3.10 Entanglement Entropy of the Kerr–Newman black hole

The geometry of the rotating black hole is more subtle than that of a static black hole: near the horizon the rotating spacetime is no longer a product of a horizon sphere S2 and a two-dimensional disk. The other difficulty with applying the technique of the heat kernel to this case is that the Euclidean version of the geometry requires the rotation parameter to be complex. Nevertheless with some care these difficulties can be overcome and the entanglement entropy of a rotating black hole can be computed along the same lines as for a static black hole [170Jump To The Next Citation Point]. In this section we briefly review the results of Mann and Solodukhin [170Jump To The Next Citation Point].

3.10.1 Euclidean geometry of Kerr–Newman black hole

First we describe the Euclidean geometry in the near-horizon limit of the Kerr–Newmann black hole. The Euclidean Kerr–Newman metric can be written in the form

2 2 ds2 = ρˆ-dr2 + ---ˆΔ-ˆρ----ω2 + ˆρ2(d𝜃2 + sin2 𝜃&tidle;ω2), (96 ) E ˆΔ (r2 − ˆa2)2
where the Euclidean time is t = ıτ and the rotation and charge parameters have also been transformed a = ıˆa, q = ıˆq, so that the metric (96View Equation) is purely real. Here ˆΔ (r) = (r − ˆr+ )(r − ˆr− ), where √ ------------- ˆr± = m ± m2 + ˆa2 + ˆq2, the quantities ω and &tidle;ω take the form
(r2 − ˆa2) (r2 − aˆ2 ) ˆa ω = ----2---(d τ − ˆasin2𝜃dϕ ), &tidle;ω = ----2----(dϕ + --2----2-dτ) (97 ) ˆρ ˆρ (r − ˆa )
with ρˆ2 = r2 − ˆa2 cos2𝜃. This spacetime has a pair of orthogonal Killing vectors
ˆa K = ∂τ − -2----2∂ϕ , K&tidle; = ˆasin2𝜃∂τ + ∂ϕ, (98 ) r − ˆa
which are the respective analogs of the vectors ∂ τ and ∂ϕ in the (Euclidean) Schwarzschild case. The horizon surface Σ defined by r = ˆr+ is the stationary surface of the Killing vector K. Near this surface the metric (96View Equation) is approximately
2 2 2 2 dsE = dsΣ + ρˆ+dsC2 , (99 )
where 2 2 2 2 ˆρ+ = ˆr+ − ˆa cos 𝜃 and
2 2 2 (ˆr2+ − ˆa2)2 2 2 dsΣ = ρˆ+d 𝜃 + -----2----sin 𝜃dψ (100 ) ˆρ+
is the metric on the horizon surface Σ up to O (x2), where variable x is defined by the relation (r − ˆr ) = γx2 + 4 and γ = 2√m2--+--ˆa2 +-qˆ2. The angle co-ordinate ψ = ϕ + --ˆa---τ (rˆ2+− ˆa2) and is well defined on Σ. The metric ds2C2 is that of a two-dimensional disk C2
γ2x2 ds2C2 = dx2 + --4-d χ2 (101 ) 4ˆρ+
attached to Σ at a point (𝜃, ψ), where χ = τ − ˆa sin2 𝜃 ϕ is an angle co-ordinate on C 2.

Regularity of the metric near the horizon implies the identifications ψ ↔ ψ + 2π and χ ↔ χ + 4π γ−1ˆρ2+. For this latter condition to hold, independently of 𝜃 on the horizon, it is also necessary to identify (τ, ϕ) with (τ + 2πβH , ϕ − 2πΩ βH ), where Ω = -2ˆa-2-- (ˆr+−ˆa ) is the (complex) angular velocity and 2 2 √ --2----2---2- βH = (ˆr+ − ˆa )∕ m + ˆa + ˆq. The identified points have the same coordinate ψ.

Therefore, near Σ we have the following description of the Euclidean Kerr–Newman geometry: attached to every point (𝜃,ψ) of the horizon is a two-dimensional disk C 2 with coordinates (x,χ). The periodic identification of points on C2 holds independently for different points on the horizon Σ, even though χ is not a global coordinate. As in the static case, there is an abelian isometry generated by the Killing vector K, whose fixed set is Σ. Locally we have K = ∂χ. The periodicity is in the direction of the vector K and the resulting Euclidean space E is a regular manifold.

Now consider closing the trajectory of K with an arbitrary period β ⁄= βH. This implies the identification (τ + 2πβ, ϕ − 2π Ωβ ), and the metric on C2 becomes

2 2 2 2 2 dsC2,α = dx + α x dχ¯ , (102 )
where 2 2 2 −1 χ = βρˆ+(ˆr+ − ˆa ) ¯χ is a new angular coordinate, with period 2π. This is the metric of a two dimensional cone with angular deficit δ = 2π(1 − α), β α ≡ βH--. With this new identification the metric (96View Equation) now describes the Euclidean conical space E α with singular surface Σ.

The difference of the Kerr–Newman metric from the static case considered above is that the Euclidean space near the bifurcation surface is not a direct product of the surface Σ and two-dimensional cone C2,α. Instead, it is a nontrivial foliation of C2,α over Σ. However, this foliation shares certain common features with the static case. Namely, the invariants constructed from quadratic combinations of extrinsic curvature of Σ vanish identically.

3.10.2 Extrinsic curvature of the horizon

In the case of a static black hole we have argued that the presence of an abelian isometry with horizon being the stationary point of the isometry guarantees that the extrinsic curvature identically vanishes on the horizon. In fact this is also true in the case of a rotating black hole. The role of the abelian isometry generated by the Killing vector K is less evident in this case. That is why, in this subsection, following the analysis of [170Jump To The Next Citation Point], we explicitly evaluate the extrinsic curvature for the Kerr–Newman black hole and demonstrate that quadratic invariants, that can be constructed with the help of the extrinsic curvature, vanish on the horizon.

With respect to the Euclidean metric (96View Equation) we may define a pair of orthonormal vectors μ {na = na∂μ , a = 1,2}:

┌ --- ││ Δˆ (r2 − ˆa2) − ˆa nr1 = ∘ -2-; n τ2 = -∘-------, nϕ2 = ∘----- . (103 ) ˆρ Δˆˆρ2 Δˆˆρ2
Covariantly, these are
∘ --- ┌│ --- ┌│ --- 1 ˆρ2 2 │∘ ˆΔ 2 │∘ Δˆ 2 nr = -ˆ-; n τ = -2 , nϕ = − --2ˆasin 𝜃 . (104 ) Δ ˆρ ρˆ
The vectors n1 and n2 are normal to the horizon surface Σ (defined as r = r+, Δ (r = r+ ) = 0), which is a two-dimensional surface with induced metric γ = g − n1n1 − n2n2 μν μν μ ν μ ν. With respect to the normal vectors a n , a = 1,2 one defines the extrinsic curvatures of the surface Σ: a α β a κμν = − γμγν ∇ αnβ. The exact expression for the components of extrinsic curvature is given in [170Jump To The Next Citation Point]. The trace of the extrinsic curvature, κa = κaμνg μν,
┌│ --- 1 2r│∘ ˆΔ-- 2 κ = − ˆρ2 ˆρ2 , κ = 0 (105 )
vanishes when restricted to the horizon surface Σ defined by condition ˆΔ (r = ˆr+) = 0. Moreover, the quadratic combinations
2r2ˆΔ 2ˆa2cos2 𝜃ˆΔ κ1μνκ μ1ν= ---6- , κ2μνκ μ2ν= -----6----- (106 ) ρˆ ˆρ
vanish on the horizon Σ. Consequently, we have κaμνκaμν = 0 on the horizon.

3.10.3 Entropy

Applying the conical singularity method to calculate the entanglement entropy of a rotating black hole we have to verify that i) the curvature singularity at the horizon of a stationary black hole behaves in the same way as in the static case and ii) there are no extra surface terms in the heat kernel expansion for the rotating black hole. The first point was explicitly checked in [170Jump To The Next Citation Point]: the curvature formulas (56View Equation) – (59View Equation) are still valid in the stationary case. Regarding the second point, it was shown by Dowker [70Jump To The Next Citation Point] that for a generic metric with conical singularity at some surface Σ the only modification of the surface terms in the heat kernel expansion (70View Equation) are due to the extrinsic curvature of Σ. For example, the surface coefficient aΣ 2 may be modified by integrals over Σ of terms a κ κa and a aμν κμνκ. Since, as was shown in Section 3.10.2, these terms identically vanish for the Kerr–Newman metric there is no modification of the surface terms in this case. Thus, the expression for the entropy (82View Equation) remains unchanged in the case of a rotating black hole. The Ricci scalar for the Kerr–Newmann metric is zero, R = 0. The integrals of the projections of Ricci and Riemann tensors over horizon surface are

∫ 2 2 2 2 2 Rijij = 8π(ˆr+-+-qˆ) + 4π ˆq--(ˆr+ −-ˆa-)ln(ˆr+-+-ˆa-) Σ ˆr2+ ˆr2+ ˆa ˆr+ ˆr+ − ˆa ∫ ˆq2( (ˆr2 − ˆa2) ˆr + ˆa ) Rii = 4π-2- 1 + --+------ ln( -+----) . (107 ) Σ ˆr+ 2ˆarˆ+ ˆr+ − ˆa
The analytic continuation of these expressions back to real values of the parameters a and q requires the substitution
2 2 2 2 ˆq = − q , ˆa = − a , ˆr+ = r+ 1- ˆr+-+-ˆa- 2- −1 a-- ˆa ln(ˆr+ − ˆa) = a tan (r+) . (108 )
With these identities the quantum entropy of the Kerr–Newman black hole reads [170Jump To The Next Citation Point]
( 2 ( 2 2 ( )) ) S = -A(Σ-)+ 1-- 1 − -3q- 1 + (r+-+-a-) tan−1 -a- ln 1-, (109 ) KN 48 π𝜖2 45 4r2+ ar+ r+ 𝜖
where 2 2 A(Σ ) = 4π(r+ + a ) is the area of the horizon Σ. In the limit a → 0 this expression reduces to that of the Reissner–Nordström black hole (87View Equation). An interesting and still somewhat puzzling feature of this result is that, in the case of the Kerr black hole, described by the Kerr–Newman metric with vanishing electric charge (q = 0), the logarithmic term in the entropy does not depend on the rotation parameter a and is the same as in the case of the Schwarzschild black hole. In particular for the extreme Kerr black hole (q = 0, m = a) one has
S = A-(Σ-)+ 1--ln r+-. (110 ) Kerr 48π 𝜖2 45 𝜖
The entropy of the Kerr black hole in the brick-wall model was calculated in [45Jump To The Next Citation Point] and a result different from Eq. (109View Equation) was found. However, the subsequent study in [92Jump To The Next Citation Point] has confirmed Eq. (109View Equation).
  Go to previous page Go up Go to next page