Observations of Hawking radiation: the Page curve and baby universes

TL;DR

The paper reframes black hole information in a Lorentzian, topology-summed gravity framework, showing that replica wormholes and baby universes can render asymptotic measurements consistent with BH unitarity while tying the black hole density of states to $S_{BH}=A/(4G)$. It develops a quantum-extremal-surface/island program to locate dominant saddles for entropy calculations, thereby producing the Page curve without violating unitarity at the level accessible to exterior observers. By disentangling observable radiation from interior degrees of freedom through a Hilbert-space of baby universes, it presents an ensemble interpretation of Hawking radiation and clarifies how correlations across identical experiments arise as classical superselection sectors. The work also discusses the Polchinski-Strominger proposal as an instructive precursor, analyzes limitations and potential UV completions, and outlines open questions related to factorization, infalling observers, and nonperturbative topology changes.

Abstract

We reformulate recent insights into black hole information in a manner emphasizing operationally-defined notions of entropy, Lorentz-signature descriptions, and asymptotically flat spacetimes. With the help of replica wormholes, we find that experiments of asymptotic observers are consistent with black holes as unitary quantum systems, with density of states given by the Bekenstein-Hawking formula. However, this comes at the cost of superselection sectors associated with the state of baby universes. Spacetimes studied by Polchinski and Strominger in 1994 provide a simple illustration of the associated concepts and techniques, and we argue them to be a natural late-time extrapolation of replica wormholes. The work aims to be self-contained and, in particular, to be accessible to readers who have not yet mastered earlier formulations of the ideas above.

Figures (20)

• Figure 1: The Page curve for the entropy of Hawking radiation emitted before time $u$ (solid curve). For a while this is increasing, given by the thermal entropy (dotted curve). But under BH unitarity it is bounded by the Bekenstein Hawking entropy $S_{BH}\sim \frac{A}{4G}$ (dashed curve). Consequently, after the Page time $u_\mathrm{Page}$ the entropy must decrease, approximately saturating that bound.
• Figure 2: The final event horizon $\mathcal{H}^+$ is shown as a dashed line, and the singularity as a jagged line. Future and past null infinity are labelled by $\mathscr{I}^\pm$. The vertical line marked $r=0$ is a regular origin of spherical polar coordinates. The state is chosen as the vacuum of quantum fields in the flat asymptotic region $\mathscr{I}^-$, thus $|\psi\rangle = |0\rangle_{\mathscr{I}^-}$. In the shaded region of (b) near $\mathscr{I}^+$, our spacetime is nearly stationary. There, the backwards-propagation reduces to scattering in a fixed potential and results in a transmitted part $T$ and a reflected part $R$. For modes localized at late (retarded) times on $\mathscr{I}^+$, the reflected part $R$ will remain in the nearly stationary region and transmitted part $T$ will be localized very close to $\mathcal{H}^+$. In particular, the wavelength of $T$ becomes very short in the reference frame shown. This allows us to complete the backwards-propagation of $T$ from the near-horizon region to $\mathscr{I}^-$ using geometric optics.
• Figure 3: A path integral that computes the matrix elements $\langle\phi^{(2)}_-|\mathcal{O}_+|\phi^{(1)}_- \rangle$. The right copy of the spacetime contains fields $\phi^{(1)}$ and is weighted by $e^{iI[\phi^{(1)}]}$, while the left copy contains fields $\phi^{(2)}$ and is weighted by $e^{-iI[\phi^{(2)}]}$ (or more generally by the CPT conjugate of the action on the left copy). This conjugation is associated with the fact that the initial conditions for the right copy (fixing the field on $\Sigma_-$) are defined by the ket-state $|\phi^{(1)}_- \rangle$ while those for the left copy are defined by the bra-state $\langle \phi^{(2)}_- |$.
• Figure 4: An in-in (or Schwinger-Keldysh) contour in the complex time-plane that computes the expectation value of $\mathcal{O}_+$ at Lorentzian time $t$ in the vacuum state $|0\rangle$. The contour begins at negative infinite Euclidean time and follows the Euclidean axis to the origin. This part of the contour computes $|0\rangle$ in terms of fields at $t=0$. The contour then proceeds along the Lorentzian axis (this part of the contour corresponding to the right spacetime of figure \ref{['fig:ininPI']}) until $\mathcal{O}_+$ is inserted at $t$, whence it returns to the origin (the left spacetime of figure \ref{['fig:ininPI']}). Finally, it proceeds from the origin to positive infinite Euclidean time to compute $\langle 0|$. For clarity, the various parts of the contour have been slightly displaced from the axes in the figure.
• Figure 5: Path integrals computing (a) the expectation value of an operator at $\mathscr{I}^+$ and (b) components of the density matrix on $\mathscr{I}^+$.