From the BTZ black hole to JT gravity: geometrizing the island

TL;DR

The work constructs a geometric bridge between JT gravity and higher-dimensional AdS$_3$/BTZ spacetimes by partially reducing AdS$_3$ to obtain a 2D JT black hole coupled to a holographic bath. It shows how extremal and finite-temperature JT black holes emerge from AdS$_3$ reductions, and computes entropies of radiation via 3D geodesics, with the Schwarzian boundary dynamics governing evolution. By introducing time dependence in the reduction parameter, the authors model evaporation, demonstrating a linear-to-exponential transition under a BTZ parameter map and yielding a Page-curve-like entropy for the radiation. The results illustrate how island-like structures and unitarity can emerge in a purely geometric, higher-dimensional setting without invoking the island formula explicitly, offering a framework that could extend to other spacetimes such as de Sitter.

Abstract

We study the evaporation of two-dimensional black holes in JT gravity from a three-dimensional point of view. A partial dimensional reduction of AdS$_3$ in Poincaré coordinates leads to an extremal 2D black hole in JT gravity coupled to a 'bath': the holographic dual of the remainder of the 3D spacetime. Partially reducing the BTZ black hole gives us the finite temperature version. We compute the entropy of the radiation using geodesics in the three-dimensional spacetime. We then focus on the finite temperature case and describe the dynamics by introducing time-dependence into the parameter controlling the reduction. The energy of the black hole decreases linearly as we slowly move the dividing line between black hole and bath. Through a re-scaling of the BTZ parameters we map this to the more canonical picture of exponential evaporation. Finally, studying the entropy of the radiation over time leads to a geometric representation of the Page curve. The appearance of the island region is explained in a natural and intuitive fashion.

Figures (5)

• Figure 1: AdS$_3$-Poincaré, partially reduced over the angle $2\pi \alpha$. The purple region is the 2D JT extremal black hole, and the green region is dual to the bath 2D CFT. The blue geodesic computes the entropy of the region $[0,b]$ in the CFT, including the quantum mechanical system.
• Figure 2: We have done a partial reduction over the angle $2\pi\alpha$. The value of $\alpha$ determines if the black hole is before (a) or after (b) the Page time.
• Figure 3: To find the entropy of the double interval $[0,b]$, including the quantum mechanical degrees of freedom, we need to compute the length of the blue geodesics.
• Figure 4: The entropy of the radiation follows a Page curve (red line).
• Figure 5: The qualitative behavior of the entropy for $\Delta \varphi = 2\pi \frac{A}{2}\tilde{t}$ (blue) and $\Delta \varphi = 2\pi ( 1 - \frac{A}{2} \tilde{t} )$ (orange) shows that it follows the Page curve (red line). To plot we set $\ell = G = 1$, $r_{\infty} = 10^{100.000}$, the evaporation rate $A = 5$ and $R = 2 \ell$ on the left, $R = 10 \ell$ on the right. The dashed green line indicates the UV cutoff.