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

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 AdS3 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.

Page curve for entanglement negativity through geometric evaporation

“…This nice feature of double holography naturally leads to a geometric perspective of the black hole formation and evaporation. Since this exciting development, a variety of very interesting models have explored different techniques to geometrize the island formulation [10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25]. In this article we focus on another distinct recent geometric construction proposed in [20].…”

“…Since this exciting development, a variety of very interesting models have explored different techniques to geometrize the island formulation [10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25]. In this article we focus on another distinct recent geometric construction proposed in [20]. The authors in [20] motivated by [26], demonstrated that the evaporation of a two dimensional Jackiw-Teitelboim black hole could be modelled by a partial dimensional reduction of a three dimensional BTZ black hole.…”

“…In this article we focus on another distinct recent geometric construction proposed in [20]. The authors in [20] motivated by [26], demonstrated that the evaporation of a two dimensional Jackiw-Teitelboim black hole could be modelled by a partial dimensional reduction of a three dimensional BTZ black hole. In this model the extent to which the JT black hole evaporates is controlled by the parameter describing the partial dimensional reduction.…”

“…In this article we address this significant issue by computing the entanglement negativity of various mixed state configurations in a bath coupled to a two dimensional non-extremal JT black hole obtained by a partial dimensional reduction of a three dimensional BTZ black hole as described in [20]. We will demonstrate that the entanglement negativity of various bipartite systems in a bath coupled to a non-extremal JT black hole exactly reproduces the analogues of the Page curve for the entanglement negativity of random mixed states which were obtained in [70].…”

“…The article is organized as follows. In section 2 we review the model involving the partial dimensional reduction described in [20] to obtain the Page curve for the entanglement entropy of the entire bath/radiation coupled to a JT black hole. Following this we briefly review [70] where the authors obtain the entanglement negativity of a random bipartite mixed state and then go on to describe the holographic entanglement negativity construction proposed in [43,45].…”