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Holographic duals of evaporating black holes

Emparan, Roberto, Luna, Raimon, Suzuki, Ryotaku, Tomašević, Marija, Way, Benson·2023·Journal of High Energy Physics
DOI 10.1007/jhep05(2023)182
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Abstract

We describe the dynamical evaporation of a black hole as the classical evolution in time of a black hole in an Anti-de Sitter braneworld. A bulk black hole whose horizon intersects the brane yields the classical bulk dual of a black hole coupled to quantum conformal fields. The evaporation of this black hole happens when the bulk horizon slides off the brane, making the horizon on the brane shrink. We use a large-D effective theory of the bulk Einstein equations to solve the time evolution of these systems. With this method, we study the dual evaporation of a variety of black holes interacting with colder radiation baths. We also obtain the dual of the collapse of holographic radiation to form a black hole on the brane. Finally, we discuss the evolution of the Page curve of the radiation in our evaporation setups, with entanglement islands appearing and then shrinking during the decreasing part of the curve.

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Figures

Figure 2 Upper : A bulk black hole is thrown towards the brane and sticks to it. Lower: dual view of the collapse of CFT radiation into a black hole. In this and subsequent figures, we show first the bulk classical evolution, and then the dual view of black holes coupled to a holographic CFT.
Figure 2 Upper : A bulk black hole is thrown towards the brane and sticks to it. Lower: dual view of the collapse of CFT radiation into a black hole. In this and subsequent figures, we show first the bulk classical evolution, and then the dual view of black holes coupled to a holographic CFT.CC-BY · From the PDF
Figure 3 Two scenarios for funnel black hole evaporation, triggered by different perturbations. Upper: Both brane black holes evaporate into the bulk. Lower: One brane black hole evaporates into the other.
Figure 3 Two scenarios for funnel black hole evaporation, triggered by different perturbations. Upper: Both brane black holes evaporate into the bulk. Lower: One brane black hole evaporates into the other.CC-BY · From the PDF
Figure 4 Small brane black hole connected to a large, colder black bath which induces black hole evaporation.
Figure 4 Small brane black hole connected to a large, colder black bath which induces black hole evaporation.CC-BY · From the PDF
Figure 5 Large AdS black hole collapsing onto a black hole the brane, and subsequently evaporating into the colder asymptotic bath (an infinite reservoir), to which the system is connected by a thin but stable funnel.
Figure 5 Large AdS black hole collapsing onto a black hole the brane, and subsequently evaporating into the colder asymptotic bath (an infinite reservoir), to which the system is connected by a thin but stable funnel.CC-BY · From the PDF
Figure 6 Temperature T H (normalized by n ) as a function of blob width r 0 and height m 0 (2.13). To leading order in 1 /n , the temperature is given by r 0 , with T H ∼ 1 /r 0 for small AdS black holes ( r 0 < 1) and T H ∼ r 0 for large AdS black holes ( r 0 > 1). Within each of these branches, the temperature varies with the height of the blob, m 0 , by a next order contribution ∝ 1 /n .
Figure 6 Temperature T H (normalized by n ) as a function of blob width r 0 and height m 0 (2.13). To leading order in 1 /n , the temperature is given by r 0 , with T H ∼ 1 /r 0 for small AdS black holes ( r 0 < 1) and T H ∼ r 0 for large AdS black holes ( r 0 > 1). Within each of these branches, the temperature varies with the height of the blob, m 0 , by a next order contribution ∝ 1 /n .CC-BY · From the PDF
Figure 7 Some static solutions for black holes (blobs) in the braneworld: (a) uniform funnel; (b) double droplet; (c) single droplet, and excited-state droplet; (d) central black hole. In this and all subsequent figures, we show m ( t, x ) and the branes are represented as green boundaries. These solutions are obtained with r 0 = . 4. Compare (a) and (c) to figure 1.
Figure 7 Some static solutions for black holes (blobs) in the braneworld: (a) uniform funnel; (b) double droplet; (c) single droplet, and excited-state droplet; (d) central black hole. In this and all subsequent figures, we show m ( t, x ) and the branes are represented as green boundaries. These solutions are obtained with r 0 = . 4. Compare (a) and (c) to figure 1.CC-BY · From the PDF
Figure 8 Evolution of collapse to form a black hole on the brane. An initial bulk black hole, shown here as a Gaussian blob, is given a boost towards the brane. When the blob hits the brane, it sticks to it, forming a stable droplet. This simulation is performed with r 0 = . 5 and initial velocity 3. Compare to figure 2.
Figure 8 Evolution of collapse to form a black hole on the brane. An initial bulk black hole, shown here as a Gaussian blob, is given a boost towards the brane. When the blob hits the brane, it sticks to it, forming a stable droplet. This simulation is performed with r 0 = . 5 and initial velocity 3. Compare to figure 2.CC-BY · From the PDF
Figure 9 Alternative evolution paths for the unstable uniform funnel. Up: when the perturbation forms a central bulge, a black tsunami flows from the brane to the bulk, ending at a central black hole without any horizons on the branes. Down: an asymmetric perturbation makes the horizon flow away from one of the branes towards the other. In both cases, the direction of evolution can be inferred from the fact that, since these are small AdS black holes, the thinner parts of the horizon are hotter and radiate towards the thicker, colder regions. Both simulations are made with r 0 = 0 . 27. Compare to figure 3.
Figure 9 Alternative evolution paths for the unstable uniform funnel. Up: when the perturbation forms a central bulge, a black tsunami flows from the brane to the bulk, ending at a central black hole without any horizons on the branes. Down: an asymmetric perturbation makes the horizon flow away from one of the branes towards the other. In both cases, the direction of evolution can be inferred from the fact that, since these are small AdS black holes, the thinner parts of the horizon are hotter and radiate towards the thicker, colder regions. Both simulations are made with r 0 = 0 . 27. Compare to figure 3.CC-BY · From the PDF
Figure 10 A black droplet on the brane is connected to a larger and colder black hole in the bulk. The initial system is far from equilibrium and the droplet is quickly absorbed by the central black hole (which then wobbles around the AdS center). Here we have set r 0 = 0 . 4. Compare to figure 4.
Figure 10 A black droplet on the brane is connected to a larger and colder black hole in the bulk. The initial system is far from equilibrium and the droplet is quickly absorbed by the central black hole (which then wobbles around the AdS center). Here we have set r 0 = 0 . 4. Compare to figure 4.CC-BY · From the PDF
Image 10
CC-BY · From the PDF
Figure 12 Evolution of a perturbed uniform funnel where a singular pinch occurs in the bulk faster than the heat transfer across the funnel. This can be seen as the competition between two unstable Gregory-Laflamme modes of the black string, such that it pinches in the bulk instead of on the brane. In the dual view, the entanglement between the two black holes is severed, yielding a burst of radiation that stops the evaporation. In this simulation, r 0 = 0 . 4.
Figure 12 Evolution of a perturbed uniform funnel where a singular pinch occurs in the bulk faster than the heat transfer across the funnel. This can be seen as the competition between two unstable Gregory-Laflamme modes of the black string, such that it pinches in the bulk instead of on the brane. In the dual view, the entanglement between the two black holes is severed, yielding a burst of radiation that stops the evaporation. In this simulation, r 0 = 0 . 4.CC-BY · From the PDF
Figure 13 A different representation of the initial configuration in figure 4 (cf. App. A). Left : Each point in this Cauchy slice is a sphere S n +1 . The Einstein-Rosen bridge that connects the two asymptotic regions is represented as a black dashed line. The blue dotted line is the SO ( n + 2) rotation symmetry axis (in each region). Right : the S n +1 remain of finite size at the ER bridge, but shrink to zero on the symmetry axis. The radius of these circles is represented in figure 4, which only shows one asymptotic region.
Figure 13 A different representation of the initial configuration in figure 4 (cf. App. A). Left : Each point in this Cauchy slice is a sphere S n +1 . The Einstein-Rosen bridge that connects the two asymptotic regions is represented as a black dashed line. The blue dotted line is the SO ( n + 2) rotation symmetry axis (in each region). Right : the S n +1 remain of finite size at the ER bridge, but shrink to zero on the symmetry axis. The radius of these circles is represented in figure 4, which only shows one asymptotic region.CC-BY · From the PDF
Figure 14 HRT surfaces for the calculation of the Page curve in an evaporating droplet: Left : surfaces (red dotted and dashed) in the representation of figure 4, which should be thought of as rotated along the horizontal central axis, and only shows the half of the surfaces in one of the asymptotic regions. Middle : surfaces in the representation of figure 13, which shows the two asymptotic regions. Right : the situation after the black hole has evaporated from the brane. We are omitting HRT surfaces that would pass to the left of the black hole, which could be dominant if the bath saturates its entanglement.
Figure 14 HRT surfaces for the calculation of the Page curve in an evaporating droplet: Left : surfaces (red dotted and dashed) in the representation of figure 4, which should be thought of as rotated along the horizontal central axis, and only shows the half of the surfaces in one of the asymptotic regions. Middle : surfaces in the representation of figure 13, which shows the two asymptotic regions. Right : the situation after the black hole has evaporated from the brane. We are omitting HRT surfaces that would pass to the left of the black hole, which could be dominant if the bath saturates its entanglement.CC-BY · From the PDF
Figure 15 A different representation (analogous to figure 13) of the configuration in figure 5 (middle) for evaporation into an infinite bath. We also show the HRT surfaces in it. This shares qualitative features with the scenario in [32], in that the Page curve at late times plateaus at a constant value, but at earlier times its behavior is more complex, first growing and then decreasing until reaching the plateau.
Figure 15 A different representation (analogous to figure 13) of the configuration in figure 5 (middle) for evaporation into an infinite bath. We also show the HRT surfaces in it. This shares qualitative features with the scenario in [32], in that the Page curve at late times plateaus at a constant value, but at earlier times its behavior is more complex, first growing and then decreasing until reaching the plateau.CC-BY · From the PDF
Figure 16 Different representations of constant-time slices of the Penrose diagram for configurations in this article (without the brane included). (a) All boundary components are presented. Each point is a sphere S n +1 generated by the action of SO ( n +2) rotations around the symmetry axis (dotted blue), where the spheres shrink to zero. Dashed black lines are Einstein-Rosen bridges that connect different asymptotic regions. (b) Shape of the horizon, as shown by the size of the S n +1 spheres along the ER bridge. (c) Combination of the information in (a) and (b) but showing only a single boundary component. (d) Two antipodal copies of the geometries in (a), with bending chosen for illustrative purposes. When the SO ( n +2) rotation is applied, all the horizons, including the black funnel, are connected as in (b).
Figure 16 Different representations of constant-time slices of the Penrose diagram for configurations in this article (without the brane included). (a) All boundary components are presented. Each point is a sphere S n +1 generated by the action of SO ( n +2) rotations around the symmetry axis (dotted blue), where the spheres shrink to zero. Dashed black lines are Einstein-Rosen bridges that connect different asymptotic regions. (b) Shape of the horizon, as shown by the size of the S n +1 spheres along the ER bridge. (c) Combination of the information in (a) and (b) but showing only a single boundary component. (d) Two antipodal copies of the geometries in (a), with bending chosen for illustrative purposes. When the SO ( n +2) rotation is applied, all the horizons, including the black funnel, are connected as in (b).CC-BY · From the PDF

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Cites

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Metadata

Authors
Emparan, R., Luna, R., Suzuki, R., Tomašević, M., & Way, B.
ORCID
Venue
Journal of High Energy Physics
Publisher
Springer Science and Business Media LLC
Published
2023-05-22
Volume
2023
Issue
5
ISSN
1029-8479
DOI
10.1007/jhep05(2023)182
Cited by
24
References
4
Language
en
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