5.2 Entropy of 3D Banados–Teitelboim–Zanelli (BTZ) black hole

5.2.1 BTZ black-hole geometry

The black-hole solution in three-dimensional gravity with negative cosmological constant was first obtained in [6] (see also [5] for global analysis of the solution). We start with the black-hole metric written in a form that makes it similar to the four-dimensional Kerr metric. Since we are interested in its thermodynamic behavior, we write the metric in the Euclidean form:

ds2 = f(r)dτ2 + f−1(r)dr2 + r2(dϕ + N (r)dτ)2 , (224 )
where the metric functions f (r) and N (r) read
r2 j2 (r2 −-r2+-)(r2-+-|r− |2) j- f(r) = l2 − r2 − m = l2r2 , N (r) = − r2 (225 )
and we use the notation
2 ∘ ---------- 2 ∘ ---------- r2+ = ml--(1 + 1 + (2j-)2) , |r− |2 = ml--( 1 + (2j-)2 − 1) (226 ) 2 ml 2 ml
Obviously one has that r+|r− | = jl. The coordinate ϕ in Eq. (224View Equation) is assumed to be periodic with period 2π.

In order to transform the metric (224View Equation) to a Lorentzian signature we need to make the analytic transformation τ → it, j → − ij so that

∘ -----( ∘ ----------) 1∕2 ml2 2j r+ → rL+ = ---- (1 + 1 − (---)2 ) , 2 ml ∘ -----( ∘ ----------) 1∕2 |r | → ı rL = ml2- (1 − 1 − (2j-)2) , (227 ) − − 2 ml
where rL+ and rL− are the values in the Lorentzian spacetime. These are the respective radii of the outer and inner horizons of the Lorentzian black hole in (2 + 1 ) dimensions. Therefore, we must always apply the transformation (227View Equation) after carrying out all calculations in the Euclidean geometry in order to obtain the result for the Lorentzian black hole. The Lorentzian version of the metric (224View Equation) describes a black hole with mass m and angular momentum J = 2j. The outer horizon is located at r = r+; the respective inverse Hawking temperature is
4π 2πr l2 βH = -′---- = -2---+---2 . (228 ) f (r+) r+ + |r− |
In the (τ,r) sector of the metric (224View Equation) there is no conical singularity at the horizon if the Euclidean time τ is periodic with period β H. The horizon Σ is a one-dimensional space with metric ds2 = l2dψ2 Σ, where r+ |r−| ψ = -l ϕ − -l2-τ is a natural coordinate on the horizon.

The BTZ space is obtained from the three-dimensional maximally-symmetric hyperbolic space H3 (sometimes called the global Euclidean anti-de Sitter space) by making certain identifications. In order to see this one may use the coordinate transformation

r |r | r |r | ψ = -+ϕ − -−--τ , 𝜃 = -+-τ + --−-ϕ l l2 l l2 − 1 r2+-+-|r− |2 1∕2 cosh ρ = (r2 + |r− |2 ) . (229 )
In new coordinates (ρ, 𝜃,ψ) the BTZ metric takes the form
( ) ds2 = l2 dρ2 + cosh2ρd ψ2 + sinh2 ρd 𝜃2 , (230 )
which is the metric on the hyperbolic space H3. In this metric the BTZ geometry is defined by identifications

i) 𝜃 → 𝜃 + 2π

ii) 𝜃 → 𝜃 + 2π |r−| , ψ → ψ + 2 πr+ . l l

The outer horizon r = r + in the coordinate system (ρ,𝜃,ψ) is located at ρ = 0 and ψ is the angular coordinate on the horizon. Notice that the geodesic distance σ between two points with coordinates (ρ,ψ, 𝜃) and (ρ,ψ ′,𝜃′) is

′ ′ sinh2 σ- = cosh2 ρ sinh2 ψ-−-ψ-+ sinh2 ρ sin2 𝜃 −-𝜃-. (231 ) 2l 2 2

5.2.2 Heat kernel on regular BTZ geometry

Consider a scalar field with the operator 2 2 𝒟 = − (∇ + ξ∕l ). The maximally-symmetric constant-curvature space is a nice example of a curved space in which the heat equation (∂s + 𝒟 )K (x,x ′,s) = 0 has a simple, exact, solution. The heat kernel in this case is a function of the geodesic distance σ between two points x and x′. On the global space H3 one finds

---1-------σ∕l--- − σ24s− μ s2 KH3 (σ,s) = (4 πs)3∕2 sinh (σ∕l)e l , (232 )
where μ = 1 − ξ. The regular BTZ geometry is defined by identifications i and ii defined above. As is seen from Eq. (231View Equation) the geodesic distance and the heat kernel (232View Equation), expressed in coordinates (ρ,ψ,𝜃 ), are automatically invariant under identification i. Thus, it remains to maintain identification ii. This is done by summing over images
+∑∞ KBTZ (x,x′,s) = KH (ρ, ρ′, ψ − ψ ′ + 2π r+n, 𝜃 − 𝜃′ + 2π|r− |n) . (233 ) n=−∞ 3 l l
Using the path integral representation of the heat kernel we would say that the n = 0 term in Eq. (233View Equation) is due to the direct way of connecting points x and x′ in the path integral. On the other hand, the n ⁄= 0 terms are due to uncontractible winding paths that go n times around the circle.

5.2.3 Heat kernel on conical BTZ geometry

The conical BTZ geometry, which is relevant to the entanglement entropy calculation, is obtained from global hyperbolic space H3 by replacing identification i as follows

′ i) 𝜃 → 𝜃 + 2 πα
and not changing identification ii. For α ⁄= 1 this Euclidean space has a conical singularity at the horizon (ρ = 0). The heat kernel on the conical BTZ geometry is constructed via the heat kernel (233View Equation) on the regular BTZ space by means of the Sommerfeld formula (22View Equation)
′ ′ 1 ∫ w ′ KBTZ α(x,x ,s) = KBTZ (x,x ,s) + ---- cot --- KBTZ (𝜃 − 𝜃 + w, s) dw , (234 ) 4πα Γ 2α
where KBTZ is the heat kernel (233View Equation). The contour Γ is defined in Eq. (22View Equation).

For the trace of the heat kernel (234View Equation) one finds [171Jump To The Next Citation Point] after computing by residues the contour integral

(VBT Zα A+ ) e−μ¯s TrKBTZ α = ---3-- + ----(2πα)c2(α ) ¯s -----3∕2 l l (4π¯s) 2 e−μ¯s A ∞∑ sinh Δψn- e− Δψ4n¯s +2π -----3∕2--+-¯s ------α-- -----2 Δ-ψn------2 γn-, (235 ) (4π¯s) l n=1 sinh Δψn (sinh -2α-+ sinh 2α)
where ¯s = s∕l2, γ = A n ∕l n − and Δ ψ = A n ∕l n + (A = 2πr + + and A = 2πr − −). Notice that we have already made the analytical continuation to the values of r+ and r− in the Lorentzian geometry.

5.2.4 The entropy

When the trace of the heat kernel on the conical geometry is known one may compute the entanglement entropy by using the replica trick. Then, the entropy is the sum of UV divergent and UV finite parts [171Jump To The Next Citation Point]

S = S + S , (236 ) ent div fin
where the UV divergent part is
∫ √ --- -A+--- ∞ -ds- −μs∕l2 -A+--- −1 --μ-π Sdiv = 24√ π 𝜖2 s3∕2e = 12√ π(𝜖 − l ). (237 )
This divergence is renormalized by the standard renormalization of Newton’s constant
∫ ∞ ----1--- = ---1--- + -1- ---1--- -ds-e−μs∕l2 (238 ) 16 πGren 16 πGB 12 (4π)3∕2 𝜖2 s3∕2
in the three-dimensional gravitational action.

The UV finite part in the entropy is

∑∞ Sfin = sn, n=1 √- 1 e− μA¯+n (A¯+n sinhA¯+n − ¯A− nsinh ¯A− n) sn = 2n-(cosh-¯A--n-−-cosh-¯A--n)(1 + ¯A+n coth A¯+n − -----(coshA¯-n-−--coshA¯-n-)----),(239 ) + − + −
where ¯A± = 2πr± ∕l.

After the renormalization of Newton’s constant, the complete entropy of the BTZ black hole, S = S + S BTZ BH ent, is a rather complicated function of the area of inner and outer horizons. Approximating in Eq. (239View Equation) the infinite sum by an integral one finds [171Jump To The Next Citation Point]

A ∫ ∞ S = ---+--+ s(x) dx , 4Gren ¯A+√ - ( ) 1 e− μx (x sinh x − kx sinhkx ) s(x) = 2x-(cosh-x −-coshkx-) 1 + xcoth x − --(cosh-x-−-coshkx-)-- , (240 )
where k = A − ∕A+. The second term on the right-hand side of Eq. (240View Equation) can be considered to be the one-loop quantum (UV-finite) correction to the classical entropy of a black hole.

For large enough ¯A ≡ A+-> > 1 + l the integral in Eq. (240View Equation) goes to zero exponentially and we have the classical Bekenstein–Hawking formula for entropy. On the other hand, for small ¯ A+, the integral in Eq. (240View Equation) behaves logarithmically so that one has [171Jump To The Next Citation Point]

√ -- S = A+--+ --μA+--− 1ln A+--+ O ((A+-)2). (241 ) BTZ 4G 6 l 6 l l
This logarithmic behavior for small values of A + (provided the ratio k = A ∕A − + is fixed) is universal and independent of the constant ξ (or μ) in the field operator and the area of the inner horizon (A −) of the black hole. Hence, the rotation parameter J enters Eq. (241View Equation) only via the area A+ of the outer horizon.

The other interesting feature of the entropy (240View Equation) is that it always develops a minimum, which is a solution to the equation

l A+ 4G----= s(-l-). (242 ) ren
This black hole of minimal entropy may be interesting in the context of the final stage of the Hawking evaporation in three dimensions. As follows from the analysis of Mann and Solodukhin [171], the minimum of the entropy occurs for a black hole whose horizon area is of the Planck length, A ∼ l + PL (in threee dimensions lPL ∼ Gren).
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