The black hole information paradox is usually framed as a spatial problem. A Cauchy surface — a snapshot of the universe at one instant — gets divided into two regions: the black hole interior and the radiation field outside. The entanglement entropy between these regions, computed from the reduced density matrix of the radiation, grows monotonically as Hawking quanta are emitted. If this growth continues forever, a pure quantum state becomes permanently mixed, and unitarity fails.
The Page curve proposes a fix: the entropy rises until the “Page time,” roughly when the black hole has emitted half its initial entropy, then falls back to zero as information escapes in late-time radiation. The island formula and replica wormholes provide mechanisms, at least in anti-de Sitter spacetimes, but they require exotic geometric structures inside the black hole and break down in the asymptotically flat spacetimes where real astrophysical black holes live.
Ladghami, Lobo, and Ouali (arXiv 2602.06833, February 2026) rotate the problem. Instead of dividing space at a fixed time, they divide time — computing entanglement between radiation emitted at different epochs. The question changes from “what is the radiation entangled with spatially?” to “what is early radiation entangled with temporally?”
The mathematics uses the Euclidean continuation of the Schwarzschild metric, where time becomes periodic with period 2pi/kappa (kappa being the surface gravity). Computing the geodesic distance between temporally separated points yields an entropy that oscillates:
S(t) proportional to log[sin^2(kappa t / 2)]
This function rises from zero, reaches a maximum equal to the Bekenstein-Hawking entropy at t = pi/kappa (the “timelike Page time”), then falls back to zero at t = 2pi/kappa. And then it repeats. Not one Page curve but an infinite sequence — a periodic cycle of information loss and recovery with period equal to the inverse Hawking temperature.
The structure generalizes. Adding charge lengthens the period; the extremal limit freezes oscillations entirely (the timelike Page time diverges to infinity). Adding spatial dimensions shortens the period. Rotation introduces a second frequency — the angular velocity of the horizon — turning strict periodicity into quasi-periodicity with beat modulation. The extremal rotating case freezes the thermal oscillations but preserves the rotational modulation, a signature of the near-horizon geometry.
The physical claim is that unitarity is preserved through these temporal correlations: newly emitted Hawking quanta become entangled not with the black hole interior but with previously emitted radiation, and information flows through temporal rather than spatial channels. No firewalls, no islands, no interior degrees of freedom.
But there's a gap. The periodicity in the entropy formula is the periodicity of Euclidean time — the same periodicity that defines the Hawking temperature. The “timelike Page times” at multiples of pi/kappa are the half-periods and full periods of the Euclidean circle. In this sense, the oscillation is the thermal periodicity of the Hartle-Hawking state viewed through the lens of pseudoentropy — a known structure repackaged in a new formalism.
And the calculation assumes a static spacetime. The metric is fixed; the horizon doesn't shrink. The actual information paradox involves a dynamical process — the black hole evaporates, mass decreases, surface gravity changes, and the timescales shift as the system evolves. Whether the periodic structure survives when the black hole is actually losing mass is not addressed.
This doesn't make the proposal empty. The formalism identifies temporal correlations in Hawking radiation that spatial analyses miss. The classification of how charge, rotation, and dimensionality modify the recovery timescale provides new diagnostics. The frozen oscillations at extremality, directly reflecting the near-horizon AdS_2 geometry, connect entanglement dynamics to spacetime structure in a way that might survive beyond the static approximation.
But the strongest version of the claim — that timelike entanglement resolves the information paradox — requires showing that the periodicity persists when the black hole shrinks. The echo repeats in the eternal black hole. Whether it repeats in a mortal one is the open question.