Orthogonality of Kerr quasinormal modes Stephen R. Green (AEI - PowerPoint PPT Presentation

Orthogonality of Kerr quasinormal modes Stephen R. Green (AEI Potsdam) with Stefan Hollands and Peter Zimmerman August 2, 2019 “Timelike Boundaries in General Relativistic Evolution Problems” Casa Matemática Oaxaca Mexico � 1


 
 Motivation Mode expansions are useful tools as foundations for nonlinear and variational • studies. 
 E.g., talk by Oleg on modes of global AdS Normal modes of self-adjoint systems are complete and orthonormal. We can project • equations into mode space. 
 “bound states” With outgoing radiation condition imposed at boundaries, obtain quasinormal modes • with � . 
 ω ∈ ℂ “resonance states” Physically relevant boundary conditions for black holes 
 and asymptotically flat spacetimes. 
 L 2 Not in general complete, and not in � . � 2


 
 
 
 
 Motivation Although not complete, for much of 
 • black hole ringdown, quasinormal modes 
 Quasinormal modes dominate the evolution. 
 Late time power law tail Initial pulse Would like to develop perturbation theory 
 in terms of quasinormal modes. Credit: Nollert (1999) Possible applications: • Near-extreme Kerr • Superradiant instability of massive fields in Kerr • Kerr-AdS • � 3


 � Summary of results Main development: inner product � bilinear form 
 ⟶ • Consider perturbations of a background Kerr spacetime. We define a symmetric bilinear form � on Weyl scalars (complex linear in both entries) with the ⟨⟨ ⋅ , ⋅ ⟩⟩ following properties: the time-evolution operator is symmetric with respect to � , ⟨⟨ ⋅ , ⋅ ⟩⟩ • is finite on quasi-normal modes. ⟨⟨ ⋅ , ⋅ ⟩⟩ • It follows that quasinormal modes with di ff erent frequencies are orthogonal • with respect to � . ⟨⟨ ⋅ , ⋅ ⟩⟩ Our bilinear form is based on earlier work by Leung, Liu and Young (1994) on • quasinormal modes of open systems. � 4

Outline 1. GHP formalism and Teukolsky equation 2. Lagrangian and Hamiltonian 3. Bilinear form 4. Quasinormal mode orthogonality 5. Extras • Relation to Wronskian • Excitation coe ffi cients • Complex scaling regularization 6. Example: Near-extreme Kerr quasinormal mode orthogonality � 5

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 Kerr geometry dt 2 + 4 Mar sin 2 θ ✓ ◆ dtd φ − Σ ∆ dr 2 − Σ d θ 2 − Λ 1 − 2 Mr Σ sin 2 θ d φ 2 ds 2 = Σ Σ ∆ = r 2 + a 2 − 2 Mr, Σ = r 2 + a 2 cos 2 θ , Λ = ( r 2 + a 2 ) 2 − ∆ a 2 sin 2 θ Two commuting continuous symmetries. Generated by Killing vectors 
 • t a = ( ∂ / ∂ t ) a , φ a = ( ∂ / ∂ ϕ ) a � Discrete � — � reflection symmetry � J : ( t , r , θ , ϕ ) → ( − t , r , θ , − ϕ ) t ϕ • Acting by the push-forward on tensors, � anti-commutes as an operator with J symmetries, 
 � . £ t J = − J £ t , £ φ J = − J £ φ � 6

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 Geroch-Held-Penrose (GHP) formalism • Kerr is Petrov type D � 2 repeated principle null directions. 
 ⟺ ( l a , n a , m a , ¯ m a ) Defines Newman-Penrose null tetrad � aligned with PNDs. • GHP (1973) developed a framework for writing the Einstein equation such that it transforms covariantly with respect to remaining tetrad freedom. 
 η → λ p ¯ η has GHP weights { p, q } λ q η ⇐ ⇒ • Key GHP covariant operators: n b ∇ a l b + p − q Θ a = ∇ a − p + q m b ∇ a m b • Derivative : � ¯ 2 2 L ξ η = £ ξ − p + q n a £ ξ l a + p − q • Lie derivative : m a £ ξ m a ¯ 2 2 • � — � reflection : = ordinary reflection combined with GHP transformation 
 J ∗ t ϕ = GHP prime � 7

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 Teukolsky equation • Perturbations of Kerr described by � or � . Teukolsky (1972) showed that linearized ψ 0 ψ 4 equations decouple and separate. 
 ⇥ ⇤ ρ )( þ 0 − ρ 0 ) − ( ð − 4 τ − ¯ τ 0 )( ð 0 − τ 0 ) − 3 Ψ 2 ( þ − 4 ρ − ¯ ψ 0 = 0 In terms of � (Bini et al, 2002) Θ a g ab ( Θ a + 4 B a )( Θ b + 4 B b ) − 16 Ψ 2 ⇥ ⇤ O ( ψ 0 ) ≡ ψ 0 = 0 B a = − ( ρ n a − τ ¯ m a ) • Resembles equation for charged scalar field Ψ − 4/3 • � satisfies adjoint equation 
 ψ 4 2 ( Ψ − 4 / 3 g ab ( Θ a − 4 B a )( Θ b − 4 B b ) − 16 Ψ 2 O † ( ψ 0 ) = ⇥ ⇤ ψ 4 ) = 0 2 � 8