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77) t→±∞ t→±∞ t→±∞ t→±∞ exist and satisfy ˜ ± = (Ω± )∗ , Ω H H ˜ ± = (Ω± )∗ , Ω ∞ ∞ ± ˜± ± ˜± ± ˜ ± Ω± + Ω ˜± Ω ∞ Ω∞ = ΩH ΩH + Ω∞ Ω∞ = IdH . 6. 56). The scattering properties of the vector Φ (describing the physical Weyl field φA in the spin-frame (oA , ιA )), are obtained from the results of these theorems via the identifying operator J : L2 ((Σ; dVol); C2 ) −→ H , J Φ := ∆σ 2 ρ2 (r2 + a2 )2 1/4 UΦ . 56), corresponds the wave operator J −1 WJ , J −1 ΩJ , or J −1 W J , for the vector Φ. 7. The theorems above show that the solutions of Eq.

E. the rotation speed of the horizon as perceived by an observer static at infinity. The operator D ∞ induces the same radial translation as DH without the rotation. Both Hamiltonians have the same spaces of incoming (respectively outgoing) data: H− = {Ψ = (ψ0 , ψ1 ) ∈ H; ψ1 = 0} (respectively H+ = {Ψ = (ψ0 , ψ1 ) ∈ H; ψ0 = 0}) . a The constructions of Sec. 8 will indeed be based on asymptotic profiles, but they will be slightly different from the ones used here, so as to make their geometric significance more obvious.

I ∆ρ2 2M ra sin θ  σ2 σ2 We then modify Eq. 53) by isolating the Dirac operator D /S 2 on the 2-sphere S 2 from the rest of the angular terms: ∂t Ψ + Ar ∂r Ψ + AS 2 iD /S 2 Ψ + Aϕ ∂ϕ Ψ + BΨ = 0 , ∆ Ar = Mr = σ  −1 0 0 1 2M ra σ2   Aϕ =  √  i ∆ σ sin θ 2 ρ −1 σ , AS 2 √ − ∆ = Id2 , σ √ −i ∆ σ sin θ  ρ2 −1  σ  ,  2M ra 2 σ ˜ −1 P˜ . 3. The matrix AS 2 is now diagonal and furthermore AS 2 D /S 2 is a shortrange perturbation of A0S 2 D /S 2 , where √ − ∆ 0 AS 2 = 2 Id2 . 4. As was remarked at the end of the previous subsection, the conserved quantity takes a considerably simplified form with respect to the new tetrad, namely T AA φA φ¯A dVol = T AA φA φ¯A σ 2 ρ2 drdω ∆ T AA φA φ¯A ∆σ 2 ρ2 dr∗ dω (r2 + a2 )2 Σt Σt = Σt = Σt = ¯ T AA φ˜A φ˜A dr∗ dω Ψ, Ψ C2 Σt dr∗ dω .