6.4 Local SL (2, ℝ) × SL (2,ℝ ) symmetries

We will now make explicit the local SL (2,ℝ ) × SL (2, ℝ) symmetries of the near-horizon scalar field equations (209View Equation). For this purpose it is convenient to define the “conformal” coordinates (ω ±,y) defined in terms of coordinates (t,r,ϕ′) by (see [68Jump To The Next Citation Point] and [210Jump To The Next Citation Point] for earlier relevant work)
∘ ------- ω+ = r −-r+-e2πTR(ϕ′−ΩRt), r − r− ∘ -r −-r- ′ ω − = ----+-e2πTL(ϕ −ΩLt), (210 ) ∘ -r −-r−- r+ − r− πT (ϕ′− Ω t)+πT (ϕ′−Ω t) y = -------e L L R R . r − r−
The change of coordinates is locally invertible if Δ Ω = ΩL − ΩR ⁄= 0. We choose the chirality Δ Ω > 0, as it will turn out to match the chirality convention in the description of extremal black holes in Section 4.4.

Several choices of coordinate ′ ′ ϕ ∼ ϕ + 2π will lead to independent SL (2,ℝ ) × SL (2,ℝ) symmetries. For the Kerr black hole, there is only one meaningful choice: ϕ′ = ϕ. For the Reissner–Nordström black hole, we identify ϕ′ = χ∕R χ, where χ is the Kaluza–Klein coordinate that allows one to lift the gauge field to higher dimensions, as done in Section 4.3. For the Kerr–Newman black hole, we use, in general, a coordinate system ′ ′ ′ ′ ′ ′ (ϕ ,χ ) ∼ (ϕ ,χ + 2π) ∼ (ϕ + 2π,χ ) parameterized by a SL (2,ℤ ) transformation

′ ϕ = p1ϕ + p2χ∕R χ, χ′ = p3ϕ + p4χ∕R χ, (211 )
with p1p4 − p2p3 = 1 so that
∂ϕ′ = p4∂ϕ − p3R χ∂χ, (212 ) ∂ χ′ = − p2∂ϕ + p1R χ∂χ. (213 )

Let us define locally the vector fields

H1 = i∂+, 1 H0 = i(ω+ ∂+ + -y ∂y), (214 ) 2 H −1 = i(ω+2∂+ + ω+y ∂y − y2∂− ),
and
¯ H1 = i∂− , ¯H = i(ω− ∂ + 1-y∂ ), (215 ) 0 − 2 y H¯−1 = i(ω− 2∂− + ω− y∂y − y2∂+).
These vector fields obey the SL (2,ℝ ) Lie bracket algebra,
[H0, H±1 ] = ∓iH ±1, [H −1,H1 ] = − 2iH0, (216 )
and similarly for (H¯ , ¯H ) 0 ±1. Note that
¯ -i- TLH0 + TRH0 = 2π ∂ϕ′. (217 )
The SL (2,ℝ ) quadratic Casimir is
1 ℋ2 = ℋ¯2 = − H20 + -(H1H − 1 + H − 1H1 ) (218 ) 2 = 1-(y2∂2− y∂y) + y2∂+∂− . (219 ) 4 y
In terms of the coordinates (r,t,ϕ ′), the Casimir becomes
( )2 ℋ2 = − -----r+ −-r−---- ∂ ϕ′ + TL-+-TR-(∂t + ΩR ∂ϕ′) (r − r+ )(4πTR )2 TL Δ Ω r − r ( T − T )2 + ------+----−---- ∂ ϕ′ +-L-----R(∂t + ΩR ∂ϕ′) + ∂rΔ ∂r, (r − r− )(4πTR )2 TL Δ Ω
where Δ (r ) = (r − r )(r − r ) + −.

We will now match the radial wave equation around the Kerr–Newman black hole in the near region (209View Equation) with the eigenvalue equation

ℋ2Φ = l(l + 1)Φ. (220 )
The scalar field has the following eigenvalues ∂tΦ = − iωΦ and ∂ϕ Φ = im Φ. In the case where an electromagnetic field is present, one can perform the uplift (2View Equation) and consider the five-dimensional gauge field (6View Equation). In that case, the eigenvalue of the five-dimensional gauge field under ∂ χ is the electric charge ∂χ Φ = iqeΦ. Let us denote the eigenvalue along ∂ ϕ′ as im ′ ≡ i(p4m − p3qeR χ). Eqs. (209View Equation) and (220View Equation) will match if and only if the two following equations are obeyed
( ) r+ − r− ′ TL ± TR ′ α(r±) = -------- − m + --------(ω − ΩRm ) , (221 ) 4πTR TLΔ Ω
where α(r) has been defined in (200View Equation).

For simplicity, let us first discuss the case of zero probe charge q = 0 e and non-zero probe angular momentum m ⁄= 0. The matching equations then admit a unique solution

a ΩR = 0, ΩL = ----------, (222 ) 2M 2 − Q2 2M 2 − Q2 M (r+ − r− ) TL = ----------, TR = ------------, 4πJ 4πJ
upon choosing ′ ϕ = ϕ (and ′ χ = χ ∕Rχ). This shows in particular that the Kerr black hole has a hidden symmetry, as derived originally in [68Jump To The Next Citation Point].

For probes with zero angular momentum m = 0, but electric charge qe ⁄= 0, there is also a unique solution,

Ω = --Q---, Ω = -----M-Q-------, (223 ) R 2M R χ L (2M 2 − Q2 )Rχ 2 2 TL = (2M---−-Q--)Rχ-, TR = M-(r+-−-r−-)Rχ-, 2πQ3 2πQ3
upon choosing ϕ′ = χ∕R χ (and χ ′ = − ϕ). This shows, in particular, that the Reissner–Nordström black hole admits a hidden symmetry, as pointed out in [81, 77].

Finally, one can more generally solve the matching equation for any probe scalar field whose probe angular momentum and probe charge are related by

p2m − p1qeRχ = 0. (224 )
In that case, one chooses the coordinate system (211View Equation) and the unique solution is then
p2Q p1a + p2M Q ∕Rχ ΩR = 2M--R--, ΩL = ----2M-2 −-Q2----, (225 ) χ -----2M--2 −-Q2------ ----M--(r+-−-r− )---- TL = 2π (2p1J + p2Q3 ∕R χ), TR = 2π (2p1J + p2Q3∕R χ) .
When p1 = 0 and Q ⁄= 0 or p2 = 0 and J ⁄= 0, one recovers the two previous particular cases. The condition (224View Equation) is equivalent to the fact that the scalar field has zero eigenvalue along ∂χ′. Since m and qeR χ are quantized, as derived in (7View Equation), there is always (at least) one solution to (224View Equation) with integers p1 and p 2.

In conclusion, any low energy and low mass scalar probe in the near region (206View Equation) of the Kerr black hole admits a local hidden SL (2,ℝ ) × SL (2,ℝ ) symmetry. Similarly, any low energy, low mass and low charge scalar probe in the near region (206View Equation) of the Reissner–Nordström black hole admits a local hidden SL (2,ℝ ) × SL (2, ℝ) symmetry. In the case of the Kerr–Newman black hole, we noticed that probes obeying (205View Equation) also admit an SL (2,ℝ ) × SL (2,ℝ) hidden symmetry, whose precise realization depends on the ratio between the angular momentum and the electric charge of the probe. For a given ratio (224View Equation), hidden symmetries can be constructed using the coordinate ϕ ′ = p1ϕ + p2χ ∕R χ. Different choices of coordinate ϕ′ are relevant to describe different sectors of the low energy, low mass and low charge dynamics of scalar probes in the near region of the Kerr–Newman black hole. The union of these descriptions cover the entire dynamical phase space in the near region under the approximations (205View Equation) – (206View Equation).


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