Evaporating Black Holes Coupled to a Thermal Bath

TL;DR

This paper extends the AEM$^4$Z doubly holographic model to a finite-temperature auxiliary bath and demonstrates that unitary Page curves persist across evaporation, equilibration, and heating regimes. Using holographic entanglement entropy and quantum extremal surfaces in a JT gravity context, it analyzes how the QES can lie outside or inside the horizon depending on bath temperature and time, and how purification of the bath affects interior reconstruction. The work reveals a bath-temperature dependent threshold $T_p$ and a minimal purifier-length requirement for interior recovery, highlighting the intricate interplay between Hawking radiation, bath purification, and island dynamics in the information-flow problem. Its insights into the Page curve and interior reconstruction have implications for understanding information recovery in dynamical black hole settings and broader island-type proposals in holography.

Abstract

We study the doubly holographic model of [arXiv:1908.10996] in the situation where a black hole in two-dimensional JT gravity theory is coupled to an auxiliary bath system at arbitrary finite temperature. Depending on the initial temperature of the black hole relative to the bath temperature, the black hole can lose mass by emitting Hawking radiation, stay in equilibrium with the bath or gain mass by absorbing thermal radiation from the bath. In all of these scenarios, a unitary Page curve is obtained by applying the usual prescription for holographic entanglement entropy and identifying the quantum extremal surface for the generalized entropy, using both analytical and numeric calculations. As the application of the entanglement wedge reconstruction, we further investigate the reconstruction of the black hole interior from a subsystem containing the Hawking radiation. We examine the roles of the Hawking radiation and also the purification of the thermal bath in this reconstruction.

Figures (14)

• Figure 1: In the AEM$^4$Z model, the holographic principle is invoked twice, resulting in three different pictures of the same physical system. In our present analysis of this model, we include a thermal bath at finite temperature. In the top picture (a), there are two quantum mechanical systems (QM$_{\textrm{\tiny L}}$ and QM$_{\textrm{\tiny R}}$), as well as a two copies of the field theory (CFT$_2$) on a half-line (dashed and dotted). Both of the quantum mechanical and field theory systems are prepared in independent thermofield double states. The middle picture (b) introduces the two-dimensional holographic geometry (JT gravity) dual to the entangled state of QM$_{\textrm{\tiny L}}$ and QM$_{\textrm{\tiny R}}$. This gravitating region also supports the same CFT$_2$ that appears in the bath region. The last picture (c) contains the doubly-holographic description, where the holographic CFT is replaced by an AdS$_3$ bulk, and in particular, the thermofield double is replaced by a bulk region with the geometry of an AdS$_3$ black hole.
• Figure 2: A cartoon illustration of the three phases for the entanglement entropy of $\mathrm{QM}_{\textrm{\tiny R}}$ or of $\mathrm{QM}_{\textrm{\tiny L}}$, (a semi-infinite interval in) the thermal bath, and the (entire) bath purifier, after the quench where $\mathrm{QM}_{\textrm{\tiny R}}$ is connected to the bath. The darker colors indicate the true generalized entropy, while the lighter colors indicate the general shape of each of the branches slightly beyond the regime where it provides the minimal value for the generalized entropy. Below the plot is a sketch of the shape of the extremal HRT surfaces in AdS$_3$ which contribute to the generalized entropy in each phase.
• Figure 3: The Penrose diagram for the AdS$_2$ black hole coupled with a thermal bath and its purification in flat spacetime at time $u=0$. The (thick) pink lines are the shock waves propagating into the gravitating and bath regions, which are generated by this joining quench. The bifurcation surface of the initial equilibrium black hole is indicated by the red dot. The new horizon is indicated by the black dashed line, ${\it i.e.,}\ y^+=\infty$. Note that only the blue and red shaded regions are covered by the $y^\pm, \tilde{y}^\pm$ coordinates, respectively. The evolution of quantum extremal surface in three phases is presented by the corresponding colored curves, as indicated in figure \ref{['fig:sertraline']}.
• Figure 4: Competing channels computing the generalized entropy of various subsystems (solid red) and the corresponding bulk RT surfaces (dashed red) and entanglement wedges (light red). In each case, the R-channel where the black hole interior is recoverable or reconstructible is shown on the left. On the right, we show the N-channel where the interior is non-recoverable or non-reconstructible. The corresponding generalized entropies for these channels are denoted $S_{\textrm{\tiny R}}$ and $S_{\textrm{\tiny N}}$, respectively. In the top row (a), we consider the generalized entropy of $\mathrm{QM}_{\textrm{\tiny L}}$, the thermal bath, and the bath's purifier. In row (b), we keep only a finite interval $[\sigma_1,\sigma_2]$ of the bath. In row (c), we further trace out the purifier. Finally, in row (d), we include a finite interval $[0,\tilde{\sigma}_3]$ of the purifier. Note that in this last case, we can also vary $\tilde{u}_3$, the time slice of the purifier interval, and we find the minimal $\tilde{\sigma}_3$ depends on $\tilde{u}_3$ --- see section \ref{['sec:smoke']}.
• Figure 5: The bath and purifier subsystems. The central panel shows a Penrose diagram of various coordinate patches of the bath and purifier subsystems. The left panel shows two examples, sharing the same $y_2^-$, of an interval $[\sigma_1,\sigma_2]$ of the bath system after the Page time: the shorter blue interval is just barely above the critical length $\Delta_\mathrm{turn}$ needed to recover the black hole interior; the green interval is much longer. Red wavy lines show thermal radiation leaving the bath prior to $y^-=y_2^-$. The right panel shows the corresponding intervals $[0,\tilde{\sigma}_3]$ needed in conjunction with the bath intervals (plus $\mathrm{QM}_{\textrm{\tiny L}}$) to recover the black hole interior. The phase boundaries of $\tilde{\sigma}_3$ for recoverability is shown in light blue and green. The dashed wavy lines show the thermal quanta of the purifier that are most entangled with the radiation marked in the left panel.