7.3 The holographic entanglement entropy of 2D black holes

In order to check the proposal (304View Equation) one needs a solution to the bulk Einstein equations that describes a black hole on the boundary of AdS. In three dimensions a solution of this type is known explicitly [194],
⌊ ( )2 ( ′ )2 ⌋ 2 d-ρ2 1- -1-f′2 −-b2 2 -1--- 1- ′′ -1-f-2 −-b2- 2 ds = 4 ρ2 + ρ ⌈f(x) 1 + 16 f ρ dτ + f(x) 1 + 4 f ρ − 16 f ρ dx ⌉ , (305 )
where the AdS radius is set to unity. At asymptotic infinity (ρ = 0) of the metric (305View Equation) one has the 2D black hole metric
ds22d = f(x)dτ2 + f− 1(x )dx2, (306 )
where f(x) has a simple zero in x = x+. The cavity ΣL is placed at x = L so that x+ ≤ x ≤ L. The regularity of the metric (306View Equation) at the horizon x = x+ requires that 0 ≤ τ ≤ βH, ′ βH = 4π∕f (x+ ). Note that Eq. (305View Equation) is a vacuum solution of the Einstein equations for any function f(x). The regularity of the 3D metric (305View Equation) requires that the constant b should be related to the Hawking temperature of the two-dimensional horizon by b = f′(x+). The geodesic Γ lies in the hypersurface of constant time τ. The induced metric on the hypersurface (ρ, x) has a constant curvature equal to − 2 for any function f(x) and is, thus, related by a coordinate transformation to the metric
dr2 1 ds2τ = --2-+ -dw2. (307 ) 4r r
The exact relation between the two coordinate systems is
( ) 1--z(x) 16f-(x) −-(b2-−-f′x2)ρ w = 8be 16f (x) + (b − f ′)2ρ , ρ x r = f(x)e2z(x)-----------------′2--2, (308 ) (16f (x) + (b − fx) ρ)
where z(x) = b ∫x-dx- 2 L f(x). The equation for the geodesic in the metric (307View Equation) is r = -12-− (w − w0)2 C. The geodesic length between two points lying on the geodesic with radial coordinates r1 and r2 is
( [ √ --------]) 1 1 − 1 − C2r r2 γ(1,2) = -- ln ----√-------2- . 2 1 + 1 − C r r1
The saddle point of the geodesic is at 2 rm = 1∕C. The constant C is determined from the condition that the geodesic Γ joins points 2 (x = x+, ρ = 𝜖 ) and 2 (x = L, ρ = 𝜖 ) lying on the regularized (with regularization parameter 𝜖) boundary. In the limit of small 𝜖 one finds that
C2r+ b2 ∫ L dx -----= 𝜖2------exp (− b -----). 4 f(x+ ) x+f (x)
The length of the geodesic Γ h joining point r+ corresponding to (x = x+, ρ = 𝜖) and thus the saddle point is 1 C2r+ γ(Γ h) = − 2 ln 4.

Now, one has to take into account that, in the AdS/CFT correspondence, the value of Newton’s constant in the bulk is related to the number of quantum fields living in the boundary ∂M. In the AdS3/CFT2 case, one has that -1- = 2c GN 3, where c is the central charge of boundary CFT. Thus, one finds that the holographic entropy (304View Equation)

1 S = -----γ(Γ h) (309 ) 4GN [ ] c 1 c ∫ L dx ′ 2 = --ln--+ --- ----(b − f ) + lnf (L) − ln b , 6 𝜖 12 x+ f(x)
where b = f′(x+ ), indeed coincides with expression (219View Equation) for the holographic entanglement entropy of the 2D black hole in conformal field theory. In particular, for large values of L the holographic formula for the entropy correctly reproduces the entropy of thermal gas cπ Sth = 3 TH L (we remember that TH = f′(x+)∕4 π). This is a consequence of the choice of the minimal surface Γ h made in the proposal (304View Equation).
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