5.1 Entropy of a 2D black hole

In two dimensions the conformal symmetry plays a special role. This has many manifestations. In particular, the conformal symmetry can be used in order to completely reproduce, for a conformal field theory (CFT), the UV finite part of the corresponding gravitational effective action. This is done by integration of the conformal anomaly. For regular two-dimensional spacetimes, the result is the well-known non-local Polyakov action. In the presence of a conical singularity the derivation is essentially the same, although one has to take into account the contribution of the singularity. Consider a two-dimensional CFT characterized by a central charge c. For a regular two-dimensional manifold, the Polyakov action can be written in the form
-c--∫ 1- 2 WPL [M ] = 48π M (2(∇ ψ ) + ψR ) , (213 )
where the field equation for the field ψ is 2 ∇ ψ = R. On a manifold α M with a conical singularity with angle deficit δ = 2π (1 − α ) the Polyakov action is modified by the contribution from the singularity at the horizon Σ (which is just a point in two dimensions) so that [196Jump To The Next Citation Point, 112Jump To The Next Citation Point]
W [M α] = W [M α∕Σ ] + c-(1 − α)ψ + O (1 − α)2, (214 ) PL PL 12 h
where ψ h is the value of the field ψ on the horizon. Applying the replica method to the Polyakov action (214View Equation) one obtains that the corresponding contribution to the entanglement entropy from the UV finite term in the effective action is
c Sfin = ---ψh . (215 ) 12
This result agrees with a derivation of Myers [179] who used the Noether charge method of Wald [221] in order to calculate the entropy. The easiest way to compute the function ψ is to use the conformal gauge gμν = e2σδμν in which ψ = 2σ. Together with the UV divergent part, the complete entanglement entropy in two dimensions is
c c Λ S = --σh + --ln --, (216 ) 6 6 𝜖
where Λ is an IR cut-off.

Let the black-hole geometry be described by a 2D metric

1 ds2bh = f(x)dτ 2 +-----dx2, (217 ) f (x)
where the metric function f (x ) has a simple zero at x = x+. Assume that this black hole is placed inside a box of finite size L so that x ≤ x ≤ L +. In order to get a regular space, one closes the Euclidean time τ with period βH, --4π-- βH = f′(x+). It is easy to see that Eq. (217View Equation) is conformal to the flat disk of radius z0 (∫ lnz = 2βπ-Lxfd(xx) H):
2 2σ 2 2 2 2 dsbh = e z0(dz + z d&tidle;τ∫ ) , (218 ) 1- 2π- x -dx-- βH--- σ = 2 lnf (x) − βH L f(x) + ln 2πz0,
where &tidle;τ = 2πτ βH (0 ≤ &tidle;τ ≤ 2π), 0 ≤ z ≤ 1. So that the entanglement entropy of the 2D black hole takes the form [199Jump To The Next Citation Point, 98]
c ∫ L dx 4π c ( β f 1∕2(L )) S = --- ----(--- − f ′) + --ln -H-------- , (219 ) 12 x+ f(x) βH 6 𝜖
where we omit the irrelevant term that is a function of (Λ, z ) 0 but not of the parameters of the black hole and have retained dependence on the UV regulator 𝜖.

As was shown in [199], the entanglement entropy (219View Equation) is identical to the entropy of the thermal atmosphere of quantum excitations outside the horizon in the “brick-wall” approach of ’t Hooft [214].

The black hole resides inside a finite-sized box and L is the coordinate of the boundary of the box. The coordinate invariant size of the subsystem complimentary to the black hole is ∫ ∘ ----- Linv = Lx+ dx∕ f(x). Two limiting cases are of interest. In the first, the size of the system Linv is taken to infinity. Then, assuming that the black-hole spacetime is asymptotically flat, we obtain that the entanglement entropy (219View Equation) approaches the entropy of the thermal gas,

cπ- S = 3 LinvTH . (220 )
This calculation illustrates an important feature of the entanglement entropy of a black hole placed in a box of volume V. Namely, the entanglement entropy contains a contribution of the thermal gas that, in the limit of large volume in dimension d, takes the form (7View Equation). This is consistent with the thermal nature of the reduced density matrix obtained from the Hartle–Hawking state by tracing over modes inside the horizon.

The other interesting case is when Linv is small. In this case, we find the universal behavior

( ) c Linv R (x+ ) 2 3 S = 6- ln--𝜖- + --24---Linv + O (Linv) . (221 )
The universality of this formula lies in the fact that it does not depend on any characteristics of the black hole (mass, temperature) other than the value of the curvature R(x+ ) at the horizon.

Consider two particular examples.

2D de Sitter spacetime is characterized by the metric function f(x) = 1 − x2 l2 and the Hawking temperature TH = 1∕2πl. In this spacetime the size of the box is bounded from above, Linv ≤ πl. The corresponding entanglement entropy

c ( 1 ) SdS2 = -ln ------tan(TH πLinv) (222 ) 6 πTH 𝜖
is a periodic function of Linv.

The string inspired black hole [222, 169] is described by the metric function f(x) = 1 − e−λx. It described an asymptotically-flat spacetime. The Hawking temperature is TH = λ- 4π. The entanglement entropy in this case is

c ( 1 ) Sstr = 6-ln 2πT--𝜖-sinh (2πTH Linv) . (223 ) H

The entropy in these two examples resembles the entanglement entropy in flat spacetime at zero temperature (11View Equation) and at a finite temperature (12View Equation) respectively.

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