The Page curve of Hawking radiation from semiclassical geometry

TL;DR

This work shows that Hawking radiation can exhibit a Page curve in a semiclassical gravity setup when the matter sector has a holographic dual, by mapping quantum extremal surface calculations to ordinary RT/HRT surfaces in a higher-dimensional bulk. The key mechanism is an extra dimension that connects the black hole interior to the radiation, enabling islands in the entanglement wedge and a Page-time transition driven by area terms. The authors introduce a general rule for computing entropies of systems entangled with gravity, allowing for quantum extremal islands and linking interior regions to exterior radiation through holographic geometry. The approach extends the RT/HRT framework to gravity-entangled settings and has potential implications for information recovery, bulk reconstruction, and higher-dimensional generalizations.

Abstract

We consider a gravity theory coupled to matter, where the matter has a higher-dimensional holographic dual. In such a theory, finding quantum extremal surfaces becomes equivalent to finding the RT/HRT surfaces in the higher-dimensional theory. Using this we compute the entropy of Hawking radiation and argue that it follows the Page curve, as suggested by recent computations of the entropy and entanglement wedges for old black holes. The higher-dimensional geometry connects the radiation to the black hole interior in the spirit of ER=EPR. The black hole interior then becomes part of the entanglement wedge of the radiation. Inspired by this, we propose a new rule for computing the entropy of quantum systems entangled with gravitational systems which involves searching for "islands" in determining the entanglement wedge.

Figures (12)

• Figure 1: On the left, we have a 2d dilaton-gravity theory coupled to a matter CFT$_2$. The fields of the matter CFT$_2$ are denoted collectively by $\chi$, and this CFT$_2$ is assumed to be holographic. On the right, we display a 3d geometry obtained by replacing the matter CFT$_2$ with its 3d dual. This is a version of the Randall-Sundrum setup Randall:1999vfGubser:1999vj. On the 2d boundary of this 3d geometry, we have the dilaton-gravity action. The boundary fields $\phi$ and $g_{ij}^{(2)}$ are also integrated over in the functional integral.
• Figure 2: We sketch three different pictures of the same system. The first is a 2d dilaton-gravity theory, plus a matter CFT$_2$, coupled to a bath consisting of the same CFT$_2$. This CFT$_2$ is assumed to have a holographic dual. The second is 3d gravity, where we replace the CFT$_2$ by its holographic dual. It contains a dynamical boundary metric on the Planck brane. More details about the state of the CFT are encoded deeper inside the 3d geometry. The third is the fully quantum mechanical description, where we replace the 2d gravity+matter theory by its quantum mechanical dual. This quantum mechanical system lives at the boundary of the bath CFT. In all cases, the thick dot represents the point $\sigma_y=0$.
• Figure 3: (a) We have shown, from the 2d perspective, the two contributions $\frac{\phi(y)}{4G_N^{(2)}}$ and $S_\text{Bulk-2d}[\mathcal{I}_y]$ to $S_\text{gen}(y)$ from equation (\ref{['GenEN']}). (b) Since the matter CFT$_2$ is holographic, the quantity $S_{\mathrm{Bulk}\text{-}2d} [{\cal I}_y]$ can be computed using a 3d RT formula.
• Figure 4: The entanglement wedge for the black hole at late times. We show a spatial slice $\Sigma_\text{Late}$ at some late time that passes through the quantum extremal surface, (see also figure \ref{['latetimeslice']}). In the leftmost picture, we have drawn $\Sigma_\text{Late}$ in the 2d geometry. The middle picture is a spatial slice of the three dimensional geometry that ends on $\Sigma_{\rm Late}$ and contains the RT/HRT surface, the pink region being the entanglement wedge. In the rightmost picture, we have an interval that contains the left boundary and whose entropy we are trying to compute.
• Figure 5: The spacetime diagram describing the coupling of the black hole to the bath, the energy pulse coming from the moment they are coupled, the formation of the black hole and its subsequent evaporation. We pick some late time nice slice $\Sigma_{\text{Late}}$ and we compute the entanglement wedge for what is to the left of $\sigma_0$. This contains only a portion of the time slice in the interior. We have also displayed the Wheeler de Witt patch, or causal domain of dependence that describes the full entanglement wedge.