2.4 Explicit near-horizon geometries

Let us now present explicit examples of near-horizon geometries of interest. We will discuss the cases of the extremal Kerr and Reissner–Nordström black holes as well as the extremal Kerr–Newman and Kerr–Newman–AdS black holes. Other near-horizon geometries of interest can be found, e.g., in [88, 121Jump To The Next Citation Point, 203Jump To The Next Citation Point].

2.4.1 Near-horizon geometry of extremal Kerr

The near-horizon geometry of extremal Kerr with angular momentum π’₯ = J can be obtained by the above procedure, starting from the extremal Kerr metric written in usual Boyer–Lindquist coordinates; see the original derivation in [33] as well as in [156Jump To The Next Citation Point, 54]. The result is the NHEK geometry, which is written as (25View Equation) without matter fields and with

α (πœƒ) = 1, Γ (πœƒ ) = J (1 + cos2πœƒ), 2sinπœƒ γ(πœƒ) = ---------, k = 1. (37 ) 1 + cos2πœƒ
The angular momentum only affects the overall scale of the geometry. There is a value πœƒ = arcsin(√3-− 1) ∼ 47 ⋆ degrees for which ∂ t becomes null. For πœƒ < πœƒ < π − πœƒ ⋆ ⋆, ∂ t is spacelike. This feature is a consequence of the presence of the ergoregion in the original Kerr geometry. Near the equator we have a “stretched” AdS3 self-dual orbifold (as the 1 S fiber is streched), while near the poles we have a “squashed” AdS3 self-dual orbifold (as the S1 fiber is squashed).

2.4.2 Near-horizon geometry of extremal Reissner–Nordström

The extremal Reissner–Nordström black hole is determined by only one parameter: the electric charge Q. The mass is β„³ = Q and the horizon radius is r = r = Q + −. This black hole is static and, therefore, its near-horizon geometry takes the form (21View Equation). We have explicitly

ν1 = Q2, ν2 = Q2, e = Q, p = 0. (38 )

2.4.3 Near-horizon geometry of extremal Kerr–Newman

It is useful to collect the different functions characterizing the near-horizon limit of the extremal Kerr–Newman black hole. We use the normalization of the gauge field such that the Lagrangian is proportional to R − FabF ab. The black hole has mass ∘ -------- β„³ = a2 + Q2. The horizon radius is given by r = r = ∘a2--+-Q2- + −. One finds

α(πœƒ) = 1, Γ (πœƒ) = r2 + a2cos2 πœƒ, + (r2+-+-a2)sin-πœƒ -2ar+--- γ(πœƒ) = r2 + a2cos2πœƒ , k = r2 + a2, (39 ) +( 2 2) 2 2 2+ 3 f(πœƒ) = Q r+-+-a-- r+-−-a-cos--πœƒ, e = --Q----. 2ar+ r2+ + a2cos2 πœƒ r2+ + a2
In the limit Q → 0, the NHEK functions (37View Equation) are recovered. The near-horizon geometry of extremal Kerr–Newman is therefore smoothly connected to the near-horizon geometry of Kerr. In the limit a → 0 one finds the near-horizon geometry of the Reissner–Nordström black hole (38View Equation). The limiting procedure is again smooth.

2.4.4 Near-horizon geometry of extremal Kerr–Newman–AdS

As a last example of near-horizon geometry, let us discuss the extremal spinning charged black hole in AdS or Kerr–Newman–AdS black hole in short. The Lagrangian is given by 2 2 L ∼ R + 6βˆ•l − F where l2 > 0. It is useful for the following to start by describing a few properties of the non-extremal Kerr–Newman–AdS black hole. The physical mass, angular momentum, electric and magnetic charges at extremality are expressed in terms of the parameters (M, a,Q ,Q ) e m of the solution as

M aM β„³ = -2-, π’₯ = --2-, (40 ) ΠΠ𝒬 = Qe-, 𝒬 = Qm-, (41 ) e Ξ m Ξ
where Ξ = 1 − a2βˆ•l2 and Q2 = Q2 + Q2 e m. The horizon radius r+(r− ) is defined as the largest (smallest) root, respectively, of
Δr = (r2 + a2)(1 + r2βˆ•l2) − 2M r + Q2. (42 )
Hence, one can trade the parameter M for r+. If one expands Δr up to quadratic order around r+, one finds
Δr = Δ0(r+ − r⋆)(r − r+) + Δ0 (r − r+)2 + O (r − r+)3, (43 )
where Δ0 and r⋆ are defined by
2 2 2 2 Δ0 = 1 +(a βˆ•l + 6r+βˆ•l , ) a2 3r2+ a2 + Q2 Δ0 (r+ − r⋆) = r+ 1 + -2-+ --2-− ----2--- . (44 ) l l r+
In AdS, the parameter r⋆ obeys r− ≤ r⋆ ≤ r+, and coincides with r− and r+ only at extremality. In the flat limit l → ∞, we have Δ0 → 1 and r⋆ → r−. The Hawking temperature is given by
Δ0 (r+ − r⋆) TH = ----2-----2-. (45 ) 4π(r+ + a )
The extremality condition is then r+ = r⋆ = r− or, more explicitly, the following constraint on the four parameters (r+,a,Qe, Qm ),
2 2 2 2 1 + a--+ 3r+-− a-+--Q--= 0. (46 ) l2 l2 r2+

The near-horizon geometry was obtained in [159Jump To The Next Citation Point, 71Jump To The Next Citation Point] (except the coefficient e given here). The result is

ρ2 Δ1 βˆ•2 2ar Ξ Γ (πœƒ) =-+-, α(πœƒ) = --01βˆ•2, k = ----2-+---2-, Δ0 Δ πœƒ Δ0 (r+ + a ) 1βˆ•2 1βˆ•2 2 2 2 2 γ(πœƒ) = Δ-πœƒ--Δ0--(r+-+-a-)sinπœƒ-, e = Qe- r+ −-a-, (47 ) ρ2+Ξ Δ0 r2+ + a2 (r2 + a2)[Q (r2 − a2 cos2πœƒ) + 2Q ar cos πœƒ] f(πœƒ) = --+---------e-+-----2------------m--+-------, 2 ρ+Ξar+
where we defined
a2- 2 2 2 2 2 Δ πœƒ = 1 − l2 cos πœƒ, ρ+ = r+ + a cos πœƒ, 2 2 r2 = r+-+-a--. (48 ) 0 Δ0
The near-horizon geometry of the extremal Kerr–Newman black hole is recovered in the limit l → ∞.
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