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Test black holes, Amplitudes, and perturbations of Kerr Justin - - PowerPoint PPT Presentation

Oct 06, 2023 326 likes •770 views

Test black holes, Amplitudes, and perturbations of Kerr Justin Vines Max Planck Institute for Gravitational Physics (AEI) Potsdam QCD meets Gravity 2019 Bhaumik Institute, UCLA, December 10 Justin Vines Max Planck Institute for Gravitational

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Test black holes, Amplitudes, and perturbations of Kerr

Justin Vines Max Planck Institute for Gravitational Physics (AEI) Potsdam QCD meets Gravity 2019 Bhaumik Institute, UCLA, December 10

Justin Vines Max Planck Institute for Gravitational Physics (AEI) Potsdam Test black holes, Amplitudes, and perturbations of Kerr

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Amplitudes for massive scalars exchanging gravitons + + +. . . − → “ → 0” − → Interacting mass monopoles in classical general relativity (e.g. nonspinning black holes)

Justin Vines Max Planck Institute for Gravitational Physics (AEI) Potsdam Test black holes, Amplitudes, and perturbations of Kerr

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Amplitudes for massive scalars exchanging gravitons + + +. . . ...to 2-loops! [to O(G3)]:

Bern, Cheung, Roiban, Shen, Solon, Zeng 1901.,1906.

− → “ → 0” − → Interacting mass monopoles in classical general relativity (e.g. nonspinning black holes)

Justin Vines Max Planck Institute for Gravitational Physics (AEI) Potsdam Test black holes, Amplitudes, and perturbations of Kerr

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Amplitudes for massive scalars exchanging gravitons + + +. . . ...to 2-loops! [to O(G3)]:

Bern, Cheung, Roiban, Shen, Solon, Zeng 1901.,1906.

− → “ → 0” − → Interacting mass monopoles in classical general relativity (e.g. nonspinning black holes) Amplitudes for (minimally coupled) massive spin-s particles exchanging gravitons

Guevara, JV, Steinhoff, Buonanno, Ochirov, Chung, Huang, Kim, Lee, Maybee, O’Connell, Arkani-Hamed, Siemonsen ++ ’17–’19

− → “ → 0 while s → ∞” − → Interacting spinning black holes in classical GR

Justin Vines Max Planck Institute for Gravitational Physics (AEI) Potsdam Test black holes, Amplitudes, and perturbations of Kerr

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SLIDE 5

Amplitudes for massive scalars exchanging gravitons + + +. . . ...to 2-loops! [to O(G3)]:

Bern, Cheung, Roiban, Shen, Solon, Zeng 1901.,1906.

− → “ → 0” − → Interacting mass monopoles in classical general relativity (e.g. nonspinning black holes) Amplitudes for (minimally coupled) massive spin-s particles exchanging gravitons

Guevara, JV, Steinhoff, Buonanno, Ochirov, Chung, Huang, Kim, Lee, Maybee, O’Connell, Arkani-Hamed, Siemonsen ++ ’17–’19

← −? − → “ → 0 while s → ∞” − → ?← − Interacting spinning black holes in classical GR

Justin Vines Max Planck Institute for Gravitational Physics (AEI) Potsdam Test black holes, Amplitudes, and perturbations of Kerr

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Amplitudes for massive scalars exchanging gravitons + + +. . . ...to 2-loops! [to O(G3S0)]:

Bern, Cheung, Roiban, Shen, Solon, Zeng 1901.,1906.

− → “ → 0” − → Interacting mass monopoles in classical general relativity (e.g. nonspinning black holes) Amplitudes for (minimally coupled) massive spin-s particles exchanging gravitons

Guevara, JV, Steinhoff, Buonanno, Ochirov, Chung, Huang, Kim, Lee, Maybee, O’Connell, Arkani-Hamed, Siemonsen ++ ’17–’19

← −? − → “ → 0 while s → ∞” − → ?← − Interacting spinning black holes in classical GR G1S∞ (tree, all spins) generic + G2S1 (1-loop, spin-orbit) generic

Bini-Damour +

G2S≥2 generic X G2S2,3,4 aligned ?? G2S≥5 ?X? G3S≥1 X

Justin Vines Max Planck Institute for Gravitational Physics (AEI) Potsdam Test black holes, Amplitudes, and perturbations of Kerr

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Quantum spin and spinning black holes

A key observation (at leading PN orders) from Vaidya 1410.: for minimally coupled massive spin-s particles exchanging gravitons,

Justin Vines Max Planck Institute for Gravitational Physics (AEI) Potsdam Test black holes, Amplitudes, and perturbations of Kerr

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Quantum spin and spinning black holes

A key observation (at leading PN orders) from Vaidya 1410.: for minimally coupled massive spin-s particles exchanging gravitons, spin-0 — (universal) monopole coupling

Justin Vines Max Planck Institute for Gravitational Physics (AEI) Potsdam Test black holes, Amplitudes, and perturbations of Kerr

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Quantum spin and spinning black holes

A key observation (at leading PN orders) from Vaidya 1410.: for minimally coupled massive spin-s particles exchanging gravitons, spin-0 — (universal) monopole coupling spin- 1

2

— adds (universal) dipole/spin-orbit coupling

Justin Vines Max Planck Institute for Gravitational Physics (AEI) Potsdam Test black holes, Amplitudes, and perturbations of Kerr

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Quantum spin and spinning black holes

A key observation (at leading PN orders) from Vaidya 1410.: for minimally coupled massive spin-s particles exchanging gravitons, spin-0 — (universal) monopole coupling spin- 1

2

— adds (universal) dipole/spin-orbit coupling spin-1 — adds black-hole quadrupole ∝ spin2 coupling

Justin Vines Max Planck Institute for Gravitational Physics (AEI) Potsdam Test black holes, Amplitudes, and perturbations of Kerr

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Quantum spin and spinning black holes

A key observation (at leading PN orders) from Vaidya 1410.: for minimally coupled massive spin-s particles exchanging gravitons, spin-0 — (universal) monopole coupling spin- 1

2

— adds (universal) dipole/spin-orbit coupling spin-1 — adds black-hole quadrupole ∝ spin2 coupling spin-2 — adds BH (octupole ∝ S3 and) hexadecapole ∝ S4

Justin Vines Max Planck Institute for Gravitational Physics (AEI) Potsdam Test black holes, Amplitudes, and perturbations of Kerr

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SLIDE 12

Quantum spin and spinning black holes

A key observation (at leading PN orders) from Vaidya 1410.: for minimally coupled massive spin-s particles exchanging gravitons, spin-0 — (universal) monopole coupling spin- 1

2

— adds (universal) dipole/spin-orbit coupling spin-1 — adds black-hole quadrupole ∝ spin2 coupling spin-2 — adds BH (octupole ∝ S3 and) hexadecapole ∝ S4

  • btain the complete BH multipole series
  • Iℓ + iJℓ = m
  • i S

m ℓ as s → ∞?

Justin Vines Max Planck Institute for Gravitational Physics (AEI) Potsdam Test black holes, Amplitudes, and perturbations of Kerr

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SLIDE 13

Quantum spin and spinning black holes

A key observation (at leading PN orders) from Vaidya 1410.: for minimally coupled massive spin-s particles exchanging gravitons, spin-0 — (universal) monopole coupling spin- 1

2

— adds (universal) dipole/spin-orbit coupling spin-1 — adds black-hole quadrupole ∝ spin2 coupling spin-2 — adds BH (octupole ∝ S3 and) hexadecapole ∝ S4

  • btain the complete BH multipole series
  • Iℓ + iJℓ = m
  • i S

m ℓ as s → ∞? “Minimally coupled” “amplitudes for all masses and spins” using massive spinor-helicity, Arkani-Hamed, Huang, Huang (AHH) 1709.: “3-point”: for any s, “Compton”: for s ≤ 2.

Justin Vines Max Planck Institute for Gravitational Physics (AEI) Potsdam Test black holes, Amplitudes, and perturbations of Kerr

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Quantum spin and spinning black holes

A key observation (at leading PN orders) from Vaidya 1410.: for minimally coupled massive spin-s particles exchanging gravitons, spin-0 — (universal) monopole coupling spin- 1

2

— adds (universal) dipole/spin-orbit coupling spin-1 — adds black-hole quadrupole ∝ spin2 coupling spin-2 — adds BH (octupole ∝ S3 and) hexadecapole ∝ S4

  • btain the complete BH multipole series
  • Iℓ + iJℓ = m
  • i S

m ℓ as s → ∞? “Minimally coupled” “amplitudes for all masses and spins” using massive spinor-helicity, Arkani-Hamed, Huang, Huang (AHH) 1709.: “3-point”: for any s, “Compton”: for s ≤ 2. Guevara 1706.: = + + + O(G3).

Justin Vines Max Planck Institute for Gravitational Physics (AEI) Potsdam Test black holes, Amplitudes, and perturbations of Kerr

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Amplitudes for massive spin-s particles and gravitons

p′

a

pa ma, sa p′

b

pb mb, sb = + + + O(G3) “Minimal coupling” (high-energy limit) [AHH ↓] [Guevara ↑] 1 2 3+ = 1 mP ζ|p1|3] ζ3 2 12 m 2s , for any s,

Justin Vines Max Planck Institute for Gravitational Physics (AEI) Potsdam Test black holes, Amplitudes, and perturbations of Kerr

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Amplitudes for massive spin-s particles and gravitons

p′

a

pa ma, sa p′

b

pb mb, sb = + + + O(G3) “Minimal coupling” (high-energy limit) [AHH ↓] [Guevara ↑] 1 2 3+ = 1 mP ζ|p1|3] ζ3 2 12 m 2s , for any s, 4 1 3− 2+ for s ≤ 2 = −3|p1|2]4 m2

Pt(s − m2)(u − m2)

43[12] + 13[42] 3|p1|2] 2s .

Justin Vines Max Planck Institute for Gravitational Physics (AEI) Potsdam Test black holes, Amplitudes, and perturbations of Kerr

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Amplitudes for massive spin-s particles and gravitons

p′

a

pa ma, sa p′

b

pb mb, sb = + + + O(G3) “Minimal coupling” (high-energy limit) [AHH ↓] [Guevara ↑] 1 2 3+ = 1 mP ζ|p1|3] ζ3 2 12 m 2s , for any s, 4 1 3− 2+ for s ≤ 2 [s > 2? : Chung, Huang, Kim, Lee] = −3|p1|2]4 m2

Pt(s − m2)(u − m2)

43[12] + 13[42] 3|p1|2] 2s .

Justin Vines Max Planck Institute for Gravitational Physics (AEI) Potsdam Test black holes, Amplitudes, and perturbations of Kerr

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Amplitudes for massive spin-s particles and gravitons

p′

a

pa ma, sa p′

b

pb mb, sb = + + + O(G3) “Minimal coupling” (high-energy limit) [AHH ↓] [↓ GOV ↑ CHKL ↓ AHO] 1 2 3+ = 1 mP ζ|p1|3] ζ3 2 12 m 2s

s→∞

− → ∝ exp p3 ∗ ˆ S12 m . 4 1 3− 2+ for s ≤ 2 [s > 2? : Chung, Huang, Kim, Lee] = −3|p1|2]4 m2

Pt(s − m2)(u − m2)

43[12] + 13[42] 3|p1|2] 2s .

Justin Vines Max Planck Institute for Gravitational Physics (AEI) Potsdam Test black holes, Amplitudes, and perturbations of Kerr

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Elastic aligned-spin scattering of two black holes

masses m1, m2 (c = 1) spins S1 = m1 a1 = Gm2

1 ˆ

a1 S2 = m2 a2 = Gm2

2 ˆ

a2 “proper” impact parameter b(cov) relative velocity v(∞) : 1 √ 1 − v2 = γ s = E2 = m2

1 + m2 2 + 2m1m2γ

PM scattering angle χ: [..., Westpfahl ’85, ..., ↓ Bini-Damour ’17-18] χ = GE b

  • 21 + v2

v2 + 3π 4 + v2 4v2 G(m1 + m2) b {↓ spin-orbit: aligned ↔ generic} −4 v a1 + a2 b − π 2 + 3v2 v3 G(4m1 + 3m2)a1 b2 + (1 ↔ 2)

  • + . . .
  • Justin Vines

Max Planck Institute for Gravitational Physics (AEI) Potsdam Test black holes, Amplitudes, and perturbations of Kerr

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Two-BH χ through O(G2σ4) from a test BH in Kerr?

test-BH limit: m1 → 0 at fixed a1. χ = GE v2

±

(1 ± v)2 b ± a1 ± a2 +

  • Gm2F(v, b, a1, a2) + (1 ↔ 2) + O(a5)
  • + . . .
  • =
  • +

+ + . . .

  • eikonal

O(G1a∞)generic. O(G2a4)aligned. Guevara, Ochirov, Vines, Steinhoff, Buonanno,

Chung,Huang,Kim,Lee,Arkani-Hamed,Kosower, Maybee, O’Connell, ’17-’19

Justin Vines Max Planck Institute for Gravitational Physics (AEI) Potsdam Test black holes, Amplitudes, and perturbations of Kerr

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Two-BH χ through O(G2σ4) from a test BH in Kerr?

test-BH limit: m1 → 0 at fixed a1. χ = GE v2

±

(1 ± v)2 b ± a1 ± a2 +

  • Gm2F(v, b, a1, a2) + (1 ↔ 2) + O(a5)
  • + . . .
  • =
  • +

+ + . . .

  • eikonal

FGOV(v, b, a1, a2) = π 2v2 ∂ ∂b

  • Γ

dζ 2πi (1 − vζ)4 (ζ2 − 1)3/2

  • b−ζa2 − ζ − v

1 − vζ a1 −1 + O(a5

1)?

O(G1a∞)generic. O(G2a4)aligned. Guevara, Ochirov, Vines, Steinhoff, Buonanno,

Chung,Huang,Kim,Lee,Arkani-Hamed,Kosower, Maybee, O’Connell, ’17-’19

Justin Vines Max Planck Institute for Gravitational Physics (AEI) Potsdam Test black holes, Amplitudes, and perturbations of Kerr

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That aligned-spin scattering angle χGOV through O(G2S4) — — agrees with all known PN results (... NNLO S2 [Levi-Steinhoff]) — predicts new results at 4.5PN (NLO S3) and 5PN (NLO S4) ...

PN order 1.5 2.5 3.5 4.5 5.5 6.5 1 2 3 4 5 6 need up to N 1PN 2PN 3PN 4PN 5PN 1PM / tree LO SO NLO SO NNLO SO NNNLO SO 2PM / 1-loop LO S^2 NLO S^2 NNLO S^2 NNNLO S^2 3PM / 2-loop LO S^3 NLO S^3 4PM / 3-loop LO S^4 NLO S^4 5PM / 4-loop LO S^5 NLO S^5 6PM / 5-loop LO S^6

... which are in agreement with all available self-force results

[Siemonsen, JV 1909.07361] [Kavanagh, Ottewill, Wardell (redshift); Bini, Damour, Geralico, Kavanagh, van de Meent (precession)]

Justin Vines Max Planck Institute for Gravitational Physics (AEI) Potsdam Test black holes, Amplitudes, and perturbations of Kerr

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“self force”: linear perturbations of an exact Kerr spacetime (mass M, spin GM 2ˆ a) (governed by the Teukolsky equation) sourced by a small body:

Justin Vines Max Planck Institute for Gravitational Physics (AEI) Potsdam Test black holes, Amplitudes, and perturbations of Kerr

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“self force”: linear perturbations of an exact Kerr spacetime (mass M, spin GM 2ˆ a) from a small body in a circular equatorial orbit with frequency Ω, the 1SF contribution δU to Detweiler’s redshift invariant: y = (GMΩ)2/3

δU y = − 1 − 2y + 7

3 ˆ

ay1.5 − 5y2 − ˆ a2y2 + 46

3 ˆ

ay2.5 . . . − 86

9 ˆ

a2y3 + 77ˆ ay3.5 + ˆ a3y3.5 . . . − 577

9 ˆ

a2y4 + 0ˆ a4y4 . . . + 1526

81 ˆ

a3y4.5 . . . . . . −2ˆ a4y5 + O(y5.5)

Justin Vines Max Planck Institute for Gravitational Physics (AEI) Potsdam Test black holes, Amplitudes, and perturbations of Kerr

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χGOV produces aligned-circular results at NLO-PN S3,4 matching 1SF , but this does not fully confirm χGOV through G2S4v∞. A unique χ through G2S4v∞ would be determined by the mass scaling χ = GE

  • Fi(v, b, a1+a2)+
  • Gm2Fii(v, b, a1, a2)+(1 ↔ 2) + O(a5)
  • +O(G3)
  • a consistent test-BH limit (finite ring-radius a, negligible mass):

following from a generally covariant effective action with the minimum (translational and rotational) d.o.f.s — matched to (unperturbed) (linearized) Kerr — unique up to G2a3, and with three effective Wilson C’s at G2a4, p2 m2 = 1 − 2

2!Ruaua + 2 3!R∗ uaua;a + 2 4!Ruaua;aa + . . .

+ C4.. · R2a4 ⊕ . . . two equations from aligned-circular 1SF ... and a third equation

Justin Vines Max Planck Institute for Gravitational Physics (AEI) Potsdam Test black holes, Amplitudes, and perturbations of Kerr

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A unique χ through G2S4v∞ would be determined by the mass scaling χ = GE

  • Fi(v, b, a1+a2)+
  • Gm2Fii(v, b, a1, a2)+(1 ↔ 2) + O(a5)
  • +O(G3)
  • a consistent test-BH limit (finite ring-radius a, negligible mass):

following from a generally covariant effective action with the minimum (translational and rotational) d.o.f.s — matched to (unperturbed) (linearized) Kerr — unique up to G2a3, and with three effective Wilson C’s at G2a4, p2 m2 = 1 − 2

2!Ruaua + 2 3!R∗ uaua;a + 2 4!Ruaua;aa + . . .

+ C4.. · R2a4 ⊕ . . . two equations from aligned-circular 1SF ... and a third equation If the 3rd eq. is χ(v, b, atest; MKerr, aKerr) → finite as v → ∞, then χGOV.

Justin Vines Max Planck Institute for Gravitational Physics (AEI) Potsdam Test black holes, Amplitudes, and perturbations of Kerr

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supplementary: (the spinning test black hole limit; self-force results for perturbations of Kerr)

Justin Vines Max Planck Institute for Gravitational Physics (AEI) Potsdam Test black holes, Amplitudes, and perturbations of Kerr

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Nonspinning black holes as point masses (monopoles)

Effective action W for spacetime metric g and masses ma with worldlines x = za(τa): W = 1 16πG

  • d4x √−g R −
  • a

ma

  • dτa.

δW δza = 0 ⇒ geodesic eqs. for worldlines za. δW δg = 0 ⇒ Einstein field eq. for metric gµν, with T µν =

  • a
  • dτa ˙

zµ ˙ zν δ4(x, za) (when gµν ˙ zµ ˙ zν = −1) ... good for test bodies in a background. ... for self-gravitating bodies? ...

Justin Vines Max Planck Institute for Gravitational Physics (AEI) Potsdam Test black holes, Amplitudes, and perturbations of Kerr

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Perturbative approximation schemes

... point masses: W = 1 16πG

  • d4x√−gR −
  • a

ma

  • dτa

... post-Newtonian: gµνdxµdxν = −(c2 + 2U)dt2 + δijdxidxj + O(c−2). post-Minkowskian: gµν = ηµν + hµν + O(h2), h ∼ O(G). post-test-body: gµν = gKerr

µν

+ hµν + O(h2), h ∼ O(m1), (“self-force”) mKerr ∼ m2 ≫ m1. — self-force results: motion is geodesic in an effective regularized perturbed metric satisfying the vacuum field eq. [..., Detweiler, Whiting, Wald, Gralla, Poisson, Pound, Harte, ...]

Justin Vines Max Planck Institute for Gravitational Physics (AEI) Potsdam Test black holes, Amplitudes, and perturbations of Kerr

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Spinning bodies and higher multipoles

Minimal degrees of freedom (others “integrated out”): worldline z(τ) plus local Lorentz frame (“body-fixed” vierbein) Λaµ(τ). Internal Lorentz symmetry ⇒ dependence only on Ωµν := Λaµ DΛaν dτ , W =

  • dτ L(z, ˙

z, Ω)[g] + constraints. : equivalently, phase-space action : general covariance W =

  • dτ
  • pµ ˙

zµ + 1 2SµνΩµν − βµSµνpν − α 2

  • p2 + M2(p, S, z)
  • ,

momentum pµ and spin Sµν varied along with z, Λaµ (and α, βµ). Dynamical mass function: −p2 = M2 pµ, Sµν, gµν(z),

  • Rµνκλ;N(z)
  • .

−gµνpµpν =

N=(α1···αn)

Justin Vines Max Planck Institute for Gravitational Physics (AEI) Potsdam Test black holes, Amplitudes, and perturbations of Kerr

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Spinning bodies and higher multipoles

Dynamical mass: −p2 = M2 pµ

  • −p2 , Sµν, gµν(z),
  • Rµνκλ;N(z)
  • .

Equations of motion: 0 = Sµνpν, D dτ pµ + 1 2Rµνκλ ˙ zνSκλ = p · ˙ z 2 D Dzµ ln M2, D dτ Sµν − 2p[µ ˙ zν] = p · ˙ z

  • p[µ ∂

∂pν] + 2S[µ

ρ

∂ ∂Sν]ρ

  • ln M2

: Mathisson-Papapetrou-Dixon-Harte ⇔ ∇µT µν = 0 with T µν(x) =

  • dσ
  • p(µ ˙

zν)δ4(x, z) − ∇ρ

  • Sρ(µ ˙

zν)δ4(x, z)

  • + p · ˙

z 2

∞

  • n=0

∂ ln M2 ∂Rαβγδ;N

  • G(µν)Rαβγδ;Nδ4(x, z) +

2 √−g δRαβγδ;N(z) δgµν(x)

  • .

Justin Vines Max Planck Institute for Gravitational Physics (AEI) Potsdam Test black holes, Amplitudes, and perturbations of Kerr

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Spinning bodies and higher (spin-induced) multipoles

−p2 = M2 uµ, σµ, z

  • [g] = m2 + O(R).

uµ := pµ

  • −p2

Bare-rest-mass-rescaled spin (ring-radius) vector: σµ = − 1 2mǫµνκλuνSκλ ⇔ Sµν = m ǫµνκλuκσλ. Minimal MPDH eqs.: D dτ pµ + 1 2Rµνκλ ˙ zνSκλ = p · ˙ z 2 D Dzµ ln M2, D dτ Sµν − 2p[µ ˙ zν] = p · ˙ z

  • u[µ

∂ ∂uν] + σ[µ ∂ ∂σν]

  • ln M2.

[..Goldberger..Rothstein..Porto.. ..Marsat.. Levi-Steinhoff ...]: Consider M2 m2 = 1 + C2Ruσuσ + C3R∗

uσuσ;σ + C4Ruσuσ;σσ + . . . + O(∼R2).

Justin Vines Max Planck Institute for Gravitational Physics (AEI) Potsdam Test black holes, Amplitudes, and perturbations of Kerr

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Relevant couplings for a spinning test black hole

Suppressing indices, ∗’s, u’s, dimensionless coefficients, M2 m2 = 1 + R σ2 + ∇R σ3 ⊕ ∇2R σ4 ⊕ ∇3R σ5 ⊕ ∇4R σ6 ⊕ . . . ⊕ R2σ4 ⊕ ∇R2σ5 ⊕ ∇2R2σ6 ⊕ . . . ⊕ R3σ6 ⊕ . . . plus many other R≥2 terms with nonzero powers of m, ⊕

  • m∇

k σ∇ l σ m n m4R2 ⊕ m6R3 ⊕ . . .

  • ,

e.g.: m4R2 terms, k, l, n = 0: leading adiabatic quadrupolar tidal effects Reasonable conjecture?: If a spinning test black hole limit exists, it should have only m0 terms.

Justin Vines Max Planck Institute for Gravitational Physics (AEI) Potsdam Test black holes, Amplitudes, and perturbations of Kerr

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SLIDE 34

Linearized matching to Kerr

Exact Kerr in Kerr-Schild form (lµlµ = 0) : gKerr

µν

= ηµν + ϕ lµ lν.

Justin Vines Max Planck Institute for Gravitational Physics (AEI) Potsdam Test black holes, Amplitudes, and perturbations of Kerr

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SLIDE 35

Linearized matching to Kerr

Exact Kerr in Kerr-Schild form (lµlµ = 0) : gKerr

µν

= ηµν + ϕ lµ lν = ηµν + hµν + 2∂(µξν), where hµν is an exact solution (off the disk) of ¯ hµν = 0 = ∂µ¯ hµν : ¯ hµν = uρu(µ exp(σ ∗ ∂)ν)

ρ

4Gm r , (σ ∗ ∂)µ

ν = ǫµ νκλσκ∂λ.

Take T µν[z, u, σ, M2, g] with gµν = ηµν + hµν + O(h2) and z = geod.(η) and solve ¯ hµν = −16πG Tµν + O(h2) for ¯ hµν = hµν − 1

2ηµνhρρ.

Match, fixing the ∼linear-in-curvature couplings: M2 − m2 2m2 =

  • − 1

2!Ruσuσ + 1 3!R∗

uσuσ;σ + 1

4!Ruσuσ;σσ + . . .

  • + O(∼R2).

Justin Vines Max Planck Institute for Gravitational Physics (AEI) Potsdam Test black holes, Amplitudes, and perturbations of Kerr

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SLIDE 36

Couplings quadratic in curvature and quartic in spin

M2 m2 = 1 − 2

2!Ruσuσ + 2 3!R∗ uσuσ;σ + 2 4!Ruσuσ;σσ + . . .

⊕ ∼ R2σ4 ⊕ . . . In terms of the STF tidal tensors, Eµν = Rµκνλuκuλ, Bµν = R∗

µκνλuκuλ,

δ(M2 m2 )4 = C4A (Eσσ)2 + C4D (Bσσ)2 + C4B EσµEσ

µσ2 + C4E BσµBσ µσ2

+ C4C EµνEµνσ4 + C4F BµνBµνσ4 + C4G Ruσuσ;uuσ2. Aligned-spin scattering in Kerr at O(G2σ4) depends only on C4a = C4A + C4B, C4c = C4C, C4e = C4E + C4F 2 .

Justin Vines Max Planck Institute for Gravitational Physics (AEI) Potsdam Test black holes, Amplitudes, and perturbations of Kerr

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SLIDE 37

Couplings quadratic in curvature and quartic in spin

Effective test black hole: M2 m2 = 1 − 2

2!Ruσuσ + 2 3!R∗ uσuσ;σ + 2 4!Ruσuσ;σσ + . . .

+ C4.. · R2σ4 ⊕ . . . Aligned-spin scattering angle for the test black hole (σ) in a Kerr background (m, a), at the leading order in the ultrarelativistic limit v → 1: χ = −315π(Gm)2(5b − 4a)σ4 256(b + a)4(b2 − a2)3/2 C4a + 2C4c + C4e 1 − v2 + O(1 − v2)0 + O(G2σ5) + O(G3). ⇒ we want C4a + 2C4c + C4e ¨ = 0 ?

Justin Vines Max Planck Institute for Gravitational Physics (AEI) Potsdam Test black holes, Amplitudes, and perturbations of Kerr

slide-38
SLIDE 38

Matching to perturbed Kerr (self-force)

[Siemonsen-JV 1909.07361] For an arbitrary-mass two-black-hole system, assume the validity of χ ˙ = GE v2

±

(1 ± v)2 b ± a1 ± a2 +

  • Gm2F(v, b, a1, a2) + 1 ↔ 2 + O(a5

1,2)

  • + . . .
  • ˙

= + + + . . . with F(v, b, σ, a) determined by a consistent test-black-hole limit

Justin Vines Max Planck Institute for Gravitational Physics (AEI) Potsdam Test black holes, Amplitudes, and perturbations of Kerr

slide-39
SLIDE 39

Matching to perturbed Kerr (self-force)

[Siemonsen-JV 1909.07361] For an arbitrary-mass two-black-hole system, assume the validity of χ ˙ = GE v2

±

(1 ± v)2 b ± a1 ± a2 +

  • Gm2F(v, b, a1, a2) + 1 ↔ 2 + O(a5

1,2)

  • + . . .
  • with F(v, b, σ, a) determined by a consistent test-black-hole limit

Assume the existence of a local-in-time canonical Hamiltonian H for the two-body conservative dynamics—determined modulo gauge by χ—, and employ the associated first law of spinning binary mechanics [Le Tiec+] (circular, aligned-spin), dE = Ω dL +

  • a
  • za dma + Ωa dSa
  • ,

to compute the (Detweiler) redshift invariants za = ∂H ∂ma and the spin precession frequencies Ωa = ∂H ∂Sa

Justin Vines Max Planck Institute for Gravitational Physics (AEI) Potsdam Test black holes, Amplitudes, and perturbations of Kerr

slide-40
SLIDE 40

Matching to perturbed Kerr (self-force)

[Siemonsen-JV 1909.07361] For an arbitrary-mass two-black-hole system, assume the validity of χ ˙ = GE v2

±

(1 ± v)2 b ± a1 ± a2 +

  • Gm2F(v, b, a1, a2) + 1 ↔ 2 + O(a5

1,2)

  • + . . .
  • with F(v, b, σ, a) determined by a consistent test-black-hole limit.

... local-in-time canonical Hamiltonian ... first law (aligned, circular) ... dE = Ω dL +

  • a
  • za dma + Ωa dSa
  • ,

Take the m2 ≫ m1 limit and compare to self-force results for z1 [Kavanagh, Ottewill, Wardell] and Ω1 [Bini, Damour, Geralico, Kavanagh, van de Meent] ⇒ C4a + 6C4c ˙ = 0, C4a + 3C4e ˙ = 0.

Justin Vines Max Planck Institute for Gravitational Physics (AEI) Potsdam Test black holes, Amplitudes, and perturbations of Kerr

slide-41
SLIDE 41

Matching to perturbed Kerr (self-force)

χ ˙ = GE v2

±

(1 ± v)2 b ± a1 ± a2 +

  • Gm2F(v, b, a1, a2) + 1 ↔ 2 + O(a5

1,2)

  • + . . .
  • The AHH-Guevara-GOV result for the aligned-spin BBH scat. angle,

F(v, b, σ, a) = π 2v2 ∂ ∂b

  • dζ

2πi (1 − vζ)4 (ζ2 − 1)3/2

  • b − ζa − ζ − v

1 − vζ σ −1 + O(σ5)? corresponds precisely to C4a ¨ = C4c ¨ = C4e ¨ = 0, which is the unique solution of the two equations from circular self-force matching and the

  • ne from a finite ultrarelativistic limit (v → 1).

Further comments: — self-force results with nonzero eccentricity e, at O(e≥2a3,4) would provide new equations yielding an overdetermined system — applicability of the first law — higher orders in spin — √ Kerr — G3

Justin Vines Max Planck Institute for Gravitational Physics (AEI) Potsdam Test black holes, Amplitudes, and perturbations of Kerr

slide-42
SLIDE 42

Summary

Solving the classical relativistic two-body problem ... via quantum amplitudes ? ... ... has surpassed traditional classical methods. Perturbative expansions ... and attempts to resum them ... ... with some success in v/c and in spin. Amplitudes for massive higher-spin particles exchanging gravitons ... “minimal coupling” “understood” up to spin-2; at spin-5/2 and beyond?? A spinning test black hole (?) moving in an exact Kerr background ... makes sense (and matches above) up to spin4; at spin5 and beyond???

Justin Vines Max Planck Institute for Gravitational Physics (AEI) Potsdam Test black holes, Amplitudes, and perturbations of Kerr