Entanglement islands and information recovery from near-extremal regular black holes

Abstract: We investigate the Page curve and information recovery in a near-extremal regular black hole inspired by T-duality, in which the central singularity is resolved by a minimal length scale. By integrating the first law of thermodynamics at fixed minimal length, we obtain a black-hole entropy containing an intrinsically quantum logarithmic correction, while the area-law contribution vanishes in the extremal limit. Consequently, the extremal remnant carries a finite entropy of purely quantum origin. Using the island prescription in the near-horizon regime, we evaluate the generalized entropy of radiation and compare the standard area functional with an alternative functional constructed from the corrected thermodynamic entropy of the black hole. In the near-extremal limit, the latter reduces analytically to the classical extremization problem with an effective coupling, and it places the physical island closer to the outer horizon. Although both prescriptions produce a Page transition, they predict different saturation values and Page times. While the standard functional yields a plateau controlled by the area term, the corrected prescription saturates at the full thermodynamic entropy of the outer horizon and, in the extremal limit, at the logarithmic remnant entropy. Including evaporation and backreaction, the Page curve develops the expected descending branch and asymptotes to the entropy of the cold extremal remnant rather than to zero. Our results indicate that a thermodynamically consistent description of information recovery from regular near-extremal black holes requires incorporating the intrinsic quantum correction to the gravitational entropy.

• The paper shows that using the first-law-corrected entropy functional produces a higher Page plateau and delayed information recovery than the standard area prescription, while remaining consistent with the vanishing area contribution at extremality.
• The analysis finds that the corrected island extremization is equivalent near extremality to rescaling the coupling as λ_eff ≈ 0.544λ, bringing the island 8/27 as far from the horizon as in the area-based treatment.
• Evaporation drives the black hole toward extremality without reaching it, and the late-time radiation entropy approaches the quantum remnant entropy S_ext rather than zero, subject to the unresolved choice of renormalization scale μ.

The paper studies information recovery from a regular black hole whose central singularity is resolved by a minimal length $\ell_0$ of stringy origin, derived from T-duality of a compactified bosonic string (2608.18603). Its central contribution is a comparison between two choices of gravitational entropy functional in the island prescription: the standard area term $\mathrm{Area}(\partial I)/4G$ and an alternative functional built from the thermodynamically corrected black-hole entropy obtained by integrating the first law at fixed $\ell_0$. The two prescriptions yield Page curves of the same qualitative shape but different plateaus and Page times, and only the corrected functional is consistent with the vanishing of the area-law entropy at extremality.

Geometry and quantum-corrected thermodynamics

The model derives from the center-of-mass propagator of a closed string on a circle near the self-dual radius, which takes the Bessel form $- \ell_0 K_1(\ell_0\sqrt{k^2+m^2})/\sqrt{k^2+m^2}$ and acts as an invariant UV cutoff. The resulting static potential $V(r) = -M/\sqrt{r^2+\ell_0^2}$ corresponds to a smeared source density, giving the lapse function

$f(r) = 1 - \frac{2Mr^2}{(r^2+\ell_0^2)^{3/2}}.$

Extremality occurs at $r_e = \sqrt{2}\,\ell_0$, $M_{\rm ext} = (3\sqrt{3}/4)\ell_0$. Integrating the first law $dM = T\,dS$ at fixed $\ell_0$ yields an entropy consisting of an area-like piece proportional to $\mathrm{Area}(\partial I)/4G$0 plus a logarithmic piece $\mathrm{Area}(\partial I)/4G$1, where $\mathrm{Area}(\partial I)/4G$2 is an integration constant. The key structural fact is that the area-like prefactor vanishes identically at extremality, so the extremal remnant carries a finite entropy

$\mathrm{Area}(\partial I)/4G$3

of purely quantum-gravitational origin. The extremal near-horizon geometry is $\mathrm{Area}(\partial I)/4G$4 with radii $\mathrm{Area}(\partial I)/4G$5 and $\mathrm{Area}(\partial I)/4G$6, and the near-extremal temperature scales as $\mathrm{Area}(\partial I)/4G$7 in the mass above extremality. This vanishing of the area term at extremality — rather than its persistence as in Reissner–Nordström — is what makes the choice of island functional consequential.

No-island phase

Working in Kruskal-like coordinates adapted to the near-extremal throat, the no-island radiation entropy for two exterior endpoints separated by conformal distance is

$\mathrm{Area}(\partial I)/4G$8

with surface gravity $\mathrm{Area}(\partial I)/4G$9. It shares the universal $\ell_0$0 structure of the Schwarzschild result, growing linearly at late times. Crucially, it diverges logarithmically as $\ell_0$1: the semiclassical no-island approximation breaks down near extremality, and the island prescription is required to regulate this divergence. This distinguishes the regular black hole from Schwarzschild, where the divergence is driven by evaporation to zero mass.

Island phase: two functionals

With the standard area functional, the generalized entropy in the right wedge is extremized numerically in dimensionless variables $\ell_0$2. Islands exist only above a threshold in $\ell_0$3 that grows as $\ell_0$4 increases ($\ell_0$5 for $\ell_0$6 at $\ell_0$7). On the physical branch the island lies parametrically close to the outer horizon,

$\ell_0$8

with displacements below $\ell_0$9 throughout the near-extremal window, mirroring the Schwarzschild pinning of Ref. (Hashimoto et al., 2020). The plateau saturates at the Bekenstein–Hawking entropy of the outer horizon, tending to $- \ell_0 K_1(\ell_0\sqrt{k^2+m^2})/\sqrt{k^2+m^2}$0 as $- \ell_0 K_1(\ell_0\sqrt{k^2+m^2})/\sqrt{k^2+m^2}$1.

The corrected prescription replaces $- \ell_0 K_1(\ell_0\sqrt{k^2+m^2})/\sqrt{k^2+m^2}$2 by $- \ell_0 K_1(\ell_0\sqrt{k^2+m^2})/\sqrt{k^2+m^2}$3 evaluated on the island boundary. The essential simplification is that $- \ell_0 K_1(\ell_0\sqrt{k^2+m^2})/\sqrt{k^2+m^2}$4 with $- \ell_0 K_1(\ell_0\sqrt{k^2+m^2})/\sqrt{k^2+m^2}$5 algebraic, and $- \ell_0 K_1(\ell_0\sqrt{k^2+m^2})/\sqrt{k^2+m^2}$6 precisely because $- \ell_0 K_1(\ell_0\sqrt{k^2+m^2})/\sqrt{k^2+m^2}$7 is the extremality condition. Consequently, throughout the near-extremal window the corrected extremization problem reduces to the classical one with rescaled coupling

$- \ell_0 K_1(\ell_0\sqrt{k^2+m^2})/\sqrt{k^2+m^2}$8

verified numerically to better than 1.5% at $- \ell_0 K_1(\ell_0\sqrt{k^2+m^2})/\sqrt{k^2+m^2}$9. The island moves closer to the horizon by the universal factor $V(r) = -M/\sqrt{r^2+\ell_0^2}$0 (numerically 0.289), and the renormalization scale $V(r) = -M/\sqrt{r^2+\ell_0^2}$1 drops out of the extremization entirely since it enters only additively.

Page curves, evaporation, and the remnant

Both prescriptions produce a rising branch followed by saturation, but they differ sharply in the plateau height. For representative parameters ($V(r) = -M/\sqrt{r^2+\ell_0^2}$2, $V(r) = -M/\sqrt{r^2+\ell_0^2}$3, $V(r) = -M/\sqrt{r^2+\ell_0^2}$4), the area functional saturates at $V(r) = -M/\sqrt{r^2+\ell_0^2}$5 while the corrected functional saturates at $V(r) = -M/\sqrt{r^2+\ell_0^2}$6 — a difference of roughly a factor of two — with corresponding Page times $V(r) = -M/\sqrt{r^2+\ell_0^2}$7 and $V(r) = -M/\sqrt{r^2+\ell_0^2}$8. The delay grows without bound as extremality is approached, since $V(r) = -M/\sqrt{r^2+\ell_0^2}$9 times the finite difference $f(r) = 1 - \frac{2Mr^2}{(r^2+\ell_0^2)^{3/2}}.$0.

Including quasi-static evaporation via $f(r) = 1 - \frac{2Mr^2}{(r^2+\ell_0^2)^{3/2}}.$1, the deviation from extremality decays exponentially with timescale $f(r) = 1 - \frac{2Mr^2}{(r^2+\ell_0^2)^{3/2}}.$2, so the black hole approaches extremality asymptotically without ever reaching it. The Page curve then develops a descending branch that asymptotes not to zero but to $f(r) = 1 - \frac{2Mr^2}{(r^2+\ell_0^2)^{3/2}}.$3 under the corrected functional, versus $f(r) = 1 - \frac{2Mr^2}{(r^2+\ell_0^2)^{3/2}}.$4 under the area functional. The adiabatic treatment is self-consistent only when $f(r) = 1 - \frac{2Mr^2}{(r^2+\ell_0^2)^{3/2}}.$5, restricting the analysis to $f(r) = 1 - \frac{2Mr^2}{(r^2+\ell_0^2)^{3/2}}.$6; outside this window the black hole evaporates before the island saddle dominates. Backreaction of the Hawking flux on the Schwarzian mode of the throat is not included.

Limitations and open questions

Three limitations are stated explicitly. First, the substitution $f(r) = 1 - \frac{2Mr^2}{(r^2+\ell_0^2)^{3/2}}.$7 extending the horizon entropy to an arbitrary sphere is an assumption, not a theorem: it is justified only if the logarithmic term descends from a local covariant (Wald-like) entropy functional rather than a global one-loop determinant, and verifying this from the underlying action remains open. Second, the plateau height inherits the renormalization ambiguity of $f(r) = 1 - \frac{2Mr^2}{(r^2+\ell_0^2)^{3/2}}.$8. Positivity of the entropy imposes the sharp bound $f(r) = 1 - \frac{2Mr^2}{(r^2+\ell_0^2)^{3/2}}.$9; at the boundary value $r_e = \sqrt{2}\,\ell_0$0 exactly, the remnant is a unique state satisfying the Nernst form of the third law, but then it can store no information and the radiation must be asymptotically pure. Choosing between $r_e = \sqrt{2}\,\ell_0$1 (finite remnant entropy) and $r_e = \sqrt{2}\,\ell_0$2 (zero-entropy ground state) requires microstate counting or a Euclidean evaluation beyond what the first law supplies — identified by the authors as the sharpest open question. Third, the semiclassical formalism is weakest precisely where it matters most: near extremality the classical entropy vanishes and the relevant term is itself a quantum correction, so a complete treatment would require the Euclidean path integral including the zero modes of the near-extremal throat.

Conclusion

The paper establishes that for a T-duality-inspired regular black hole, building the island functional from the first-law entropy rather than the bare area term resolves an internal tension in the semiclassical description: the resulting extremal plateau equals the purely logarithmic $r_e = \sqrt{2}\,\ell_0$3 instead of the horizon area, consistent with the vanishing of the area-law contribution at extremality. Analytically, the correction amounts to a coupling rescaling $r_e = \sqrt{2}\,\ell_0$4, an island displacement reduced by $r_e = \sqrt{2}\,\ell_0$5, and a delayed, higher Page transition whose descending branch terminates at the remnant entropy. Whether the vanishing of the area term at extremality generalizes to other minimal-length geometries such as Bardeen or Hayward solutions, and how $r_e = \sqrt{2}\,\ell_0$6 is fixed by microscopic physics, remain open.