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

SLIDE 1

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

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SLIDE 2

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.


• With outgoing radiation condition imposed at boundaries, obtain quasinormal modes

with . 
 
 Physically relevant boundary conditions for black holes 
 and asymptotically flat spacetimes.
 Not in general complete, and not in .

ω ∈ ℂ L2

“resonance states” “bound states”

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Motivation

• Although not complete, for much of 


black hole ringdown, quasinormal modes 
 dominate the evolution.
 
 
 
 
 
 Would like to develop perturbation theory
 in terms of quasinormal modes.

• Possible applications:
• Near-extreme Kerr
• Superradiant instability of massive fields in Kerr
• Kerr-AdS

Credit: Nollert (1999)

Initial pulse Quasinormal modes Late time power law tail

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Summary of results

• Main development: inner product

bilinear form
 
 Consider perturbations of a background Kerr spacetime. We define a symmetric bilinear form

• n 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 different 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.

⟶ ⟨⟨ ⋅ , ⋅ ⟩⟩ ⟨⟨ ⋅ , ⋅ ⟩⟩ ⟨⟨ ⋅ , ⋅ ⟩⟩ ⟨⟨ ⋅ , ⋅ ⟩⟩

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Outline

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

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Kerr geometry

• Two commuting continuous symmetries. Generated by Killing vectors

• Discrete — reflection symmetry


 
 Acting by the push-forward on tensors, anti-commutes as an operator with symmetries,


• .

ta = (∂/∂t)a, φa = (∂/∂ϕ)a t ϕ J : (t, r, θ, ϕ) → (−t, r, θ, − ϕ) J £tJ = − J£t, £φJ = − J£φ

ds2 = ✓ 1 − 2Mr Σ ◆ dt2 + 4Mar sin2 θ Σ dtdφ − Σ ∆dr2 − Σdθ2 − Λ Σ sin2 θdφ2

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∆ = r2 + a2 − 2Mr, Σ = r2 + a2 cos2 θ, Λ = (r2 + a2)2 − ∆a2 sin2 θ

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Geroch-Held-Penrose (GHP) formalism

• Kerr is Petrov type D

2 repeated principle null directions.
 
 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.


• Key GHP covariant operators:
• Derivative:
• Lie derivative:
• — reflection: = ordinary reflection combined with GHP transformation


= GHP prime

⟺ (la, na, ma, ¯ ma) Θa = ∇a − p + q 2 nb∇alb + p − q 2 ¯ mb∇amb t ϕ

η → λp¯ λqη ⇐ ⇒ η has GHP weights {p, q}

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Lξη = £ξ − p + q 2 na£ξla + p − q 2 ¯ ma£ξma

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J∗

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Teukolsky equation

• Perturbations of Kerr described by
• r

. Teukolsky (1972) showed that linearized equations decouple and separate.
 
 
 
 
 
 
 


• Resembles equation for charged scalar field

satisfies adjoint equation
 
 


ψ0 ψ4 Ψ−4/3

2

ψ4

O†(ψ0) = ⇥ gab(Θa − 4Ba)(Θb − 4Bb) − 16Ψ2 ⇤ (Ψ−4/3

2

ψ4) = 0

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⇥ (þ − 4ρ − ¯ ρ)(þ0 − ρ0) − (ð − 4τ − ¯ τ 0)(ð0 − τ 0) − 3Ψ2 ⇤ ψ0 = 0

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O(ψ0) ≡ ⇥ gab(Θa + 4Ba)(Θb + 4Bb) − 16Ψ2 ⇤ ψ0 = 0

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In terms of (Bini et al, 2002)

Θa

Ba = −(ρna − τ ¯ ma)

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Lagrangian and symplectic form

and equations derive from Lagrangian (Toth, 2018)
 
 
 
 by independently varying .

• Given Cauchy surface and Lagrangian obtain symplectic form



 
 
 
 
 
 


conserved on solutions, independent of precise choice of .

𝒫 𝒫† Υ ≡ Ψ−4/3

2

ψ4, ˜ Υ ≡ ψ0 Σ ΠΣ[ ˜ Υ, Υ] Σ

Σ

I +

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H+

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i0

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WΣ[g; (Υ1, ˜ Υ1), (Υ2, ˜ Υ2)] = Z

Σ

✏dabc h ˜ Υ2(Θd − 4Bd)Υ1 − Υ1(Θd + 4Bd)˜ Υ2 −˜ Υ1(Θd − 4Bd)Υ2 + Υ2(Θd + 4Bd)˜ Υ1 i ≡ ΠΣ[˜ Υ2, Υ1] − ΠΣ[˜ Υ1, Υ2]

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L(˜ Υ, Υ) = h gab(Θa + 4Ba)˜ Υ(Θb − 4Bb)Υ + 16Ψ2 ˜ ΥΥ i ✏

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Phase space and Hamiltonian

• Boyer-Lindquist slices, with

Kerr time-translation Killing vector field.
 Use GHP covariant Lie derivative .

• Canonical momentum



 


• Legendre transform

Hamiltonian 


• Hamilton’s equations 



 
 
 with
 
 


ta ⟶ ⟶

$ = @L @( Lt ˜ ⌥) = √ −h⌫a (⇥a − 4Ba) ⌥

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Lt ✓⌥ $ ◆ = H ✓⌥ $ ◆

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H = sM 1/3(Ψ2/3

2

− 2ξaBa) + N a(Θa + 2sBa)

N √−h

− √ −h ⇥ hab(Θa + 2sBa)N(Θb + 2sBb) − 4s2NΨ2 ⇤ sM 1/3(Ψ2/3

2

− 2ξaBa) + (Θa + 2sBa)N a !

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Σ

I +

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H+

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i0

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νa

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Lt

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10

SLIDE 11

Bilinear form

• For GHP scalars

with , we have the conserved quantity
 
 
 
 
 


• We would like to define bilinear form on two weight

scalars.
 
 Require mapping from . — reflection

Υ ≗ {−4,0}, ˜ Υ ≗ {4,0} 𝒫†(Υ) = 𝒫( ˜ Υ) = 0 {−4,0} ker 𝒫 → ker 𝒫† t ϕ

ΠΣ[˜ Υ, Υ] = Z

Σ

⇡(˜ Υ, Υ) = Z

Σ

✏dabc h ˜ Υ(Θd − 4Bd)Υ − Υ(Θd + 4Bd)˜ Υ i = Z

(3)e

⇣ ˜ Υ$ − Υ ˜ $ ⌘

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OΨ4/3

2

J ∗ = Ψ4/3

2

J ∗O†

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11

SLIDE 12

— reflection

t ϕ

• Show :



 
 
 
 
 
 
 


• So

OΨ4/3

2

J ∗ = Ψ4/3

2

J ∗O†

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OΨ4/3

2

J = ⇥ gab(Θa + 4Ba)(Θb + 4Bb) − 16Ψ2 ⇤ Ψ4/3

2

J = J ⇥ gab(Θa + 4B0

a)(Θb + 4B0 b) − 16Ψ2

⇤ Ψ4/3

2

= Ψ4/3

2

J ⇥ gab(Θa − 4Ba)(Θb − 4Bb) − 16Ψ2 ⇤ = Ψ4/3

2

J O†

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12

Ψ4/3

2

J ∗ : ker O† → ker O

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SLIDE 13

Bilinear form (compact support)

• For
• f compact support on in

, define bilinear form
 
 
 
 
 
 


• It can be shown that:

(i) (ii) (iii) is independent of precise choice of

• But

is divergent on quasinormal modes!

Υ1, Υ2 ≗ {−4,0} Σ ker 𝒫† Σ ⟨⟨ ⋅ , ⋅ ⟩⟩

hhΥ1, Υ2ii ⌘ ΠΣ[Ψ4/3

2

J ∗Υ1, Υ2] = Z

Σ

✏dabcΨ4/3

2

⇥ (J ∗Υ1)(Θd 4Bd)Υ2 Υ2J ∗(Θd 4Bd)Υ1 ⇤ = Z

Σ

Ψ4/3

2

[(J ∗Υ1)$2 + Υ2J ∗$1]

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hh Lt⌥1, ⌥2ii = hh⌥1, Lt⌥2ii

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hhΥ1, Υ2ii = hhΥ1, Υ2ii

<latexit sha1_base64="RhxwH9h0I79ajNcg5bSvX7SDqA=">ACf3icnVFNS8NAEN3Er1q/oh69LBbRg5SkCupBKHrxWMG0hSaUzXbSLt1swu5GKU/0qO/xJPgtgmoLXhwYJnHe/Ng9k2Uca0675b9tr6xuZWZbu6s7u3f+AcHrVmksKPk15KrsRUcCZAF8zaGbSBJxKETjR/neucVpGKpeNGTDMKEDAWLGSXaUH1nHAihzKhgM/U4wbxbv8xg0cyGKqaPge/8fWd2pu3V0UXgVeCWqorFbf+QwGKc0TEJpyolTPczMdTonUjHKYVYNcQUbomAyhZ6AgCahwughlhs8M8BxKs0TGi/Yn4pSZSaJGZTIgeqWVtTv6p/SLnjFSxWlpKx7fhlIks1yBosVOc6xTPD8GHjAJVPOJAYRKZr6F6YhIQrU5WdXk5S2nswrajbp3VW8X9eaD2VyFXSCTtEF8tANaqIn1EI+ougNfVjIsmzLPrfrtluM2lbpOUa/yr7Als/w6Y=</latexit>

hhΥ1, Υ2ii

<latexit sha1_base64="apT8F+IX8ersDtU/A9bn6BmY6K4=">ACSXicfVBNS8NAEJ20ftT6VfXoZbEIHqQkVdFj0YvHCqYtNKVstpt2cbMJuxuhP4Sf41X+wv8Gd5ENy0AW0LPhjm8d4MzDw/5kxp2363CsW19Y3N0lZ5e2d3b79ycNhSUSIJdUnEI9nxsaKcCepqpjntxJLi0Oe07T/dZX7mUrFIvGoxzHthXgoWMAI1kbqV648jsWQ07whz40V48Zxzn95HXlyPjVvqF+p2jV7BrRKnJxUIUezX/n2BhFJQio04ViprmPHupdiqRnhdFL2EkVjTJ7wkHYNFTikqpfO3pugU6MUBJU0Kjmfp3I8WhUuPQN5Mh1iO17GXiv96CmClSBWrpKB3c9FIm4kRTQeY3BQlHOkJZrGjAJCWajw3BRDLzFiIjLDHRJvyctZTmeVtOo156JWf7isNm7z5EpwDCdwBg5cQwPuoQkuEHiBV3iDqTW1PqxP62s+WrDynSNYQKH4A6eUs6Y=</latexit>

13

SLIDE 14

Bilinear form (noncompact support)

• For noncompact support data, try to prove symmetry



 


• n solutions.



 Must keep track of boundary terms.

• On solutions, Cartan’s magic formula

• since

.
 
 Integrate over partial Cauchy surface

• Obtain



 


⟹ £tπ = d(t ⋅ π) dπ = 0 ⟹ ∫S £tπ(Ψ4/3

2 𝒦Υ1, Υ2) = ∫∂S

t ⋅ π(Ψ4/3

2 𝒦Υ1, Υ2)

I +

<latexit sha1_base64="UuoNd5n3xrIcO/5/cZLQZoMrk=">ACHnicfZBLSwMxFIUz9VXHV9Wlm2ARBKHM1IVuxKIb3VWwD+iMJZNm2tBMJiQZoZT+DbcW/C+6EpfqjxHMTLuwLXgcPjuvXByAsGo0o7zbeWldW1/Lr9sbm1vZOYXevruJEYlLDMYtlM0CKMpJTVPNSFNIgqKAkUbQv07njUciFY35vR4I4keoy2lIMdIGeV6EdE9hCW8fTtqFolNyMsF405N8fLNvhAvn3a1XfjxOjFOIsI1ZkiplusI7Q+R1BQzMrK9RBGBcB91SctYjiKi/GWeQSPDOnAMJbmcQ0z+vdiCKlBlFgNrOM87MU/jubgSmRKlRzoXR47g8pF4kmHE8yhQmDOoZpV7BDJcGaDYxBWFLzLYh7SCKsTaO26cudb2fR1Msl97RUvnOKlSswUR4cgENwDFxwBirgBlRBDWAgwBN4BmNrbL1a79bHZDVnTW/2wYysr19H56a2</latexit>

H+

<latexit sha1_base64="rxrHvRem5ImGNmur3wc6cWf0S4=">ACHnicfZBLSwMxFIUz9VXHV9Wlm2ARBKHM1IVuxKbLivYB3TGkzbWgmE5KMUIb+DbcW/C+6EpfqjxHMtF3YFjwQOHz3Xjg5gWBUacf5tnIrq2vrG/lNe2t7Z3evsH/QUHEiManjmMWyFSBFGOWkrqlmpCUkQVHASDMY3Gbz5iORisb8Xg8F8SPU4zSkGmDPC9Cuo8Rg9WHs06h6JScieCycWemeP1mX4mXT7vWKfx43RgnEeEaM6RU23WE9lMkNcWMjGwvUQgPEA90jaWo4goP51kHsETQ7owjKV5XMJ/XuRokipYRSYzSyjWpxl8N/ZHMyIVKFaCKXDSz+lXCSacDzNFCYM6hmXcEulQRrNjQGYUnNtyDuI4mwNo3api93sZ1l0yiX3PNS+c4pVm7AVHlwBI7BKXDBaiAKqiBOsBAgCfwDMbW2Hq13q2P6WrOmt0cgjlZX78dTKad</latexit>

i0

<latexit sha1_base64="yD7JXdbEo+qWQvYt1P8Vr/TDLWk=">ACFXicfZDNSgMxFIXv+FvHv6pLN8EiuCozdaEbsejGZUX7A+1YMmDc1khiQjlKGP4NY+gM/hTnTpUnwYwUzbhW3BA4HDd+6Fm+PHnCntON/W0vLK6tp6bsPe3Nre2c3v7dUlEhCqyTikWz4WFHOBK1qpjltxJLi0Oe07vevs7z+SKVikbjXg5h6Ie4KFjCtUF37MFp5wtO0RkLRp3agqXH/ZF/PJlV9r5n1YnIklIhSYcK9V0nVh7KZaEU6HditRNMakj7u0azAIVeOj51iI4N6aAgkuYJjcb070aKQ6UGoW8mQ6x7aj7L4L/ZDMyIVIGaO0oH517KRJxoKsjkpiDhSEcoqwh1mKRE84ExmEhmvoVID0tMtCnSNn258+0smlqp6J4WS7dOoXwFE+XgEI7gBFw4gzLcQAWqQKALT/AMI2tkvVpv1vtkdMma7hzAjKzPX1kEoxM=</latexit>

S

<latexit sha1_base64="vFchJGr6Z+gyWiveB04FdIL/myI=">AB6HicbVDLTgJBEOzF+IL9ehlIjHxRHbRI9ELx4hyiOBDZkdemFkdnYzM2tCF/gxYPGePWTvPk3DrAHBSvpFLVne6uIBFcG9f9dnJr6xubW/ntws7u3v5B8fCoqeNUMWywWMSqHVCNgktsG4EthOFNAoEtoLR7cxvPaHSPJYPZpygH9GB5CFn1Fipft8rltyOwdZJV5GSpCh1it+dfsxSyOUhgmqdcdzE+NPqDKcCZwWuqnGhLIRHWDHUkj1P5kfuiUnFmlT8JY2ZKGzNXfExMaT2OAtsZUTPUy95M/M/rpCa89idcJqlByRaLwlQE5PZ16TPFTIjxpZQpri9lbAhVZQZm03BhuAtv7xKmpWyd1Gu1C9L1ZsjycwCmcgwdXUIU7qEDGCA8wyu8OY/Oi/PufCxac042cwx/4Hz+AK9zjNs=</latexit>

ra ra

hh Lt⌥1, ⌥2ii = hh⌥1, Lt⌥2ii

<latexit sha1_base64="NZSlg+tBuoK3TYac/jxkGY/tU=">ACbHicbVFNSwMxEM2u3/WrfhyUIgSL4kHKbhX0IhS9ePBQwarQLSWbTmswm12SWbEsPfkPvfkTvPgbTNsFa3UgzOPNvJnkJUykMOh5H47Mzs3v7C4VFheWV1bL25s3ps41RwaPJaxfgyZASkUNFCghMdEA4tCQ/h89Ww/vAC2ohY3WE/gVbEekp0BWdoqXbxLZBM9STkiQYIr5gFN4M20qCRGCFtl3/8g6s0GPFONELOj1iUvbfuOkR7WLZq3ijoH+Bn4MyaPeLr4HnZinESjkhnT9L0EWxnTKLiEQSFIDSMP7MeNC1ULALTykZmDeiBZTq0G2t7FNIRO6nIWGRMPwptZ8TwyUzXhuR/tWaK3fNWJlSIig+XtRNJcWYDp2nHaGBo+xbwLgW9q6UPzHNONr/KVgT/Okn/wX31Yp/UqnenpZrl7kdi6RE9skR8ckZqZFrUicNwsmns+7sOLvOl7vtlty9cavr5Jot8ivcw29dJbu7</latexit>

Z

S

⇡( 4/3

2

J Lt⌥1, ⌥2) − Z

∂S (2)✏N 4/3 2

⌥2J [ra(⇥a − 4Bd)⌥1] = Z

S

⇡( 4/3

2

J ⌥1, Lt⌥2) − Z

∂S (2)✏N 4/3 2

(J ⌥1)ra(⇥a − 4Bd)⌥2

<latexit sha1_base64="S8fj/KRs4QIz/V+uc49Na0CXgsw=">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</latexit>

14

SLIDE 15

Bilinear form (outgoing radiation)

• Augment bilinear form with boundary terms such that symmetry of holds.
• Outgoing radiation condition:



 
 
 i.e.,
 
 


• For

satisfying the outgoing radiation condition, 
 define bilinear form
 
 


Υ1, Υ2

Lt

<latexit sha1_base64="pVG0UMNhGce1e+E4Jz/kyS3S7wU=">AB8nicbVBNS8NAEN34WetX1aOXxSJ4KkV9Fj04sFDBfsBSib7aZdutkNuxOxhP4MLx4U8eqv8ea/cdvmoK0PBh7vzTAzL0oFN+C6387K6tr6xmZpq7y9s7u3Xzk4bBuVacpaVAmluxExTHDJWsBsG6qGUkiwTrR6Gbqdx6ZNlzJBxinLEzIQPKYUwJW8gNgT5AHd5Me9CpVt+bOgJeJV5AqKtDsVb6CvqJZwiRQYzxPTeFMCcaOBVsUg4yw1JCR2TAfEslSZgJ89nJE3xqlT6OlbYlAc/U3xM5SYwZJ5HtTAgMzaI3Ff/z/AziqzDnMs2ASTpfFGcCg8LT/3Gfa0ZBjC0hVHN7K6ZDogkFm1LZhuAtvrxM2vWad16r319UG9dFHCV0jE7QGfLQJWqgW9RELUSRQs/oFb054Lw4787HvHXFKWaO0B84nz+qmZGA</latexit>

I +

<latexit sha1_base64="UuoNd5n3xrIcO/5/cZLQZoMrk=">ACHnicfZBLSwMxFIUz9VXHV9Wlm2ARBKHM1IVuxKIb3VWwD+iMJZNm2tBMJiQZoZT+DbcW/C+6EpfqjxHMTLuwLXgcPjuvXByAsGo0o7zbeWldW1/Lr9sbm1vZOYXevruJEYlLDMYtlM0CKMpJTVPNSFNIgqKAkUbQv07njUciFY35vR4I4keoy2lIMdIGeV6EdE9hCW8fTtqFolNyMsF405N8fLNvhAvn3a1XfjxOjFOIsI1ZkiplusI7Q+R1BQzMrK9RBGBcB91SctYjiKi/GWeQSPDOnAMJbmcQ0z+vdiCKlBlFgNrOM87MU/jubgSmRKlRzoXR47g8pF4kmHE8yhQmDOoZpV7BDJcGaDYxBWFLzLYh7SCKsTaO26cudb2fR1Msl97RUvnOKlSswUR4cgENwDFxwBirgBlRBDWAgwBN4BmNrbL1a79bHZDVnTW/2wYysr19H56a2</latexit>

H+

<latexit sha1_base64="rxrHvRem5ImGNmur3wc6cWf0S4=">ACHnicfZBLSwMxFIUz9VXHV9Wlm2ARBKHM1IVuxKbLivYB3TGkzbWgmE5KMUIb+DbcW/C+6EpfqjxHMtF3YFjwQOHz3Xjg5gWBUacf5tnIrq2vrG/lNe2t7Z3evsH/QUHEiManjmMWyFSBFGOWkrqlmpCUkQVHASDMY3Gbz5iORisb8Xg8F8SPU4zSkGmDPC9Cuo8Rg9WHs06h6JScieCycWemeP1mX4mXT7vWKfx43RgnEeEaM6RU23WE9lMkNcWMjGwvUQgPEA90jaWo4goP51kHsETQ7owjKV5XMJ/XuRokipYRSYzSyjWpxl8N/ZHMyIVKFaCKXDSz+lXCSacDzNFCYM6hmXcEulQRrNjQGYUnNtyDuI4mwNo3api93sZ1l0yiX3PNS+c4pVm7AVHlwBI7BKXDBaiAKqiBOsBAgCfwDMbW2Hq13q2P6WrOmt0cgjlZX78dTKad</latexit>

i0

<latexit sha1_base64="yD7JXdbEo+qWQvYt1P8Vr/TDLWk=">ACFXicfZDNSgMxFIXv+FvHv6pLN8EiuCozdaEbsejGZUX7A+1YMmDc1khiQjlKGP4NY+gM/hTnTpUnwYwUzbhW3BA4HDd+6Fm+PHnCntON/W0vLK6tp6bsPe3Nre2c3v7dUlEhCqyTikWz4WFHOBK1qpjltxJLi0Oe07vevs7z+SKVikbjXg5h6Ie4KFjCtUF37MFp5wtO0RkLRp3agqXH/ZF/PJlV9r5n1YnIklIhSYcK9V0nVh7KZaEU6HditRNMakj7u0azAIVeOj51iI4N6aAgkuYJjcb070aKQ6UGoW8mQ6x7aj7L4L/ZDMyIVIGaO0oH517KRJxoKsjkpiDhSEcoqwh1mKRE84ExmEhmvoVID0tMtCnSNn258+0smlqp6J4WS7dOoXwFE+XgEI7gBFw4gzLcQAWqQKALT/AMI2tkvVpv1vtkdMma7hzAjKzPX1kEoxM=</latexit>

S

<latexit sha1_base64="vFchJGr6Z+gyWiveB04FdIL/myI=">AB6HicbVDLTgJBEOzF+IL9ehlIjHxRHbRI9ELx4hyiOBDZkdemFkdnYzM2tCF/gxYPGePWTvPk3DrAHBSvpFLVne6uIBFcG9f9dnJr6xubW/ntws7u3v5B8fCoqeNUMWywWMSqHVCNgktsG4EthOFNAoEtoLR7cxvPaHSPJYPZpygH9GB5CFn1Fipft8rltyOwdZJV5GSpCh1it+dfsxSyOUhgmqdcdzE+NPqDKcCZwWuqnGhLIRHWDHUkj1P5kfuiUnFmlT8JY2ZKGzNXfExMaT2OAtsZUTPUy95M/M/rpCa89idcJqlByRaLwlQE5PZ16TPFTIjxpZQpri9lbAhVZQZm03BhuAtv7xKmpWyd1Gu1C9L1ZsjycwCmcgwdXUIU7qEDGCA8wyu8OY/Oi/PufCxac042cwx/4Hz+AK9zjNs=</latexit>

ra ra hhΥ1, Υ2ii ⌘ lim

S→Σ

⇢ ΠS[Ψ4/3

2

J Υ1, Υ2] + Z

∂S

Ψ4/3

2

(J Υ1)Υ2

• <latexit sha1_base64="V0pfa6GEW9azFmIFVskV1Uk+sCc=">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</latexit>

Λ−1/4ra(Θa − 4Ba)(Λ1/4Υ) → 1 √ −h$

• n @S, as S → Σ

<latexit sha1_base64="jln506FXpfvzaUd5fsn+japKt9o=">ACpHicfZFPa9swGMZl71+X/cu643YQKy0pLKndFbpj6C479NCRui1UiXkty4moLKmSXBaMP8Bu+2xjl32R3QaTkzDWFPaC4OH3PpJePcq04NZF0c8gvHf/wcNHG487T54+e/6i+3LzKrKUJZQJZS5yMAywSVLHeCXWjDoMwEO8+uPrb98xtmLFfy1M01G5cwlbzgFJxHafcbOfbmHCZ1P947aMwEeuR0xhykgPv4AB+lsIt7f02thyTacqHkLiZOYVIYoHXc1MReG1f3Z02DyQ0YzTG5riD3JvbF1UpizUYx0Hg0bsVxmA9Hy1PGvFpCWl3KxpEi8J3RbwSW8PhD02+lt9P0u5vkitalUw6KsDayzjSbly3N1HBmg6pLNAr2DKLr2UDI7rhfJNXjbkxwXyvglHV7Qf3fUFo7LzPvLMHN7Hqvhf/t3YItMbawa0O54sO45lJXjkm6nKmoBPaRtD+Gc24YdWLuBVD/bMwnYFP3fl/7fi84vV07oqz/UH8frD/2Qd3hJa1gV6jt6iHYnSIhugTOkEJouhX8CbYDnbCnfA4HIXJ0hoGqz2v0K0KJ38AFuvSoQ=</latexit>

na(Θa − 4Ba)(Λ1/4Υ) → 0, as r → r+ la(Θa − 4Ba)(Λ1/4Υ) → 0, as r → ∞

<latexit sha1_base64="aKzEcCy/YWOjcMcGPNQgseGwZ5g=">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</latexit>

15

SLIDE 16

Bilinear form (outgoing radiation)

• Boundary terms act as a regulator!
• In asymptotic region where outgoing radiation condition holds, the volume integrand

becomes exact. Pulled back to surface ,
 
 
 


• As we take limit, any additional contribution from larger volume integration exactly

counterbalanced by pushing the boundary terms outward.

• Can show that bilinear form satisfies all the other desired properties.

S

⇡(Ψ4/3

2

J Υ1, Υ2) ≈ d h −(2)✏Ψ4/3

2

(J Υ1)Υ2 i

<latexit sha1_base64="kvmHrz8ODlTWDHyHPIl6TB4qM=">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</latexit>

boundary integrand volume integrand

hhΥ1, Υ2ii ⌘ lim

S→Σ

⇢ ΠS[Ψ4/3

2

J Υ1, Υ2] + Z

∂S

Ψ4/3

2

(J Υ1)Υ2

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16

SLIDE 17

Quasinormal modes

• Quasinormal mode with frequency

, satisfies, on phase space,
 
 
 subject to outgoing radiation condition.

• Boundary terms in bilinear form precisely cancel divergence in integral to

give finite product between quasinormal modes.

• Let

and be quasinormal modes with frequencies , . Then either

• r

.
 
 Proof: By symmetry of time-evolution operator,
 


ω Y1 Y2 ω1 ω2 ⟨⟨Y1, Y2⟩⟩ = 0 ω1 = ω2

HY = −iωY

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Y = ✓ Υ $ ◆

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0 = hhY 1, HY 2ii hhHY 1, Y 2ii = i(ω2 ω1)hhY 1, Y 2ii

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17

SLIDE 18

Quasinormal modes

• Separated form of mode solution





• Teukolsky showed we get separated angular and radial equations. With Kinnersley

tetrad,

sΥ`m! = e−i!t+im sR`m!(r)sS`m!(θ)

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 1 sin θ d dθ ✓ sin θ d dθ ◆ + ✓ K − m2 + s2 + 2ms cos θ sin2 θ − a2ω2 sin2 θ − 2aωs cos θ ◆

sS`m!(θ) = 0

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 ∆−s d dr ✓ ∆s+1 d dr ◆ + ✓H2 − 2is(r − M)H ∆ + 4isωr + 2amω − K + s(s + 1) ◆

sR`m!(r) = 0

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with H ≡ (r2 + a2)ω − am

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18

SLIDE 19

Angular equation

• Regular solutions are spin-weighted spheroidal harmonics.
• For fixed

, angular functions with different are orthogonal:

s, m, ω ℓ

Z ⇡ dθ sin θsS`m!(θ)sS`0m!(θ) = δ``0

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 1 sin θ d dθ ✓ sin θ d dθ ◆ + ✓ K − m2 + s2 + 2ms cos θ sin2 θ − a2ω2 sin2 θ − 2aωs cos θ ◆

sS`m!(θ) = 0

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19

SLIDE 20

Radial equation

• Outgoing boundary conditions





• Imposing both conditions, obtain discrete set of quasinormal modes with

frequency .

• Note: angular and radial equations both depend on
• nonlinearly. Only in

phase space, do we have

ω ∈ ℂ ω

 ∆−s d dr ✓ ∆s+1 d dr ◆ + ✓H2 − 2is(r − M)H ∆ + 4isωr + 2amω − K + s(s + 1) ◆

sR`m!(r) = 0

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Rin ∼ e−ikr∗ ∆s , r∗ → −∞, Rup ∼ eiωr∗ r2s+1 , r∗ → ∞,

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HY = −iωY

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20

SLIDE 21

Bilinear form on modes

• 2d orthogonality relation: integral does not factorize into 1d integrals, except

in special cases ( , near-NHEK, …)

• Cancellations between boundary and volume divergences.

a → 0

hhΥ`1m1!1, Υ`2m2!2ii = 8πM 4/3δm1m2e−i(!2−!1)t lim

r2→∞ r1→r+

( Z r2

r1

Z ⇡ drdθ sin θ ∆2 S1S2R1R2· · ✓ iΛ ∆ (ω1 + ω2) + 2iMra ∆ (m1 + m2) + 2  r ia cos θ + M ∆ (r2 a2) ◆ + "Z ⇡ dθ p Λ sin θ ∆2 S1S2R1R2 #

r=r1

+ "Z ⇡ dθ p Λ sin θ ∆2 S1S2R1R2 #

r=r2

) .

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21

SLIDE 22

Wronskian

• If

solutions to radial equation for fixed , then Wronskian is independent of .

• If

are linearly dependent, then .
 


• = 0 at quasinormal frequencies

.

• What about

? At , gives the norm of the quasinormal mode.

R1, R2 s, m, ℓ, ω r R1, R2 𝒳[R1, R2] = 0 ⟹ 𝒳[Rin

ω , Rout ω ]

ω = ωn d𝒳[Rin

ω , Rout ω ]/dω

ωn

W[R1, R2] = ∆1+s(r)  R1(r)dR2 dr − R2(r)dR1 dr

• <latexit sha1_base64="PFpcESPvXz1c6VtkRpF9rMvTbo4=">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</latexit>

22

SLIDE 23

Wronskian

• 1. Let

be GHP scalars in separated form, with the same , but where

• do not necessarily satisfy the radial equation. Then



 


• 2. Let

be ingoing, upgoing solutions to the radial equation at frequency . Then at a quasinormal frequency ,

Υ1, Υ2 m, ℓ, ω R1, R2 Rin

ω , Rup ω

ω ωn

8πM 4/3W[R1, R2] = Z

S2(t,r)

t · π(Ψ4/3

2

J Υ1, Υ2)

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d dω W[Rin

ω , Rup ω ]

• ω=ωn

= i 8πM 4/3 hhΥin

ωn, Υup ωnii

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W[R1, R2] = ∆1+s(r)  R1(r)dR2 dr − R2(r)dR1 dr

• <latexit sha1_base64="PFpcESPvXz1c6VtkRpF9rMvTbo4=">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</latexit>

23

SLIDE 24

Wronskian

• Sketch of proof of 2: (based on Leung et al, 1994)



 Since

• n solutions,



 
 Integrate:
 
 
 
 
 Differentiate both sides wrt , and set :

dπ = 0 ω ω → ωn

d ⇣ t · π ⇣ Ψ4/3

2

J Υin

ωn, Υup ω

⌘⌘ = £tπ ⇣ Ψ4/3

2

J Υin

ωn, Υup ω

⌘ = −i(ω − ωn)π ⇣ Ψ4/3

2

J Υin

ωn, Υup ω

⌘

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Z

∂S

t · π(Ψ4/3

2

J Υin

ωn, Υup ω ) = −i(ω − ωn)

Z

S

π(Ψ4/3

2

J Υin

ωn, Υup ω )

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I +

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H+

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i0

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S

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ra ra

d dω

• ω=ωn

right side = −i Z

S

π(Ψ4/3

2

J Υin

ωn, Υup ωn)

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24

SLIDE 25

Wronskian

• Sketch of proof (cont’d):



 
 
 
 
 
 
 Combining,
 
 
 
 
 
 
 Asymptotic behaviors of

• right side reduces to bilinear form in limit
• .

Rin

ω , Rup ω ⟹

S → Σ

d dω

• ω=ωn

left side = Z

∂S+

t · π Ψ4/3

2

J Υin

ωn, d

dω

• ω=ωn

Υup

ω

! − d dω

• ω=ωn

Z

∂S−

t · π ⇣ Ψ4/3

2

J Υin

ω , Υup ω

⌘ + Z

∂S−

t · π d dω

• ω=ωn

Ψ4/3

2

J Υin

ω , Υup ωn

!

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Wronskian

8πM 4/3 d dω W[Rin

ω , Rup ω ]

• ω=ωn

= − i Z

S

π(Ψ4/3

2

J Υin

ωn, Υup ωn)

− Z

∂S−

t · π d dω

• ω=ωn

Ψ4/3

2

J Υin

ω , Υup ωn

! − Z

∂S+

t · π Ψ4/3

2

J Υin

ωn, d

dω

• ω=ωn

Υup

ω

!

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25

SLIDE 26

Excitation coefficients

• Suppose we have compact support initial data


 
 Then quasinormal mode field response is given by
 
 
 
 where
 
 
 
 
 
 
 


• This is precisely result obtained from standard Laplace transform analysis.

(Υ, ϖ)|t=0

ΥQNM = X

`mn

c`mnΥ`mn

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c`mn = hhΥ`mn, (Υ, $)iit=0 hhΥ`mn, Υ`mniit=0 = 1 hhΥ`mn, Υ`mniit=0 Z

Σ (3)eΨ4/3 2

[(J Υ`mn)$ + ΥJ $`mn]t=0 = 1 dW/d!|!`mn 1 2⇡i Z 2⇡ Z ⇡ Z ∞

r+

sin ✓ ∆2 e−imS`mn(✓)R`mn(r) ⇢ Λ ∆(@tΥ i!`mnΥ) +4 M ∆ (r2 a2) r ia cos ✓

• Υ + 2Mra

∆ (@Υ + imΥ)

• t=0

drd✓d

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26

SLIDE 27

Complex scaling

• Numerically, can be tricky to evaluate limit in bilinear form.



 
 
 On modes, volume integrand and boundary terms as 
 


• exponential divergence if


 (Cancellations still give finite result)

• Complexify by deforming into complex-

plane such that integrals converge:

∼ e±i(ω1+ω2)r* r* → ± ∞ ⟹ ℑ(ω1 + ω2) < 0 Σ r*

hhΥ1, Υ2ii = ΠΣC[Ψ4/3

2

J Υ1, Υ2]

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r* r*

hhΥ1, Υ2ii ⌘ lim

S→Σ

⇢ ΠS[Ψ4/3

2

J Υ1, Υ2] + Z

∂S

Ψ4/3

2

(J Υ1)Υ2

• <latexit sha1_base64="V0pfa6GEW9azFmIFVskV1Uk+sCc=">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</latexit>

27

SLIDE 28

Other bilinear forms: Hertz potentials

• Fundamental identity (Wald, 1978)



 
 
 


• Adjoint identity





• If (ingoing radiation gauge) Hertz potential

satisfies , then

is a real solution to linearized Einstein, and

is a solution to equation, but not the same as

ψ ≗ {−4,0} 𝒫†(ψ) = 0 ℜ𝒯†Υ Ψ−4/3

2

𝒰†ℜ𝒯†ψ 𝒫† ψ

28

SE = OT

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Teukolsky Linearized Einstein 𝒰 : γab ↦ ψ0

ES† = T †O†

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SLIDE 29

Other bilinear forms: Hertz potentials

• If we can find a Hertz potential that generates a given Weyl scalar, then by

differentiating, can reconstruct entire metric.

• Suppose

generated by (outgoing radiation gauge) Hertz potentials , i.e.,


• Then by repeated application of Prabhu-Wald identity,



 
 


• btain

Υ1, Υ2 ˜ ψ1, ˜ ψ2

29

Υi = Ψ4/3

2

T 0<S0†Ψ4/3

2

˜ ψi, i = 1, 2

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W G

S [γ, S†Υ] = −ΠS[T γ, Υ]

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hhΥ1, Υ2ii = 1 4Π  ˜ ψ2, Ψ4/3

2

T 0S0†Ψ4/3

2

J T 0S0†Ψ4/3

2

˜ ψ1 ⇤ = 1 4Π h ˜ ψ2, Ψ4/3

2

J þ4 ⇣ ¯ Ψ4/3

2

þ04 ⇣ Ψ4/3

2

˜ ψ1 ⌘⌘i⇤ = 1 4 DD ˜ ψ2, þ4 ⇣ ¯ Ψ4/3

2

þ04 ⇣ Ψ4/3

2

˜ ψ1 ⌘⌘EE⇤

s=+2

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bilinear form on Hertz potentials

SLIDE 30

Other bilinear forms: Hertz potential

• Using a Teukolsky-Starobinsky identity, this second argument can be written



 
 


• So we obtain a relation between bilinear form on Weyl scalars and on Hertz

potentials that generate them.

• Similarly, can obtain relation with bilinear form on metric perturbations.
• Aim is to use these relations to go to nonlinear order.

30

þ4 ⇣ ¯ Ψ4/3

2

þ04 ⇣ Ψ4/3

2

˜ ψ1 ⌘⌘ = ð4 ⇣ ¯ Ψ4/3

2

ð04 ⇣ Ψ4/3

2

˜ ψ1 ⌘⌘ − 9¯ ŁξŁξ ˜ ψ1

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algebraic on modes

SLIDE 31

Example: Near-extreme Kerr quasinormal modes

• Near-extreme Kerr has long lived modes. Potential nonlinear turbulent effects,

(Yang, Zimmerman, Lehner, 2015).

• Far limit: Extreme Kerr
• Near-NHEK limit:



 Extremality parameter
 
 while holding fixed σ = r+ − r− r+ → 0

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n = 0 n = 1

accumulation point at horizon frequency

<(ω) =(ω)

surface gravity

Im(ωn) ∝ κ

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κ = p 1 − a2/M 2

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¯ t = tσ, ¯ x = x σ , ¯ θ = θ, ¯ φ = φ − t 2M

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x = r − r+ r+

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Gives enhanced near-horizon symmetry

sl2(R) × u(1)

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31

SLIDE 32

Example: Near-extreme Kerr quasinormal modes

• Modes obtained in matched asymptotic expansion



 
 
 
 
 


• To leading order, spin-weighted spheroidal harmonics evaluated at

.

• Far solution:

radial solution to extreme Kerr

• Near solution: hypergeometric functions, which reduce to terminating polynomials upon

matching

• Matching gives

ω = mΩH ω = mΩH

near-zone: x ⌧ 1 far-zone: x σ¯ ω

• verlap region: σ¯

ω ⌧ x ⌧ 1.

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where ¯ ω = 2M(ω − mΩH) σ

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¯ ωn = − i 2(h+ + n + im), h+ ∈ R+ ¯ ωn = − i 2(h− + n + im), h− = 1/2 + ir ∈ C

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h± = 1 2 ± r 1 4 + K − 2m2

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32

SLIDE 33

Example: Near-extreme Kerr quasinormal modes

• Check orthogonality
• Split bilinear form
• Near zone:



 
 
 
 
 


• Obtain orthogonality by explicit evaluation.

hhΥ1, Υ2ii = hhΥ1, Υ2iinear + hhΥ1, Υ2iifar

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hhΥ1, Υ2iinear = 2M −2/3 " 2i lim

✏→0

Z c/√

✏

d¯ x ⇣ ¯ !1 + ¯ !2 + 2ˆ e ˆ A¯

t

⌘ (x(1 + x))−3 Rnear

1

Rnear

2

+Rnear

1

(✏)Rnear

2

(✏) ✏2(1 + ✏)2

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• dependent part precisely cancels

amounts to minimal subtraction

ϵ

33

SLIDE 34

Conclusions

• We established a bilinear form on Weyl scalars with respect to which Kerr

quasinormal modes with different frequencies are orthogonal.
 
 Construction works in phase space. Relies on type D nature of Kerr and — reflection symmetry.

• Extensions:
• Alternative regularization schemes: complex scaling, minimal subtraction
• Consistency with standard calculations for excitation coefficients
• Relation of bilinear form on Weyl scalar to bilinear forms on metric

perturbations and Hertz potentials

t ϕ

Thank you

34