7.4 Holographic entanglement entropy of higher dimensional black holes

In higher dimensions there is no known exact solution similar to Eq. (305View Equation). However, a solution in the form of a series expansion in ρ is available. In the rest of this section we focus on the case of boundary dimension 4. Then one finds [131Jump To The Next Citation Point]
dρ2 1 ds2 = --2-+ --gij(x,ρ)dxidxj (310 ) 4ρ ρ g (x, ρ) = g(0)(x) + g(2)ρ + g(4)ρ2 + h(4)ρ2 ln ρ + ..,
where g(0)ij(x) is the boundary metric and coefficient [131Jump To The Next Citation Point]
g(2)ij = − 1-(Rij − 1Rg (0)ij) (311 ) 2 6
is the local covariant function of the boundary metric. Coefficient g(4) is not expressed as a local function of the boundary metric and is related to the stress-energy tensor of the boundary CFT [60Jump To The Next Citation Point], which has an essentially non-local nature. h(4) is a local, covariant, function of the boundary metric and is obtained as a variation of the integrated conformal anomaly with respect to the boundary metric [60Jump To The Next Citation Point]. Its explicit form was computed in [60]. h(4) is a traceless tensor and in mathematics literature it is known as the “obstruction” tensor [122]. The explicit form of h(4) or g(4) is not important if one wants to compute the UV divergence terms in the entropy.

One may choose the boundary metric describing a static spherically-symmetric black hole to take the form

ds2 = f (r)dτ2 + f−1(r)dr2 + r2(d𝜃2 + sin2 𝜃dĪ•2). (312 )
The minimal surface Γ lies in the hypersurface of the constant τ of 5-dimensional spacetime (310View Equation). The induced metric on the hypersurface takes the form
dρ2 1[ dr2 ] ds2τ = ----+ --F -----+ R2(d𝜃2 + sin2𝜃dĪ•2 ) , (313 ) 4ρ2 ρ f (r)
where functions F (r,ρ) = grrg (0) rr and R2 (r,ρ) = g 𝜃𝜃 have a ρ-expansion due to Eq. (310View Equation). The minimal surface Γ can be parameterized by (ρ,𝜃,Ī•). Instead of the radial coordinate r, it is convenient to introduce the coordinate ∫ √ -- y = dr ∕ f so that, near the horizon, one has r = r+ + by2∕4 + O (y4). The coordinate y measures the invariant distance along the radial direction. By spherical symmetry, the area to be minimized is
∘ --------------- ∫ dρ 1 F dy Area(Γ ) = 4 π --R2 --2-+ --(--)2, (314 ) ρ 4ρ ρ dρ
where ρm is the saddle point. In the vicinity of the horizon (y â‰Ē 1), we can neglect the dependence of the functions F(y, ρ) and R2(y,ρ ) on the coordinate y. The minimization of the area of the surface gives the equation
FR2 dy --∘------dρ------= C = const. (315 ) ρ2 41ρ2 + Fρ(ddyρ)2
The area of the minimal surface Γ h is then given by the integral
∫ ρm R2 Area (Γ h) = 2π 2 dρ 𝒜(ρ), 𝒜 = --∘-----C2ρ3. (316 ) 𝜖 ρ2 1 − F-R4
Using Eq. (310View Equation) we find that r2+ g(2)𝜃𝜃(r+) 𝒜 (ρ) = [ρ2 + ρ + ..]. Substituting this expansion into Eq. (316View Equation) we find that the first two terms produce divergences (when 𝜖 goes to zero), which, according to our proposal, are to be interpreted as UV divergences of the entanglement entropy. At the black hole horizon, one has the relation 2R 𝜃𝜃|r = r2(R − Raa ) + +. Putting everything together and applying proposal (304View Equation), one finds for the divergent part
A (Σ) N 2 ∫ 1 1 Sdiv = ----2 N 2 −--- (--Raa − --R) ln 𝜖, (317 ) 4π𝜖 2π Σ 4 6
where A(Σ ) = 4πr2+ is the horizon area.

The logarithmic term in Eq. (317View Equation) is related to the logarithmic divergence as (calculated holographically in [131])

N 2 ∫ 1 1 Wlog = ---2 (-R2μν − ---R2) ln 𝜖 4π 8 24
in the quantum effective action of boundary CFT. This relation is a particular manifestation of the general formula (280View Equation) that relates the logarithmic term in the entropy to the conformal anomalies of type A and B. One notes that in the 𝒩 = 4 superconformal SU (N ) gauge theory one has that N2- A = B = π2.

It should be noted that the UV finite terms and their dependence on the size Linv of the box can be computed in the limit of small Linv. This calculation is given in [202Jump To The Next Citation Point]. In particular, in any even dimension d one finds a universal term in the entropy that takes the form (up to numerical factor) S ∼ rd−2h (r )L2 lnL + (d)𝜃𝜃 + inv inv and is proportional to the value of the “obstruction tensor” on the black hole horizon. The direct calculation of such terms in the entanglement entropy on the CFT side is not yet available.

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