2.3 Near-horizon of extremal spinning geometries

Let us now consider extremal spinning black holes. Let us denote the axis of rotation to be ∂ϕ, where ϕ ∼ ϕ + 2π and let r = r+ be the black-hole horizon. The generator of the horizon is ξ ≡ ∂t + ΩeJxt∂ϕ where ΩeJxt is the extremal angular velocity. We choose a coordinate system such that the coordinate t diverges at the horizon, which is equivalent to the fact that gtt diverges at the horizon. As in the static case, one needs to perform a gauge transformation of parameter (19View Equation), when electrostatic fields are present. One can again interpret this change of gauge parameter as a change of coordinates in a higher-dimensional auxiliary spacetime (2View Equation). The near-horizon limit is then defined as
t- t → r0λ, r → r + λr r, (23 ) + 0 ϕ → ϕ + Ωext r0t, J λ Φexet A → A + --λ- r0dt,
with λ → 0. The scale r0 is again introduced in order to factor out the overall scale of the near-horizon geometry. The additional effect with respect to the static near-horizon limit is the shift in the angle ϕ in order to reach the frame co-moving with the horizon. The horizon generator becomes ξ = λ ∕r ∂ 0 t in the new coordinates. Including the gauge field, one has precisely the relation (22View Equation). As in the static case, any finite energy excitation of the near-horizon geometry is confined and amounts to no net charges in the original (asymptotically flat of AdS) geometry.

One is free to redefine λ as λ → αλ for any α > 0 and, therefore, the near-horizon geometry admits the enhanced symmetry generator

ζ = r∂ − t∂ , (24 ) 0 r t
in addition to ζ− 1 = ∂t and L0 = ∂ϕ. Together ζ0 and ζ− 1 form a non-commutative algebra under the Lie bracket.

Now, contrary to the static case, the existence of a third Killing vector is not guaranteed by geometric considerations. Nevertheless, it turns out that Einstein’s equations derived from the action (1View Equation) imply that there is an additional Killing vector ζ1 in the near-horizon geometry [194Jump To The Next Citation Point, 19] (see also [64] for a geometrical derivation). The vectors ζ−1,ζ0,ζ1 turn out to obey the SL (2,ℝ ) ∼ SO (2,1) algebra. This dynamical enhancement is at the origin of many simplifications in the near-horizon limit. More precisely, one can prove [194Jump To The Next Citation Point] that any stationary and axisymmetric asymptotically-flat or anti-de Sitter extremal black-hole solution of the theory described by the Lagrangian (1View Equation) admits a near-horizon geometry with SL (2,ℝ ) × U(1) isometry. The result also holds in the presence of higher-derivative corrections in the Lagrangian provided that the black hole is big, in the technical sense that the curvature at the horizon remains finite in the limit where the higher-derivative corrections vanish. The general near-horizon geometry of extremal spinning black holes consistent with these symmetries is given by

[ ] 2 2 2 dr2- 2 2 2 2 ds = Γ (𝜃) − r dt + r2 + α(𝜃) d𝜃 + γ(𝜃) (dϕ + krdt ) , e χA = χA (𝜃), AI = fI(𝜃)(dϕ + krdt) − -Id ϕ, (25 ) k
where Γ (𝜃) > 0, γ(𝜃) ≥ 0, χA (𝜃),fI(𝜃) and k, eI ∈ ℝ are fixed by the equations of motion. By inverting t and redefining AI → − AI, we can always set k ≥ 0, eI ≥ 0. The function α (𝜃) ≥ 0 can be removed by redefining 𝜃 but it is left for convenience because some near-horizon geometries are then more easily described.8

The term − eId ϕ k in (25View Equation) is physical since it cannot be gauged away by an allowed gauge transformation. For example, one can check that the near-horizon energy 𝒬 ∂t would be infinite in the Kerr–Newman near-horizon geometry if this term would be omitted. One can alternatively redefine f I(𝜃) = bI(𝜃) + eI∕k and the gauge field takes the form

AI = bI(𝜃)(dϕ + krdt ) + eIrdt. (27 )

The static near-horizon geometry (21View Equation) is recovered upon choosing only SO (3) covariant quantities with a well-defined static limit. This requires k → 0 and it requires the form

I pI b (𝜃) = − ---cos𝜃, (28 ) 4π
where I p are some pure numbers, which are the magnetic charges.

Going back to the spinning case, the SL (2,ℝ ) × U(1) symmetry is generated by

ζ1 = t, ζ0 = t∂t r∂r,
ζ1 = (1 --2- 2r + t2 -- 2)t tr∂r k -- rϕ, L0 = ϕ. (29)
In addition, the generator ζ1 should be accompanied by the gauge transformation of parameter ΛI = − eI∕r so that ℒ ζAI + ∂μΛI = 0 1 μ. Note that all of these symmetries act within a three-dimensional slice of fixed polar angle 𝜃. The metric is also invariant under discrete symmetry, which maps
(t,ϕ) → (− t,− ϕ). (30 )
This is often called the t-ϕ reflection symmetry in black-hole literature. The parity/time reversal transformation (30View Equation) reverses the electromagnetic charges of the solution.

The geometry (25View Equation) is a warped and twisted product of AdS2 × S2. The (r,t) coordinates are analogous to Poincaré coordinates on AdS2 with an horizon at r = 0. One can find global coordinates in the same way that the global coordinates of AdS2 are related to the Poincaré coordinates [33Jump To The Next Citation Point]. Let

2 1∕2 2 1∕2 r = (1 + y ) cosτ + y, tr = (1 + y ) sin τ. (31 )
The new axial angle coordinate φ is chosen so that d ϕ + krdt = dφ + kyd τ, with the result
| | | cosτ + y sin τ | ϕ = φ + k log ||1 +-(1 +-y2-)1∕2sinτ-||. (32 )
In these new coordinates, the near-horizon geometry becomes
[ ] 2 2 2 dy2 2 2 2 2 ds = Γ (𝜃 ) − (1 + y )dτ + -----2 + α(𝜃) d𝜃 + γ(𝜃) (dφ + kyd τ) , 1 + y (33 ) χA = χA (𝜃), AI = fI(𝜃)(dφ + kyd τ) − eIdφ, k
after performing an allowed gauge transformation (as the change of gauge falls into the boundary conditions (115View Equation) derived in Section 4.1). Note that the τ = 0 hypersurface coincides with the t = 0 hypersurface, and that ϕ = φ on this hypersurface. The geometry has two boundaries at y = − ∞ and y = + ∞.

Geodesic completeness of these geometries has not been shown in general, even though it is expected that they are geodesically complete. For the case of the near-horizon geometry of Kerr, geodesic completeness has been proven explicitly in [33Jump To The Next Citation Point] after working out the geodesic equations.

At fixed polar angle 𝜃, the geometry can be described in terms of 3d warped anti-de Sitter geometries; see [8] for a relevant description and [226, 158, 238, 127, 223, 175, 174, 6, 119, 43, 26, 93, 222] for earlier work on these three-dimensional geometries. Warped anti-de Sitter spacetimes are deformations of AdS3, where the S1 fiber is twisted around the AdS 2 base. Because of the identification ϕ ∼ ϕ + 2π, the geometries at fixed 𝜃 are quotients of the warped AdS geometries, which are characterized by the presence of a Killing vector of constant norm (namely ∂ϕ). These quotients are often called self-dual orbifolds by analogy to similar quotients in AdS3 [100Jump To The Next Citation Point].9

The geometries enjoy a global timelike Killing vector (which can be identified as ∂τ) if and only if

k γ(𝜃) < 1, ∀ 𝜃 ∈ [0,π ]. (34 )
If there is no global timelike Killing vector, there is at least one special value of the polar angle 𝜃⋆, where kγ (𝜃⋆) = 1. At that special value, the slice 𝜃 = 𝜃⋆ is locally an ordinary AdS3 spacetime and acquires a local SL (2, ℝ) × SL (2,ℝ ) isometry. At all other values of 𝜃, one SL (2,ℝ ) is broken to U (1). Note that there is still a global time function for each near-horizon geometry. Constant global time τ in the global coordinates (33View Equation) are spacelike surfaces because their normal is timelike,
gab∂aτ ∂bτ = gττ = − (1 + y2)− 1Γ −1(𝜃) < 0. (35 )
Hence, there are no closed timelike curves.

One can show the existence of an attractor mechanism for extremal spinning black holes, which are solutions of the action (1View Equation[17Jump To The Next Citation Point]. According to [17Jump To The Next Citation Point], the complete near-horizon solution is generically independent of the asymptotic data and depends only on the electric charges 𝒬I e, magnetic charges 𝒬I m and angular momentum 𝒥 carried by the black hole, but in special cases there may be some dependence of the near horizon background on this asymptotic data. In all cases, the entropy only depends on the conserved electromagnetic charges and the angular momentum of the black hole and might only jump discontinuously upon changing the asymptotic values of the scalar fields, as it does for static charged black holes [227, 118].

One can generalize the construction of near-horizon extremal geometries to higher dimensions. In five dimensions, there are two independent planes of rotation since the rotation group is a direct product SO (4 ) ∼ SO (3) × SO (3). Assuming the presence of two axial U (1) symmetries ∂ ϕi, i = 1,2 (with fixed points at the poles), one can prove [194] that the near-horizon geometry of a stationary, extremal black-hole solution of the five-dimensional action (1View Equation) possibly supplemented by Chern–Simons terms (4View Equation) is given by

[ ] dr2 ∑2 ds2 = Γ (𝜃) − r2dt2 + ---+ α(𝜃)2d𝜃2 + γij(𝜃 )2(dϕi + kirdt)(dϕj + kjrdt) , (36 ) r2 i,j=1 2 A A I ∑ I i eI i χ = χ (𝜃), A = fi (𝜃 )(dϕ + kirdt) − kidϕ . i=1
In particular, the solutions obtained from the uplift (2View Equation) – (3View Equation) fall into this class. In general, these solutions can be obtained starting from both black holes (with S3 horizon topology) and black rings (with 2 2 S × S horizon topology) [133].
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