Explicit near-horizon geometries

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 , 121

, 203

2.4.1 Near-horizon geometry of extremal Kerr

The near-horizon geometry of extremal Kerr with angular momentum $𝒥 = J$ 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 [156 , 54 ]. The result is the NHEK geometry, which is written as (25 ) without matter fields and with

α (𝜃) = 1, Γ (𝜃 ) = J (1 + cos2𝜃), 2sin𝜃 γ(𝜃) = ---------, k = 1. (37 ) 1 + cos2𝜃

The angular momentum only affects the overall scale of the geometry. There is a value $𝜃 = arcsin(√3-− 1) ∼ 47 ⋆$ degrees for which $∂ t$ becomes null. For $𝜃 < 𝜃 < π − 𝜃 ⋆ ⋆$ , $∂ t$ 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 $ℳ = Q$ and the horizon radius is $r = r = Q + −$ . This black hole is static and, therefore, its near-horizon geometry takes the form (21 ). We have explicitly

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 $∘ -------- ℳ = a2 + Q2$ . The horizon radius is given by $r = r = ∘a2--+-Q2- + −$ . One finds

α(𝜃) = 1, Γ (𝜃) = r2 + a2cos2 𝜃, + (r2+-+-a2)sin-𝜃 -2ar+--- γ(𝜃) = r2 + a2cos2𝜃 , k = r2 + a2, (39 ) +( 2 2) 2 2 2+ 3 f(𝜃) = Q r+-+-a-- r+-−-a-cos--𝜃, e = --Q----. 2ar+ r2+ + a2cos2 𝜃 r2+ + a2

In the limit $Q → 0$ , the NHEK functions (37

) 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 (38

). 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 $2 2 L ∼ R + 6∕l − F$ 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 ℳ = -2-, 𝒥 = --2-, (40 ) Ξ Ξ 𝒬 = Qe-, 𝒬 = Qm-, (41 ) e Ξ m Ξ

where $Ξ = 1 − a2∕l2$ and $Q2 = Q2 + Q2 e m$ . The horizon radius $r+(r− )$ is defined as the largest (smallest) root, respectively, of

Hence, one can trade the parameter $M$ for $r+$ . If one expands $Δr$ up to quadratic order around $r+$ , one finds

where $Δ0$ and $r⋆$ are defined by

2 2 2 2 Δ0 = 1 +(a ∕l + 6r+∕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