Competing Saddles and Entropy Transitions

When several admissible quantum extremal surfaces exist, the leading entropy is the smallest of their renormalized generalized entropies. A Page transition is therefore a change of dominant branch, not a singularity of either candidate geometry. This worked example compares an empty-island branch with a nonempty-island branch and isolates which assumptions make the familiar rise-and-fall curve possible.

Required background. Quantum Information supplies fine-grained entropy, purification, and mutual information.

Helpful background. Semiclassical Gravity supplies generalized entropy and evaporation approximations; Quantum Fields in Curved Spacetime supplies the regulated radiation entropy; Black-Hole Geometry and Mechanics supplies horizons and admissible codimension-two surfaces.

Two generalized-entropy branches

Fix a two-dimensional semiclassical black hole coupled transparently to a nongravitating bath. Let $R(t)$ be the radiation collected up to bath time $t$. The same state, regulator, boundary conditions, and subtraction scheme are used for both candidates.

The empty-island candidate has no new gravitational boundary:

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

In the Hawking regime it grows as outgoing quanta become entangled with interior partners. A nonempty island $I(t)$ has boundary $X(t)=\partial I(t)$ and

$$S_I(t) =\frac{\phi(X(t))}{4G_2} +S_{\mathrm{bulk}}(R(t)\cup I(t)) +S_{\mathrm{ct}}(X(t)),$$

where $\phi/(4G_2)$ is the dilaton-gravity replacement for an area term, summed over all island endpoints. In JT gravity take $\phi$ here to include the constant topological contribution: each endpoint contributes $S_0+\phi_{\mathrm{dyn}}/(4G_2)$. Omitting $S_0$ would change dominance even though it leaves fixed-topology stationarity unchanged. Each branch has already been extremized with respect to $X$ when present. The radiation prescription is

$$S(R(t)) =\min\bigl\{S_{\varnothing}(t),S_I(t),\ldots\bigr\}$$

to the stated semiclassical order. Explicit bath models producing these branches are analyzed in Almheiri, Engelhardt, Marolf, and Maxfield 2019, §§2–4 and Penington 2020, §§2–3.

Solving a transparent Page-transition model

Over a time interval short enough that the rates vary slowly, approximate the extremized branches by

$$S_{\varnothing}(t)=S_{R,0}+\gamma_R t, \qquad S_I(t)=S_{\mathrm{BH},0}+s_I-\gamma_B t,$$

with $\gamma_R\gt0$ the coarse Hawking entanglement-production rate, $\gamma_B\gt0$ the rate at which the remaining black-hole entropy decreases, and $s_I$ a finite offset for the specified state and renormalization prescription. The complete branch is cutoff independent; $s_I$ need not be independent of the prepared state. All entropies are dimensionless, so the $\gamma$ values have units of inverse time.

Equating the branches gives

$$t_{\mathrm P} = \frac{S_{\mathrm{BH},0}+s_I-S_{R,0}} {\gamma_R+\gamma_B}.$$

A physical crossing in this interval requires a positive numerator and that $t_{\mathrm P}$ lie before the approximation fails. Before $t_{\mathrm P}$, the empty island dominates and the leading entropy rises. Afterwards the island dominates and the leading entropy falls. The slope jumps from $\gamma_R$ to $-\gamma_B$:

$$\frac{dS}{dt} = \begin{cases} \gamma_R, & t\lt t_{\mathrm P},\\ -\gamma_B, & t\gt t_{\mathrm P}. \end{cases}$$

The kink comes from taking a minimum of smooth saddle values. Neither branch becomes geometrically singular at the crossing. The standard island review derives this minimization structure and its Page-curve interpretation in Almheiri et al. 2021, §§6–8.

What changes dominance

The transition is sensitive to physical inputs, not just algebra. Changing the state preparation changes $S_{R,0}$ and $s_I$. Partially reflecting the bath interface changes the flux and therefore both rates. Adding another admissible QES can preempt the two-branch crossing. At fixed geometric scale, one-loop terms are ordinarily $O(G^0)$ relative to a leading $O(1/G)$ geometric term; subsequent loops have higher powers of $G$. The retained matter entropy must not be counted again as an omitted correction. Any remaining correction can appreciably move a crossing when it is comparable to the small difference between the branches.

At finite entropy the exact answer need not have a literal cusp. A schematic smoothing scale $\varepsilon$ can be represented by

$$S_{\mathrm{smooth}} = \frac{S_{\varnothing}+S_I}{2} -\sqrt{\frac{(S_{\varnothing}-S_I)^2}{4}+\varepsilon^2}.$$

This formula is an illustration of an avoided crossing, not a derivation of gravitational nonperturbative effects. Away from the crossing it differs from the minimum by a suppressed amount; near equality it rounds the kink over a time scale of order $\varepsilon/(\gamma_R+\gamma_B)$.

Independent checks and limitations

Initial-time check. If $S_{R,0}\lt S_{\mathrm{BH},0}+s_I$, the empty island is correctly selected at $t=0$. Reversing that inequality would describe a state already on the island-dominant branch.

Rate check. If evaporation stops so $\gamma_B=0$ but radiation entanglement continues to grow, a crossing can still occur against a constant island branch. If both rates vanish, time cannot cause a branch exchange.

Scheme check. A local counterterm shift on $X$ must be compensated in the renormalized dilaton or gravitational term. Comparing branches in different schemes can manufacture a false Page time.

This calculation is an entropy benchmark, not an evaporation equation. It does not derive the flux rates, a microscopic radiation density matrix, or a decoder. It applies to the declared semiclassical bath model; generic QFT foundations and explicit higher-dimensional string constructions remain outside its scope.

The linear island branch must also be stopped before it predicts a negative entropy or a dilaton of Planckian size. Near the endpoint, slow-rate and weak-backreaction assumptions fail together, so extrapolating the late branch is not a consistency test of the island rule. A complete model must replace the local linearization with its actual endpoint dynamics and boundary conditions.

Exercises

Take $S_{R,0}=0$, $S_{\mathrm{BH},0}=100$, $s_I=2$, $\gamma_R=3$, and $\gamma_B=1$ in consistent units. Find the crossing, the entropy there, and the shift in $t_{\mathrm P}$ if a common calculation error adds $+4$ only to the island branch.

Solution

The correct crossing is $t_{\mathrm P}=102/4=25.5$, and the entropy is $S=3(25.5)=76.5$. Adding $4$ only to the island branch changes the numerator to $106$, giving $t_{\mathrm P}=26.5$, a spurious delay of one time unit. If the added term were a legitimate scheme change, its compensating gravitational shift would cancel it and leave the crossing unchanged.

References

• Almheiri et al. 2019, Ahmed Almheiri, Netta Engelhardt, Donald Marolf, and Henry Maxfield, “The Entropy of Bulk Quantum Fields and the Entanglement Wedge of an Evaporating Black Hole,” Journal of High Energy Physics 12 (2019) 063.
• Almheiri et al. 2021, Ahmed Almheiri, Thomas Hartman, Juan Maldacena, Edgar Shaghoulian, and Amirhossein Tajdini, “The Entropy of Hawking Radiation,” Reviews of Modern Physics 93 (2021) 035002.
• Penington 2020, Geoffrey Penington, “Entanglement Wedge Reconstruction and the Information Paradox,” Journal of High Energy Physics 09 (2020) 002.