‫هشتمین کنفرانس فیزیک ریاضی ایران‬

‫ دااﮕشنه ﻢﻗ ﻰتﻌنﺻ‬،1403 ‫ تیر ماه‬18-17

8th Iranian Conference on Mathematical Physics

AdS Black Hole Evaporation Modeling with emphasis on Partial-Dimensional

Reduction

Seyed Moein Doostaninejad Amin Faraji Astaneh

M.Sc student, Department of Physics, Sharif Assistant Professor, Sharif University of
University of Technology Technology
sm_doostaninejad@physics.sharif.edu faraji@sharif.edu

Abstract
As a contribution to the subject of the information loss paradox in (1+1)-dimensional gravitational systems,
we studied the evaporation of (1+1)-dimensional black holes in AdS gravity from a (3+1)-dimensional
point of view. A partial dimensional reduction of Einstein-Hilbert action with a negative cosmological
constant in four dimensions led to a black hole in AdS2 gravity. This method allowed us to effectively split
the (3+1)-dimensional spacetime into a (1+1)-dimensional black hole and a remainder, which could be
interpreted as a ‘bath’. We computed the fine-grained entropy of the radiation using geodesic lengths in the
(3+1)-dimensional spacetime. We then focus on the finite temperature case and described the dynamics by
introducing time-dependence into the parameter controlling the reduction. Finally, studying the entropy of
the radiation over time led to a geometric representation of the Page curve. The appearance of the island
region is explained in a natural and intuitive fashion. Actually, we could reproduce the Page curve simply
from the Rye-Takayanagi prescription, without making use of the island formula.

Key words: 2D Gravity, Information Paradox, Partial Dimensional Reduction, AdS Black Holes

1. Introduction
It is widely believed that the resolution of the information loss paradox could be the key to gaining a deeper
insight into the quantum nature of gravity. The standard calculation done by Hawking predicts that the
process of black hole formation and evaporation breaks the principle of unitary time evolution and leads to
a monotonic increase of the fine-grained entanglement entropy of the radiation. However, the requirement
of information conservation implies that the entanglement entropy of the quantum fields outside the black
hole should not exceed the course-grained limit and must follow the so-called Page curve instead [1]. How
this kind of behavior might arise is a topic that has received a lot of attention recently.
One proposal that stands out is based on the holographic principle: replacing the black hole by its dual
representation, it is clear that the process must be unitary. One would like to find the analogous statement
from the gravity point of view, e.g. using holographic entropy tools such as the Ryu-Takayanagi formula
and its covariant extension [2]. A recent progress in this proposal is based on the idea that the fine-grained
entropy of the Hawking radiation can receive an extra contribution from the so-called “island” [3,4].
According to this resolution, the higher-dimensional geometry connects the radiation to the black hole
interior, such that (at late times) the black hole interior becomes part of the entanglement wedge of the

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‫ دااﮕشنه ﻢﻗ ﻰتﻌنﺻ‬،1403 ‫ تیر ماه‬18-17
8th Iranian Conference on Mathematical Physics

radiation or ‘bath’. This led to a new rule for computing the entropy in gravitational systems, now known
as the “Island Rule”:
Area[𝜕𝐼]
𝑆𝑅𝐴𝐷 = min 𝐼 {ext 𝐼 [𝑆[Rad ∪ I] + ]} (1)
4𝐺

Most of the recent work takes place in two spacetime dimensions and in Anti-de Sitter space, which makes
the calculation of entanglement entropy more tractable [5,6].
In this paper, our goal is to shed light on the origin of the Page curve and the island phenomenon by
‘geometrizing’ both concepts in a relatively simple setting. This aim is followed in [7] by viewing the 2D
black hole in JT gravity as the dimensional reduction of part of the 3D BTZ geometry. They present a model
of evaporation of a two-dimensional black hole that is coupled to a heat bath consisting of a thermal CFT.
Now, we want to develop this method to a 4D spacetime. The idea is to do a dimensional reduction of part
of the 4D AdS-Schwarzschild geometry that results in a well-known 2D dilaton gravity, AdS2 ground-state
[8]. Instead of applying a full dimensional reduction, we will reduce over part of the range of the angular
coordinate; thereby, we effectively split the geometry into two parts: an AdS2 black hole, and the remainder
of the 4D geometry, that could be interpreted as a ‘bath’, and we can now compute the entanglement entropy
of an interval in the bath system using RT surfaces.
We can introduce dynamics into the finite temperature JT black hole system by giving time-dependence to
the parameter that controls the dimensional reduction. In this way, we can let the black hole ‘geometrically’
evaporate. Finally, computing the entropy of the entire bath system for the ‘geometric evaporation’, we
obtain a Page curve for the radiation entropy.
This paper is organized as follows. In section 2 we construct our setup and briefly discuss how to obtain
the 2D black hole from AdS4 by partial dimensional reduction. In section 3, we compute the generalized
entropy. Then, we will introduce dynamics in section 4, and allow the black hole to slowly evaporate.
.
2. Geometry
−3
We start with the 4D Einstein-Hilbert action with a negative cosmological constant: Λ = < 0 . This
𝑙2
action gives us the 4D AdS-Schwarzschild black hole.
−1
𝑟2 𝑅 𝑟2 𝑅
𝑑𝑠 2 = − ( 𝑙2 − 𝑟 + 1) 𝑑𝑡 2 + ( 𝑙2 − 𝑟 + 1) 𝑑𝑟 2 + 𝑙 2 𝑟 2 (𝑑𝜃 2 + 𝑠𝑖𝑛2 𝜃 𝑑𝜑 2 )

Now, consider the spherically symmetric ansatz for the metric:

𝑑𝑠 2 = 𝑔𝜇𝜐 𝑑𝑥 𝜇 𝑑𝑥 𝜐 = ℎ𝑖𝑗 𝑑𝑥 𝑖 𝑑𝑥 𝑗 + 𝑙 2 𝑒 −2𝜙 (𝑑𝜃 2 + 𝑠𝑖𝑛2 𝜃 𝑑𝜑 2 ) (2)

Where 𝜙 is the dilaton field. Integrating over 𝜃, from 0 to 𝜋, and 𝜑, from 0 to 2𝜋𝛼, leads to the Reduced-
Einstein-Hilbert action:

𝑙𝜋 2 𝛼
𝑆𝑅𝐸𝐻 = 16𝜋𝐺 (4) ∫ 𝑑 2 𝑥 √−ℎ [𝑒 −2𝜙 (𝑅 (2) + 2(∇∅)2 − 2Λ)] + Surface term + 𝑆𝑚𝑎𝑡𝑡𝑒𝑟 (3)

Here, we accounted for a partial reduction controlled by the parameter 𝛼 ∈ (0,1] for reasons that will become
clear later. Therefore, we have a partial-reduced theory with respect to the 𝜑 coordinate; while it is full-
reduced with respect to 𝜃. Now, defining a dilaton field as: X = 𝑙𝜋 2 𝛼 𝑒 −2𝜙 , leading to the general form of
2D-dilaton gravity for our reduced theory:

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8th Iranian Conference on Mathematical Physics

1
𝑆𝑅𝐸𝐻 = 16𝜋𝐺 (4) ∫ 𝑑 2 𝑥 √−ℎ [X 𝑅 (2) − U(X)(∇X)2 − 2Λ V(X)] + 𝑆𝑚𝑎𝑡𝑡𝑒𝑟 (4)
−1
Where the kinetic and potential functions will be, respectively: U(X) = (X)−1 and V(X) = X. This is a
2
well-known dilaton gravity, AdS2 ground-state [8], with solution:

1 1 −1
𝑑𝑠 2 = − (𝑎2 𝑟 2 − 𝑎𝑟 + 1) 𝑑𝑡 2 + (𝑎2 𝑟 2 − 𝑎𝑟 + 1) 𝑑𝑟 2 (5)
−2𝜙 2 2
𝑒 =𝑎 𝑟 (6)

|Λ|
With 𝑎 = √ 3 . It is easy to see that for 𝑎−1 = 𝑅, we recognize precisely the above metric in the first part
of the 4D AdS-Schwarzschild metric. Consequently, the 4D AdS-Schwarzschild black hole splits into two
parts: an AdS2 ground-state and the remainder of the 4D geometry that could be interpreted as thermal
bath. Now, it is straightforward to do the partial reduction with respect to the 𝜃-coordinate, along with 𝜑.
Integrating over 𝜃, from 0 to 𝜋𝛽, and again over 𝜑, from 0 to 2𝜋𝛼, leads to the same form for the reduced
𝛼
action as in (4); but in this case with the dilaton field X′ = 𝑙𝜋 2 (2𝜋𝛽 − sin 2𝜋𝛽) 𝑒 −2𝜙 instead of X. Note
that 𝛽 ∈ (0,1] is another parameter which control the partial reduction over 𝜃.

3. Generalized Entropy
In [9,7], the generalized entropy for an interval in the extremal black hole + bath system was computed
from the two and three dimensional point of view, respectively. Here, instead, we want to compute the
generalized entropy from the point of view of AdS4.
Based on the same logic as in the 3D case, the fine-grained entropy of the CFT/bath system in our model
will be given by the length of the geodesic. This can be easily computed using embedding coordinates. The
geodesic distance ∆𝑠 between two points 𝑠1 , 𝑠2 in terms of embedding coordinates is given by [10]:

∆𝑠
−𝑙 2 cosh ( 𝑙 ) = 𝑌𝜇 (𝑠1 ) 𝑌𝜇 ( 𝑠2 ) (7)

For the AdS-Schwarzschild black hole, the embedding coordinates are computed in [11]. Using these
coordinate we have:

∆𝑠 √𝐴(𝑟1) 𝐴(𝑟2 ) 4√𝐵(𝑟1 ) 𝐵(𝑟2 ) 𝑏

−𝑙 2 cosh ( )= 2
cos 𝑎(𝑡1 − 𝑡2 ) − 2
cos (𝑡1 − 𝑡2 )
𝑙 𝑎 𝑎 2
+ 𝑟1 𝑟2 cosh 𝜃1 cosh 𝜃2 [1 − cosh(𝜑1 − 𝜑2 )]

𝑟2 𝑅
Where 𝐴(𝑟) and 𝐵(𝑟) are chosen as: 𝐴(𝑟) = 1 + 𝑙2 , 𝐵(𝑟) = . also, a, b are two constants. Now, in a
𝑙
fixed time slice and for two points at the same radius and 𝜃 one can expand for 𝑟 → ∞ to find:

2 𝛽
∆𝑠 ≈ 𝑙 log [ 𝑙2 [ 1−𝛽 − (1 − cosh ∆𝜑)]] + UV cutoff (8)

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8th Iranian Conference on Mathematical Physics

𝛽 is another constant that is computed in terms of a and b. Here we consider the full-reduction over 𝜃, so
assuming that 𝜃 → 0. This geodesic leads to an entropy

∆𝑠 𝑙 2 𝛽
𝑆= = log [ 𝑙2 [ 1−𝛽 − (1 − cosh ∆𝜑)]] (9)
4G 4G

Where we dropped the UV cutoff. It is straightforward to generalize the above result to the case of 𝜃-partial
reduction. The entropy will be:

𝑙 2 𝛽
𝑆= log [ 𝑙2 [ 1−𝛽 − cosh 𝜃 cosh(𝜃 + 𝛾) (1 − cosℎ ∆𝜑)]] (10)
4G

Again, we dropped the UV cutoff and defined 𝛾 ≡ 𝜃2 − 𝜃1 .

4. Dynamical Evaporation
In order to find the time-dependent entropy to obtain the Page curve, we should find the dilaton as a function
of time. Therefore, we start with the Gibbons-Hawking boundary action for a 4D geometry. Following the
same procedure as for the 4D Einstein-Hilbert action, one finds the Reduced Gibbons-Hawking boundary
action (again, a partial reduction over 𝜑 and a full reduction over 𝜃):

𝑙𝜋 2 𝛼
𝑆𝑅𝐺𝐻 = 8𝜋𝐺
∫ 𝑑𝑡 √−ℎ𝑡𝑡 𝑒 −2∅𝑏 𝐾1 + surface term (11)

where 𝐾1 is the extrinsic curvature on the boundary of a 2D-manifold, 𝑒 −2∅𝑏 is the boundary value of the
dilaton 𝑒 −2𝜙 , and the surface term is exactly equal to the surface term in (3); but with a minus sign, so it
will be cancelled in the total action.
It can be shown that the kinetic term, (∇X)2, in (4) is a subdominant contribution near the horizon of an
extremal black hole, leading to reduction of the action 𝑆𝑅𝐸𝐻 to the JT action [12]. Therefore, we can follow
the same steps as for the JT, to show that our reduced boundary action takes the Schwarzian form [13]:

1
𝑆𝑅𝐺𝐻 = 8𝜋𝐺 ∫ 𝑑𝑡 Xr {𝜏(𝑡) , 𝑡} (12)

with definition Xr ≡ 𝜖 Xb and Xb = 𝑙𝜋 2 𝛼 𝑒 −2∅𝑏 is the boundary value of the time-dependent dilaton X.
Now, variation of 𝑆𝑅𝐺𝐻 with respect to 𝜏(𝑡) gives the equation of motion. To represent the interaction with
the CFT-bath, we add an extra term to the equation, equal to the incoming minus the outgoing energy flux
as in [7,14,15]. Solving the equation will give:

𝐴
Xr = 𝜖𝑙𝜋 2 𝑒 −2∅𝑏 𝛼(𝑡) = Xr0 (1 − 2 𝑡) (13)

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8th Iranian Conference on Mathematical Physics

𝑐 𝐺
with Xr0 = 𝜖𝑙𝜋 2 𝑒 −2∅𝑏 and 𝐴 = 𝜖 6 , where c is the central charge. Now, one can easily calculate the
X0r
time-dependent entropy. ∆𝜑 before and after the Page time can be written in terms of the time-dependent
parameter of reduction, 𝛼(𝑡), as: ∆𝜑 𝑡<𝑡Page = 2𝜋(1 − 𝛼(𝑡)) and ∆𝜑 𝑡>𝑡Page = 2𝜋 𝛼(𝑡). Consequently,
the entropy will be:

𝑙 2 𝛽
𝑆 𝑡<𝑡Page = log [ 𝑙2 [ 1−𝛽 − (1 − cosh 𝜋𝐴𝑡)]] (14)
4G

𝑙 2 𝛽 𝐴𝑡
𝑆 𝑡>𝑡Page = log [ 𝑙2 [ 1−𝛽 − (1 − cosh 2𝜋 (1 − ))]] (15)
4G 2

5. Discussion
In this paper, we investigated an AdS2 black hole coupled to a bath from a four-dimensional, geometrical
perspective. By performing a partial dimensional reduction, we effectively split the four-dimensional
spacetime into a two-dimensional black hole and a remainder, takes the role of the bath. This procedure
allowed us to compute the entropy of the bath/radiation by simply computing geodesic lengths in the four-
dimensional spacetime. By making the dimensional reduction parameter α time-dependent, we could model
the dynamics of the 4D AdS-Schwarzschild system and allowed the 2D black hole to evaporate.

Figure 1. The qualitative behavior of the entropy for ∆𝜑 = 2𝜋(1 − 𝛼(𝑡)) (orange) and ∆𝜑 = 2𝜋𝛼(𝑡)
(purple) shows that it follows the Page curve. To plot we set 𝑙 = 𝐺 = 1, the evaporation rate A = 2, and
𝛽 = 2.

Finally, it is easy to see that by choosing some appropriate values for constants, the entropy of the
radiation/bath system (14,15) follows a Page curve.

References

[1] D.N. Page, Information in black hole radiation, Phys. Rev. Lett. 71 (1993) 3743.
[2] S. Ryu and T. Takayanagi, Holographic derivation of entanglement entropy from AdS/CFT, Phys. Rev.
Lett. 96 (2006) 181602.

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‫ دااﮕشنه ﻢﻗ ﻰتﻌنﺻ‬،1403 ‫ تیر ماه‬18-17
8th Iranian Conference on Mathematical Physics

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