the no-island entropy grows linearly,

$$S_{\mathrm{empty}}(t)=\alpha t,$$

while the island entropy follows the decreasing black-hole entropy,

$$S_{\mathrm{island}}(t)=S_0-\beta t.$$

The island formula gives

$$S(R)=\min(\alpha t, S_0-\beta t).$$

The Page time is

$$t_{\mathrm{Page}}=\frac{S_0}{\alpha+\beta}.$$

Of course real black-hole evaporation is not exactly linear. The point of the toy model is only the saddle structure. A new extremum does not need to dominate immediately. It can be present but subdominant, then take over when the naive Hawking entropy exceeds the generalized entropy of the island saddle.

This is similar in spirit to a Hawking-Page transition: two semiclassical saddles can both exist, but the physical answer is determined by the dominant one. Here the saddles are not thermal geometries but entropy saddles in a replica computation.

What is an island?

An island is not a literal island of matter floating in the radiation bath. It is a region of the gravitating spacetime that is included in the entanglement wedge of the nongravitating radiation region.

More precisely, for a radiation region $R$, a candidate island $I$ contributes to the generalized entropy

$$S_{\mathrm{gen}}(R\cup I) = \frac{\operatorname{Area}(\partial I)}{4G_N} +S_{\mathrm{bulk}}(R\cup I).$$

The island $I$ is selected by extremizing and minimizing this quantity. Once the island saddle dominates, the entanglement wedge of $R$ contains $I$. Operators in $I$, within the appropriate code subspace, can then be reconstructed from the radiation degrees of freedom.

This is the entropy version of a strong reconstruction statement:

$$I\subset \mathcal E(R) \quad\Longrightarrow\quad \text{operators in } I \text{ are encoded in } R.$$

The island therefore resolves a counting problem. The interior Hawking partners that were naively traced out are not independent of the radiation in the fine-grained entropy calculation. They are included in the radiation wedge.

Replica trick and replica wormholes

The island formula can be motivated from quantum extremal surfaces, but its gravitational origin becomes sharper through the replica trick. For an ordinary quantum system,

$$S(R) = -\operatorname{Tr}\rho_R\log\rho_R =\lim_{n\to 1}\frac{1}{1-n}\log \operatorname{Tr}\rho_R^n.$$

In nongravitational QFT, $\operatorname{Tr}\rho_R^n$ is computed by gluing $n$ replicas cyclically along $R$. In a theory with dynamical gravity, one must also sum over geometries compatible with the replica boundary conditions.

The traditional Hawking calculation effectively uses disconnected gravitational replicas. Each copy has its own independent black-hole interior. This gives the empty-island answer.

Replica wormholes are additional saddles in which the $n$ gravitational replicas are connected through the interior. These saddles are not visible if one insists on the disconnected topology from the start. When analytically continued to $n\to 1$, the fixed locus of the replica symmetry becomes the quantum extremal surface bounding the island.

Schematic path integral:

$$\operatorname{Tr}\rho_R^n = Z_n^{\mathrm{disc}} + Z_n^{\mathrm{wormhole}}+\cdots.$$

Then

$$S(R) =\lim_{n\to 1}\frac{1}{1-n}\log \left(Z_n^{\mathrm{disc}}+Z_n^{\mathrm{wormhole}}+\cdots\right).$$

At early times the disconnected saddle dominates. At late times the replica-wormhole saddle dominates. This is the gravitational path-integral origin of the Page transition.

Two points are worth emphasizing.

First, replica wormholes are saddles of an entropy calculation, not ordinary Lorentzian wormholes through which a signal travels. They do not make the black hole traversable.

Second, the island result is semiclassical but nonperturbative in the sense relevant to entropy. The leading area term is $O(1/G_N)$, and the difference between competing saddles is also $O(1/G_N)$. Selecting the correct saddle is not captured by perturbing the Hawking saddle order by order in small local corrections.

A minimal derivation of the island rule from replicas

The derivation is conceptually parallel to the Lewkowycz-Maldacena derivation of RT, upgraded to include bulk quantum fields and nongravitating radiation regions.

Start with the replicated entropy computation. If the dominant $n$-replica gravitational saddle has a replica symmetry, one can quotient by that symmetry. The quotient geometry contains a codimension-two defect with opening angle

$$\Delta \phi = \frac{2\pi}{n}.$$

Equivalently, in the quotient one has a cosmic brane with tension

$$T_n=\frac{n-1}{4nG_N}.$$

Taking $n\to 1$ sends the brane tension to zero but keeps its first variation finite. The condition that the saddle be smooth under variations of the brane location gives extremality of the generalized entropy:

$$\delta\left[ \frac{\operatorname{Area}}{4G_N}+S_{\mathrm{bulk}} \right]=0.$$

For radiation regions coupled to gravity, the replica boundary conditions allow saddles whose fixed locus is not anchored directly to the radiation region. The corresponding region enclosed by the fixed locus is the island.

Thus the island rule is not an independent rule pasted onto Hawking’s calculation. It is the quantum extremal surface prescription applied to the correct gravitational replica problem.

Entanglement wedge interpretation

Before the Page time, the entanglement wedge of the radiation region $R$ is essentially just the radiation region in the bath. The black-hole interior is not reconstructable from $R$.

After the Page time, the island saddle dominates and

$$\mathcal E(R)=R\cup I$$

in the semiclassical description. This means that some interior operators are encoded in the radiation.

This statement resolves the monogamy tension in a very holographic way. The late outgoing Hawking mode $b$ appears, in local semiclassical QFT, to be entangled with an interior partner $a$. After the Page time, unitarity says that $b$ must also be correlated with the early radiation $R$. If $a$ and $R$ were independent systems, this would violate entanglement monogamy.

The island story says that the interior partner $a$ is not independent of $R$. It lies in the entanglement wedge of $R$ and can be reconstructed from $R$ in the appropriate code subspace:

$$a \in I\subset \mathcal E(R).$$

So the same logical information can have two semiclassical descriptions: as an interior partner mode in the bulk effective theory and as encoded information in the radiation. This is not ordinary local quantum field theory on a fixed background; it is gravitational quantum error correction.

Islands and the Page curve in one equation

A compact way to summarize the result is

$$S(R) = \min\left\{ S_{\mathrm{bulk}}(R),\, \frac{\operatorname{Area}(\partial I_*)}{4G_N} +S_{\mathrm{bulk}}(R\cup I_*) \right\},$$

where $I_*$ is the nonempty island solving

$$\delta_{I}\left[ \frac{\operatorname{Area}(\partial I)}{4G_N} +S_{\mathrm{bulk}}(R\cup I) \right]_{I=I_*}=0.$$

At early times,

$$S_{\mathrm{bulk}}(R) < \frac{\operatorname{Area}(\partial I_*)}{4G_N} +S_{\mathrm{bulk}}(R\cup I_*),$$

so the empty island dominates. At late times,

$$S_{\mathrm{bulk}}(R) > \frac{\operatorname{Area}(\partial I_*)}{4G_N} +S_{\mathrm{bulk}}(R\cup I_*),$$

so the island dominates.

The transition is a change in the dominant entropy saddle. It is not necessarily a dramatic local event at the horizon. A freely falling observer need not see anything special at the Page time in the semiclassical approximation.

What the island calculations establish

The modern Page-curve calculations are powerful because they show that semiclassical gravitational entropy calculations can reproduce unitary behavior when the entropy is computed correctly. They establish several lessons.

1. The Hawking calculation used the wrong fine-grained entropy after the Page time

The Hawking calculation correctly computes local radiation production and coarse-grained entropy flux. It does not correctly compute the fine-grained von Neumann entropy of the radiation after the Page time.

The difference is not a small correction to the local stress tensor. It is a different saddle in the entropy computation.

2. The Bekenstein-Hawking area is part of a fine-grained entropy formula

The area term is not merely thermodynamic bookkeeping. In the generalized entropy,

$$S_{\mathrm{gen}}=\frac{A}{4G_N}+S_{\mathrm{bulk}},$$

the area term competes directly with matter entanglement. It controls which regions are included in the entanglement wedge.

3. The interior can be encoded in the radiation

After the Page time, the radiation entanglement wedge contains an island. This gives a precise semiclassical version of the slogan that the black-hole interior is encoded in the Hawking radiation.

The statement is not that every observer can easily decode the interior. Reconstruction may be extraordinarily complex, code-subspace dependent, and sensitive to nonperturbative data.

4. Entropy can be corrected without large local violations of effective field theory

The Page curve does not require order-one corrections to local Hawking emission at the horizon. Instead, it requires using the correct nonlocal gravitational entropy prescription.

This is why the island story is compatible with an approximately smooth horizon in suitable states, at least at the level of entropy.

5. The information problem is not fully solved by entropy alone

Computing the Page curve is a huge achievement, but it does not automatically provide a microscopic time-dependent decoding map, a full nonperturbative bulk Hilbert space, or a complete description of the endpoint of evaporation in realistic four-dimensional quantum gravity.

The entropy calculation tells us that the information is not lost. It does not by itself hand us a practical decoding algorithm.

What remains subtle

The island formula is now a central tool, but several conceptual issues remain active research topics.

Factorization and ensemble averaging

In simple models such as Jackiw-Teitelboim gravity, wormhole contributions often resemble ensemble-averaged behavior. This raises a factorization puzzle: a single boundary theory should have factorized products of independent partition functions, while gravitational wormholes can connect apparently independent boundaries.

In AdS/CFT with a specific microscopic CFT, exact factorization must be restored by nonperturbative effects or by a more refined understanding of the bulk path integral. Replica wormholes are therefore both a solution to the entropy problem and a diagnostic of what semiclassical gravity does not fully define by itself.

State dependence and code subspaces

Entanglement-wedge reconstruction is normally a statement within a code subspace. After the Page time, interior reconstruction from radiation may depend on which family of black-hole microstates is being considered. This is not necessarily pathological, but it means one must be careful about asking for a single interior operator acting correctly on all possible black-hole states.

The endpoint of evaporation

The island formula captures the entropy curve through the Page transition and often through late semiclassical evaporation. The final Planckian endpoint still requires physics beyond the semiclassical approximation.

Operational decoding

The island formula says that the radiation contains the relevant information. It does not mean that the information is easy to extract. Hayden-Preskill-style decoding can be exponentially complex in generic systems. The existence of a reconstruction map is not the same as practical access.

Realistic black holes

Most sharp island calculations use lower-dimensional gravity, baths, branes, special couplings, or holographic matter sectors. The general principles are expected to be broader, but realistic four-dimensional evaporating black holes remain technically much harder.

Relation to AdS/CFT

AdS/CFT enters the island story in two complementary ways.

First, it provides a nonperturbative unitary definition of quantum gravity in asymptotically AdS settings. Thus the exact answer must be compatible with a Page curve.

Second, it supplies the entanglement-wedge and quantum-error-correction logic that makes islands natural. In ordinary holographic subregion duality,

$$S(A)=\min_{\gamma_A}\left[ \frac{\operatorname{Area}(\gamma_A)}{4G_N}+S_{\mathrm{bulk}}(\mathcal E_A)\right].$$

The island formula is the same logic applied when the boundary subsystem includes a nongravitating radiation region coupled to a gravitating system. Once the radiation region is allowed to have a gravitational entanglement wedge, islands are not an exotic extra ingredient; they are quantum extremal surfaces doing their job.

This is why the modern information story fits so well with the previous module on entanglement wedges and quantum error correction.

Common mistakes

Mistake 1: Thinking an island is a new physical object placed behind the horizon

The island is not a new brane, membrane, or matter distribution. It is a region selected by an entropy extremization problem.

Mistake 2: Saying Hawking radiation is locally nonthermal at leading order

The island formula does not require large local deviations from Hawking’s near-horizon calculation. It changes the fine-grained entropy computation by changing the dominant gravitational entropy saddle.

Mistake 3: Ignoring the minimization

Extremizing $S_{\mathrm{gen}}$ is not enough. One must choose the dominant extremum. The Page transition occurs because the minimum switches from the empty island to a nonempty island.

Mistake 4: Confusing coarse-grained entropy with fine-grained entropy

The radiation can carry a large thermodynamic entropy flux while its fine-grained entropy eventually decreases. This is standard in unitary evolution: a subsystem can look locally thermal while being globally purified by subtle correlations.

Mistake 5: Treating the island formula as a complete microscopic solution

The island formula computes entropy. It strongly constrains the microscopic story, but it is not itself a complete microscopic dynamics of black-hole evaporation.

Summary

The island formula gives a modern semiclassical mechanism for the Page curve:

$$S(R)=\min_I\;\operatorname*{ext}_I \left[ \frac{\operatorname{Area}(\partial I)}{4G_N} +S_{\mathrm{bulk}}(R\cup I) \right].$$

At early times, $I=\varnothing$ and the radiation entropy agrees with Hawking’s growing result. At late times, a nonempty island dominates, the radiation entanglement wedge includes part of the black-hole interior, and the entropy follows the decreasing black-hole entropy rather than growing without bound.

Replica wormholes explain how the island saddle arises in the gravitational replica trick. Entanglement-wedge reconstruction explains why the island should be interpreted as interior information encoded in the radiation. The Page curve is therefore not obtained by breaking semiclassical physics locally at the horizon, but by applying the correct fine-grained entropy prescription in quantum gravity.

The modern lesson is precise but humbling:

$$\boxed{ \text{AdS/CFT plus quantum extremal surfaces explain the Page curve; the full microscopic decoding story remains deeper.} }$$

Exercises

Exercise 1: Page’s typicality estimate

Consider a pure state on a bipartite Hilbert space

$$\mathcal H=\mathcal H_R\otimes\mathcal H_B, \qquad \dim\mathcal H_R=m, \qquad \dim\mathcal H_B=n.$$

For a typical highly entangled state, estimate the entropy $S(R)$ in the two regimes $m\ll n$ and $m\gg n$. Explain why this gives a Page-curve shape during evaporation.

Solution

For a typical pure state, the smaller subsystem is nearly maximally mixed. Thus

$$S(R)\approx \begin{cases} \log m, & m\ll n,\\ \log n, & m\gg n. \end{cases}$$

Equivalently,

$$S(R)\approx \min(\log m,\log n).$$

During evaporation, $m$ represents the effective dimension of the emitted radiation subsystem and grows with time, while $n$ represents the effective dimension of the remaining black-hole subsystem and decreases with time. Therefore $S(R)$ initially grows, reaches a maximum near $m\sim n$, and then decreases. This is the Page curve.

Exercise 2: A toy island transition

Suppose the empty-island and island entropies are approximated by

$$S_{\mathrm{empty}}(t)=\alpha t, \qquad S_{\mathrm{island}}(t)=S_0-\beta t,$$

with $\alpha,\beta,S_0>0$. Find the Page time and the physical entropy $S(R)$.

Solution

The physical entropy is the minimum of the two branches:

$$S(R)=\min(\alpha t,S_0-\beta t).$$

The transition occurs when the two branches are equal:

$$\alpha t_{\mathrm{Page}}=S_0-\beta t_{\mathrm{Page}}.$$

Therefore

$$t_{\mathrm{Page}}=\frac{S_0}{\alpha+\beta}.$$

For $t<t_{\mathrm{Page}}$, $S(R)=\alpha t$. For $t>t_{\mathrm{Page}}$, $S(R)=S_0-\beta t$. The result rises and then falls, giving a Page-curve shape.

Exercise 3: Why the island is included in $S_{\mathrm{bulk}}(R\cup I)$

The island formula contains $S_{\mathrm{bulk}}(R\cup I)$ rather than $S_{\mathrm{bulk}}(R)$. Explain why this is essential for obtaining a Page curve.

Solution

If one used only $S_{\mathrm{bulk}}(R)$, one would reproduce the Hawking calculation: the radiation entropy would keep increasing as more locally thermal radiation is emitted. Including $I$ changes the subsystem whose bulk entropy is computed. The island contains interior partners of Hawking modes, so correlations that appeared to be lost behind the horizon are included in the entropy calculation for the radiation wedge.

The area term charges a gravitational cost for including $I$:

$$S_{\mathrm{gen}}(R\cup I)=\frac{\operatorname{Area}(\partial I)}{4G_N}+S_{\mathrm{bulk}}(R\cup I).$$

At early times this cost is too large, so $I=\varnothing$ dominates. At late times the cost is smaller than the growing Hawking entropy, so the island saddle dominates. This produces the Page curve.

Exercise 4: Replica wormholes and the von Neumann entropy

Starting from

$$S(R)=\lim_{n\to 1}\frac{1}{1-n}\log\operatorname{Tr}\rho_R^n,$$

explain in words why the gravitational computation of $\operatorname{Tr}\rho_R^n$ can include saddles not present in an ordinary nongravitational QFT calculation.

Solution

In nongravitational QFT, the background spacetime is fixed. Replicas are glued according to the branch structure of the density matrix, but one does not sum over spacetime geometries. In quantum gravity, the metric itself is dynamical. Therefore the path integral for $\operatorname{Tr}\rho_R^n$ must include all gravitational saddles compatible with the replicated boundary conditions.

Some of these saddles have disconnected gravitational replicas, reproducing the Hawking answer. Others have connected interiors: replica wormholes. When continued to $n\to 1$, these connected saddles produce quantum extremal surfaces and islands. Thus the gravitational entropy calculation has more saddle points than the fixed-background QFT calculation.

Exercise 5: Monogamy and island reconstruction

Let $b$ be a late outgoing Hawking mode, $a$ its interior partner in semiclassical effective field theory, and $R$ the early radiation. Explain how the island story avoids the claim that $b$ must be independently maximally entangled with both $a$ and $R$.

Solution

If $a$ and $R$ were independent Hilbert-space factors, maximal entanglement of $b$ with both would violate monogamy of entanglement. The island story changes the factorization assumption. After the Page time, the interior region containing $a$ can lie in the entanglement wedge of the radiation region $R$:

$$a\in I\subset \mathcal E(R).$$

Therefore $a$ is not independent of $R$ in the exact quantum-gravitational description. It is encoded in $R$, in the sense of entanglement-wedge reconstruction within an appropriate code subspace. The apparent monogamy conflict arises only if one treats the semiclassical interior and the early radiation as independent microscopic systems.

Exercise 6: Extremizing a toy generalized entropy

Consider a toy candidate island labeled by a coordinate $x>0$ with generalized entropy

$$S_{\mathrm{gen}}(x) =\frac{\Phi_0+\Phi_r x}{4G_N} +\frac{c}{6}\log\frac{1}{x}.$$

Find the extremum and determine when it exists at positive $x$.

Solution

Differentiate:

$$\frac{dS_{\mathrm{gen}}}{dx} =\frac{\Phi_r}{4G_N}-\frac{c}{6x}.$$

The extremum satisfies

$$\frac{\Phi_r}{4G_N}=\frac{c}{6x_*},$$

so

$$x_* = \frac{2cG_N}{3\Phi_r}.$$

If $c>0$, $G_N>0$, and $\Phi_r>0$, then $x_*>0$. This toy model illustrates the balance between an increasing geometric area/dilaton term and a decreasing matter-entanglement term. Real island calculations have more complicated coordinate dependence, but the extremization logic is the same.

Further reading

• D. N. Page, Average Entropy of a Subsystem.
• D. N. Page, Information in Black Hole Radiation.
• G. Penington, Entanglement Wedge Reconstruction and the Information Paradox.
• A. Almheiri, N. Engelhardt, D. Marolf, and H. Maxfield, The Entropy of Bulk Quantum Fields and the Entanglement Wedge of an Evaporating Black Hole.
• A. Almheiri, R. Mahajan, J. Maldacena, and Y. Zhao, The Page Curve of Hawking Radiation from Semiclassical Geometry.
• C. Akers, N. Engelhardt, and D. Harlow, Simple Holographic Models of Black Hole Evaporation.
• G. Penington, S. H. Shenker, D. Stanford, and Z. Yang, Replica Wormholes and the Black Hole Interior.
• A. Almheiri, T. Hartman, J. Maldacena, E. Shaghoulian, and A. Tajdini, Replica Wormholes and the Entropy of Hawking Radiation.
• A. Almheiri, T. Hartman, J. Maldacena, E. Shaghoulian, and A. Tajdini, The Entropy of Hawking Radiation.
• N. Engelhardt and A. C. Wall, Quantum Extremal Surfaces: Holographic Entanglement Entropy beyond the Classical Regime.
• T. Faulkner, A. Lewkowycz, and J. Maldacena, Quantum Corrections to Holographic Entanglement Entropy.