Spinning binary black holes. effective worldline actions, post-newtonian vs. post-minkowskian gravity, and input from Amplitudes

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1 : effective worldline actions, post-newtonian vs. post-minkowskian gravity, and input from Amplitudes Max Planck Institute for Gravitational Physics (AEI) Potsdam-Golm, Germany QCD meets Gravity, UCLA/Caltech 14 Dec. 2017

2 Nonspinning black holes as effective point-masses Effective action for (monopole) point particles with worldlines x = z(τ) coupled to gravity, metric g µν (x): S[g, z] = g R d 4 x 16πG particles m dτ g µν (z)ż µ ż ν. Phase-space action (auxiliary field/lagrangian multiplier λ): S[g, z, p] = g R d 4 x 16πG + [ dτ p µ ż µ λ ] 2 (p2 + m 2 ). (δ p S = 0 p µ = m żµ ż 2 ; δ p,zs = 0 λ = ż 2 ) (-,+,+,+)

3 Nonspinning black holes as effective point-masses Effective action for (monopole) point particles with worldlines x = z(τ) coupled to gravity, metric g µν (x): S[g, z] = g R d 4 x 16πG particles m dτ g µν (z)ż µ ż ν. Phase-space action (auxiliary field/lagrangian multiplier λ): S[g, z, p] = g R d 4 x 16πG + [ dτ p µ ż µ λ ] 2 (p2 + m 2 ). (δ p S = 0 p µ = m żµ ż 2 ; δ p,zs = 0 λ = ż 2 ) (-,+,+,+)

4 Nonspinning black holes as effective point-masses Effective action for (monopole) point particles with worldlines x = z(τ) coupled to gravity, metric g µν (x): S[g, z] = g R d 4 x 16πG particles m dτ g µν (z)ż µ ż ν. Phase-space action (auxiliary field/lagrangian multiplier λ): S[g, z, p] = g R d 4 x 16πG + [ dτ p µ ż µ λ ] 2 (p2 + m 2 ). (δ p S = 0 p µ = m żµ ż 2 ; δ p,zs = 0 λ = ż 2 ) (-,+,+,+)

5 Nonspinning black holes as effective point-masses Phase-space action: S[g, z, p] = g R d 4 x 16πG + [ dτ p µ ż µ λ ] 2 (p2 + m 2 ). δs = 0 1 ( R µν 1 8πG 2 Rgµν) = T µν = dτ ż (µ p ν) δ4 (x z), g µ T µν = 0 p ν ν p µ = 0. One particle alone in empty spacetime g µν = Schwarzschild. Two particles: solve perturbatively (and integrate out the field)... (OR: take the classical limit of two minimally coupled massive scalars...)

6 Nonspinning black holes as effective point-masses Phase-space action: S[g, z, p] = g R d 4 x 16πG + [ dτ p µ ż µ λ ] 2 (p2 + m 2 ). δs = 0 1 ( R µν 1 8πG 2 Rgµν) = T µν = dτ ż (µ p ν) δ4 (x z), g µ T µν = 0 p ν ν p µ = 0. One particle alone in empty spacetime g µν = Schwarzschild. Two particles: solve perturbatively (and integrate out the field)... (OR: take the classical limit of two minimally coupled massive scalars...)

7 Nonspinning black holes as effective point-masses Phase-space action: S[g, z, p] = g R d 4 x 16πG + [ dτ p µ ż µ λ ] 2 (p2 + m 2 ). δs = 0 1 ( R µν 1 8πG 2 Rgµν) = T µν = dτ ż (µ p ν) δ4 (x z), g µ T µν = 0 p ν ν p µ = 0. One particle alone in empty spacetime g µν = Schwarzschild. Two particles: solve perturbatively (and integrate out the field)... (OR: take the classical limit of two minimally coupled massive scalars...)

8 Nonspinning black holes as effective point-masses Phase-space action: S[g, z, p] = g R d 4 x 16πG + [ dτ p µ ż µ λ ] 2 (p2 + m 2 ). δs = 0 1 ( R µν 1 8πG 2 Rgµν) = T µν = dτ ż (µ p ν) δ4 (x z), g µ T µν = 0 p ν ν p µ = 0. One particle alone in empty spacetime g µν = Schwarzschild. Two particles: solve perturbatively (and integrate out the field)... (OR: take the classical limit of two minimally coupled massive scalars...)

9 Nonspinning black holes as effective point-masses Phase-space action: S[g, z, p] = g R d 4 x 16πG + [ dτ p µ ż µ λ ] 2 (p2 + m 2 ). δs = 0 1 ( R µν 1 8πG 2 Rgµν) = T µν = dτ ż (µ p ν) δ4 (x z), g µ T µν = 0 p ν ν p µ = 0. One particle alone in empty spacetime g µν = Schwarzschild. Two particles: solve perturbatively (and integrate out the field)... (OR: take the classical limit of two minimally coupled massive scalars...)

10 Nonspinning black holes as effective point-masses Phase-space action: S[g, z, p] = g R d 4 x 16πG + [ dτ p µ ż µ λ ] 2 (p2 + m 2 ). δs = 0 1 ( R µν 1 8πG 2 Rgµν) = T µν = dτ ż (µ p ν) δ4 (x z), g µ T µν = 0 p ν ν p µ = 0. One particle alone in empty spacetime g µν = Schwarzschild. Two particles: solve perturbatively (and integrate out the field)... (OR: take the classical limit of two minimally coupled massive scalars...)

11 Solving the relativistic two-body problem Numerical relativity (solve full Einstein field eqs. on supercomputers) Analytic approximation schemes: Limit Newtonian gravity Perturbation theory post-newtonian c m 1 m 2 1, Gm rc 2 v2 c 2 1 test-body motion in a stationary background m 1 0 m 2 special relativity m 1 m 2 1, post-test-body ( self-force ) Gm rc 2 post-minkowskian v2 c 2 1 G 0 m 1 m 2 1, Gm rc 2 v2 c 2 1

12 post-newtonian (PN) vs. post-minkowskian (PM) L = Mc 2 + µv2 2 + GMµ + 1 ( ) r c }{{} ( ) c PN 1PN 2PN 3PN 4PN 5PN... 0PM: 1 v 2 v 4 v 6 v 8 v 10 v 12 1PM: 1/r v 2 /r v 4 /r v 6 /r v 8 /r v 10 /r 2PM: 1/r 2 v 2 /r 2 v 4 /r 2 v 6 /r 2 v 8 /r 2 3PM: 1/r 3 v 2 /r 3 v 4 /r 3 v 6 /r 3 4PM: 1/r 4 v 2 /r 4 v 4 /r Mc 2, v 2 v2 c 2, 1 r GM rc 2. current PN results; current PM results; overlap; unknown.

13 1PM (monopole/scalar) gravitational scattering

14 1PM (monopole/scalar) gravitational scattering

15 1PM (monopole/scalar) gravitational scattering Impact parameter: b µ = (x µ 1 xµ 2 ) min, b p 1 = b p 2 = 0,

16 1PM (monopole/scalar) gravitational scattering Impact parameter: b µ = (x µ 1 xµ 2 ) min, b p 1 = b p 2 = 0, Relative Lorentz factor: γ = p 1 p 2 m 1 m 2,

17 1PM (monopole/scalar) gravitational scattering Impact parameter: b µ = (x µ 1 xµ 2 ) min, b p 1 = b p 2 = 0, Relative Lorentz factor: γ = p 1 p 2 m 1 m 2, Deflection: p µ 2γ 2 1 b µ = 2Gm 1 m 2 γ2 1 b 2 + O(G2 ). [Portilla 79, 80; Westpfahl+ 79, 85; Bel+ 81; Damour+ 81;...]

18 Spinning black holes Effective phase-space action for a spinning black hole: ( worldline z(τ), momentum p µ (τ), ˆp µ = p µ / ) p 2 tetrad Λ µ a (τ), spin S µν (τ), S BH [z, p, Λ, S, g] = dτ {p µ ż µ + 12 S µνλ aµ DΛ a ν dτ χ µ S µν p ν λ [ p 2 + M 2 (z, ˆp, S)] }. 2 δs BH = 0 Mathisson-Papapetrou-Dixon (MPD) eqs.: Dp µ dτ DS µν dτ = 1 2 R µνρσż ν S ρσ + p ż D 2 z µ log M2, ( = 2p [µ ż ν] + p ż p [µ + 2S [µ ρ p ν] S ν]ρ ) log M 2.

19 Spinning black holes Effective phase-space action for a spinning black hole: ( worldline z(τ), momentum p µ (τ), ˆp µ = p µ / ) p 2 tetrad Λ µ a (τ), spin S µν (τ), S BH [z, p, Λ, S, g] = dτ {p µ ż µ + 12 S µνλ aµ DΛ a ν dτ χ µ S µν p ν λ [ p 2 + M 2 (z, ˆp, S)] }. 2 δs BH = 0 Mathisson-Papapetrou-Dixon (MPD) eqs.: Dp µ dτ DS µν dτ = 1 2 R µνρσż ν S ρσ + p ż D 2 z µ log M2, ( = 2p [µ ż ν] + p ż p [µ + 2S [µ ρ p ν] S ν]ρ ) log M 2.

20 Spinning black holes Effective phase-space action for a spinning BH: S BH = dτ {p µ ż µ + 12 S µνλ aµ DΛ a ν χ µ S µν p ν λ [ p 2 + M 2]}. dτ 2 Mass-rescaled (Pauli-Lubanski) spin vector: a µ = 1 2m ɛµ νρσ ˆp ν S ρσ. Dynamical mass function for a BH: ( M 2 = m 2 + 2m 2 ˆp µ ˆp ν 1 2! Rµ α ν β a α a β : quadrupole S R µ α ν β;γ a α a β a γ : octupole S 3 3! + 1 4! Rµ α ν β;γδ a α a β a γ a δ : hexadecapole S 4 ) +...

21 Spinning black holes Effective phase-space action for a spinning BH: S BH = dτ {p µ ż µ + 12 S µνλ aµ DΛ a ν χ µ S µν p ν λ [ p 2 + M 2]}. dτ 2 Mass-rescaled (Pauli-Lubanski) spin vector: a µ = 1 2m ɛµ νρσ ˆp ν S ρσ. Dynamical mass function for a BH: ( M 2 = m 2 + 2m 2 ˆp µ ˆp ν 1 2! Rµ α ν β a α a β : quadrupole S R µ α ν β;γ a α a β a γ : octupole S 3 3! + 1 4! Rµ α ν β;γδ a α a β a γ a δ : hexadecapole S 4 ) +...

22 Spinning black holes Effective phase-space action for a spinning BH: S BH = dτ {p µ ż µ + 12 S µνλ aµ DΛ a ν χ µ S µν p ν λ [ p 2 + M 2]}. dτ 2 Mass-rescaled (Pauli-Lubanski) spin vector: a µ = 1 2m ɛµ νρσ ˆp ν S ρσ. Dynamical mass function for a BH: ( M 2 = m 2 + 2m 2 ˆp µ ˆp ν 1 2! Rµ α ν β a α a β : quadrupole S R µ α ν β;γ a α a β a γ : octupole S 3 3! + 1 4! Rµ α ν β;γδ a α a β a γ a δ : hexadecapole S 4 ) +...

23 Eikonal limits of minimally coupled massive fields Scalar field geodesic motion (monopole only) Dirac field pole - dipole Spin-1 pole - dipole - (black-hole-)quadrupole (PN) [Holstein, Ross 08]... Spin-2 pole (black-hole-)hexadecapole (PN) [Vaidya 15]... Spin-s pole (black-hole-)2 2s pole [Guevara 17] The full black-hole spin-multipole structure is seemingly obtained from the eikonal limit of a minimally coupled massive infinite-spin field (how do we conclusively demonstrate that this is so?? or are there caveats?...)

24 Eikonal limits of minimally coupled massive fields Scalar field geodesic motion (monopole only) Dirac field pole - dipole Spin-1 pole - dipole - (black-hole-)quadrupole (PN) [Holstein, Ross 08]... Spin-2 pole (black-hole-)hexadecapole (PN) [Vaidya 15]... Spin-s pole (black-hole-)2 2s pole [Guevara 17] The full black-hole spin-multipole structure is seemingly obtained from the eikonal limit of a minimally coupled massive infinite-spin field (how do we conclusively demonstrate that this is so?? or are there caveats?...)

25 Eikonal limits of minimally coupled massive fields Scalar field geodesic motion (monopole only) Dirac field pole - dipole Spin-1 pole - dipole - (black-hole-)quadrupole (PN) [Holstein, Ross 08]... Spin-2 pole (black-hole-)hexadecapole (PN) [Vaidya 15]... Spin-s pole (black-hole-)2 2s pole [Guevara 17] The full black-hole spin-multipole structure is seemingly obtained from the eikonal limit of a minimally coupled massive infinite-spin field (how do we conclusively demonstrate that this is so?? or are there caveats?...)

26 Eikonal limits of minimally coupled massive fields Scalar field geodesic motion (monopole only) Dirac field pole - dipole Spin-1 pole - dipole - (black-hole-)quadrupole (PN) [Holstein, Ross 08]... Spin-2 pole (black-hole-)hexadecapole (PN) [Vaidya 15]... Spin-s pole (black-hole-)2 2s pole [Guevara 17] The full black-hole spin-multipole structure is seemingly obtained from the eikonal limit of a minimally coupled massive infinite-spin field (how do we conclusively demonstrate that this is so?? or are there caveats?...)

27 Eikonal limits of minimally coupled massive fields Scalar field geodesic motion (monopole only) Dirac field pole - dipole Spin-1 pole - dipole - (black-hole-)quadrupole (PN) [Holstein, Ross 08]... Spin-2 pole (black-hole-)hexadecapole (PN) [Vaidya 15]... Spin-s pole (black-hole-)2 2s pole [Guevara 17] The full black-hole spin-multipole structure is seemingly obtained from the eikonal limit of a minimally coupled massive infinite-spin field (how do we conclusively demonstrate that this is so?? or are there caveats?...)

28 Eikonal limits of minimally coupled massive fields Scalar field geodesic motion (monopole only) Dirac field pole - dipole Spin-1 pole - dipole - (black-hole-)quadrupole (PN) [Holstein, Ross 08]... Spin-2 pole (black-hole-)hexadecapole (PN) [Vaidya 15]... Spin-s pole (black-hole-)2 2s pole [Guevara 17] The full black-hole spin-multipole structure is seemingly obtained from the eikonal limit of a minimally coupled massive infinite-spin field (how do we conclusively demonstrate that this is so?? or are there caveats?...)

29 Eikonal limits of minimally coupled massive fields Scalar field geodesic motion (monopole only) Dirac field pole - dipole Spin-1 pole - dipole - (black-hole-)quadrupole (PN) [Holstein, Ross 08]... Spin-2 pole (black-hole-)hexadecapole (PN) [Vaidya 15]... Spin-s pole (black-hole-)2 2s pole [Guevara 17] The full black-hole spin-multipole structure is seemingly obtained from the eikonal limit of a minimally coupled massive infinite-spin field (how do we conclusively demonstrate that this is so?? or are there caveats?...)

30 Spinning black holes Effective phase-space action for a spinning BH: S BH = dτ {p µ ż µ + 12 S µνλ aµ DΛ a ν χ µ S µν p ν λ [ p 2 + M 2]}. dτ 2 Dynamical mass function for a BH: M 2 = m 2 + 2m 2 ˆp µ ˆp ν ( 1 2! Rµ α ν β a α a β : quadrupole S R µ α ν β;γ a α a β a γ : octupole S 3 3! + 1 4! Rµ α ν β;γδ a α a β a γ a δ : hexadecapole S 4 ) +... All the linear-in-riemann terms [the only ones that matter at O(G)] are fixed by symmetries and matching to the (linearized) Kerr solution...

31 Spinning black holes Effective phase-space action for a spinning BH: S BH = dτ {p µ ż µ + 12 S µνλ aµ DΛ a ν χ µ S µν p ν λ [ p 2 + M 2]}. dτ 2 Dynamical mass function for a BH: M 2 = m 2 + 2m 2 ˆp µ ˆp ν ( 1 2! Rµ α ν β a α a β : quadrupole S R µ α ν β;γ a α a β a γ : octupole S 3 3! + 1 4! Rµ α ν β;γδ a α a β a γ a δ : hexadecapole S 4 ) +... All the linear-in-riemann terms [the only ones that matter at O(G)] are fixed by symmetries and matching to the (linearized) Kerr solution...

32 Kerr-Schild structure of Schwarzschild...

33 ... vs. Kerr...

34 Spin-multipole structure of Kerr Exact Kerr: g µν = η µν + h µν + 2 (µ ξ ν). Linearized harmonic-gauge part h µν : hµν = P µν αβ h αβ h µν = P µν αβ = δ (µ (α δ ν) β) 1 2 η µνη αβ, ( 1 1 2! aα a β αβ + 1 4! aα a β a γ a δ αβγδ... ) 4m r u µu ν ( + a σ α 1 ) 4m 3! aσ a β a γ αβγ +... r u (µɛ α ν)ρσ u ρ = 4m exp(a ) α γ u γ u β, r (a ) µ ν ɛ µ ναβa α β. [Steinhoff, JV 16; Harte, JV 16; JV 17]

35 Spin-multipole structure of Kerr Exact Kerr: g µν = η µν + h µν + 2 (µ ξ ν). Linearized harmonic-gauge part h µν : hµν = P µν αβ h αβ h µν = P µν αβ = δ (µ (α δ ν) β) 1 2 η µνη αβ, ( 1 1 2! aα a β αβ + 1 4! aα a β a γ a δ αβγδ... ) 4m r u µu ν ( + a σ α 1 ) 4m 3! aσ a β a γ αβγ +... r u (µɛ α ν)ρσ u ρ = 4m exp(a ) α γ u γ u β, r (a ) µ ν ɛ µ ναβa α β. [Steinhoff, JV 16; Harte, JV 16; JV 17]

36 Spin-multipole structure of Kerr Exact Kerr: g µν = η µν + h µν + 2 (µ ξ ν). Linearized harmonic-gauge part h µν : hµν = P µν αβ h αβ h µν = P µν αβ = δ (µ (α δ ν) β) 1 2 η µνη αβ, ( 1 1 2! aα a β αβ + 1 4! aα a β a γ a δ αβγδ... ) 4m r u µu ν ( + a σ α 1 ) 4m 3! aσ a β a γ αβγ +... r u (µɛ α ν)ρσ u ρ = 4m exp(a ) α γ u γ u β, r (a ) µ ν ɛ µ ναβa α β. [Steinhoff, JV 16; Harte, JV 16; JV 17]

37 Spin-multipole structure of Kerr Exact Kerr: g µν = η µν + h µν + 2 (µ ξ ν). Linearized harmonic-gauge part h µν : hµν = P µν αβ h αβ h µν = P µν αβ = δ (µ (α δ ν) β) 1 2 η µνη αβ, ( 1 1 2! aα a β αβ + 1 4! aα a β a γ a δ αβγδ... ) 4m r u µu ν ( + a σ α 1 ) 4m 3! aσ a β a γ αβγ +... r u (µɛ α ν)ρσ u ρ = 4m exp(a ) α γ u γ u β, r (a ) µ ν ɛ µ ναβa α β. [Steinhoff, JV 16; Harte, JV 16; JV 17]

38 Spin-multipole structure of Kerr Exact Kerr: g µν = η µν + h µν + 2 (µ ξ ν). Linearized harmonic-gauge part h µν : hµν = P µν αβ h αβ h µν = P µν αβ = δ (µ (α δ ν) β) 1 2 η µνη αβ, ( 1 1 2! aα a β αβ + 1 4! aα a β a γ a δ αβγδ... ) 4m r u µu ν ( + a σ α 1 ) 4m 3! aσ a β a γ αβγ +... r u (µɛ α ν)ρσ u ρ = 4m exp(a ) α γ u γ u β, r (a ) µ ν ɛ µ ναβa α β. [Steinhoff, JV 16; Harte, JV 16; JV 17]

39 Linear interaction of two spinning black holes (see [JV 17]) L int = 1 2 ˆT µν (p 1, a 1, ) h 2µν (x) x=z1

40 Linear interaction of two spinning black holes = 2G ˆT µν (p 1, a 1, ) P µναβ ˆT αβ (p 2, a 2, ) 1 r 2 x=z1 (see [JV 17]) L int = 1 2 ˆT µν (p 1, a 1, ) h 2µν (x) x=z1

41 Linear interaction of two spinning black holes (see [JV 17]) L int = 1 2 ˆT µν (p 1, a 1, ) h 2µν (x) x=z1 = 2G ˆT µν (p 1, a 1, ) P µναβ ˆT αβ (p 2, a 2, ) 1 r 2 x=z1 = u 1µ exp(a 1 ) µ ν Q ν α exp(a 2 ) α β u β Gm 1 m 2 2 r 2 x=z1 Q µ α = 2P µ ναβu ν 1u β 2.

42 Linear interaction of two spinning black holes (see [JV 17]) L int = 1 2 ˆT µν (p 1, a 1, ) h 2µν (x) x=z1 = 2G ˆT µν (p 1, a 1, ) P µναβ ˆT αβ (p 2, a 2, ) 1 r 2 x=z1 = u 1µ exp(a 1 ) µ ν Q ν α exp(a 2 ) α β u β Gm 1 m 2 2 r 2 = u 1 exp(a1 ) Q exp(a 2 ) Gm 1 m 2 u2 r 2 x=z1 x=z1 Q µ α = 2P µ ναβu ν 1u β 2.

43 Linear interaction of two spinning black holes (see [JV 17]) L int = 1 2 ˆT µν (p 1, a 1, ) h 2µν (x) x=z1 = 2G ˆT µν (p 1, a 1, ) P µναβ ˆT αβ (p 2, a 2, ) 1 r 2 x=z1 = u 1µ exp(a 1 ) µ ν Q ν α exp(a 2 ) α β u β Gm 1 m 2 2 r 2 = u 1 exp(a1 ) Q exp(a 2 ) Gm 1 m 2 u2 = u 1 Q exp ( (a1 + a 2 ) ) u2 Gm 1 m 2 r 2 r 2. x=z1 x=z1 x=z1 Q µ α = 2P µ ναβu ν 1u β 2.

44 Linear interaction of two spinning black holes (see [JV 17]) L int = 1 2 ˆT µν (p 1, a 1, ) h 2µν (x) x=z1 = 2G ˆT µν (p 1, a 1, ) P µναβ ˆT αβ (p 2, a 2, ) 1 r 2 x=z1 = u 1µ exp(a 1 ) µ ν Q ν α exp(a 2 ) α β u β Gm 1 m 2 2 r 2 = u 1 exp(a1 ) Q exp(a 2 ) Gm 1 m 2 u2 = u 1 Q exp ( (a1 + a 2 ) ) u2 Gm 1 m 2 r 2 r 2. x=z1 x=z1 x=z1 Q µ α = 2P µ ναβu ν 1u β 2. This depends on the spins only through a 1 + a 2!

45 Linear interaction of two spinning black holes L int = u 1 exp(a1 ) Q exp(a 2 ) Gm 1 m 2 u2 = u 1 Q exp ( (a1 + a 2 ) ) u2 Gm 1 m 2 r 2 r 2. x=z1 x=z1 The double sum over BH1-multipole BH2-multipole couplings factorizes into a single sum over the multipoles of one effective spinning BH coupled to an effective point-mass at linear order in G.

46 Post-Minkowskian scattering angles {amplitudes} Scattering angle χ (encodes the full dynamics...) Impact parameter b ; γ = p 1 p 2 /m 1 m 2 ; M = m 1 + m 2, ν = m 1 m 2 /M 2 Schw.-Schw. 1PM/tree: χ = 2 GM b 1 + 2ν(γ 1) γ 2 (2γ 2 1) + O(G 2 ). 1 Kerr-Kerr (aligned-spins) 1PM/tree: {O(a ) Guevara 17} [O(a 1 ) Bini, Damour 17] [O(a ) JV 17] χ = 2 GM b 1 + 2ν(γ 1) γ 2 1 2γ 2 1 2γ γ 2 1 a 1 + a 2 b ( ) 2 + O(G 2 ). a1 + a 2 1 b

47 Post-Minkowskian scattering angles {amplitudes} Scattering angle χ (encodes the full dynamics...) Impact parameter b ; γ = p 1 p 2 /m 1 m 2 ; M = m 1 + m 2, ν = m 1 m 2 /M 2 Schw.-Schw. 1PM/tree: χ = 2 GM b 1 + 2ν(γ 1) γ 2 (2γ 2 1) + O(G 2 ). 1 Kerr-Kerr (aligned-spins) 1PM/tree: {O(a ) Guevara 17} [O(a 1 ) Bini, Damour 17] [O(a ) JV 17] χ = 2 GM b 1 + 2ν(γ 1) γ 2 1 2γ 2 1 2γ γ 2 1 a 1 + a 2 b ( ) 2 + O(G 2 ). a1 + a 2 1 b

48 Post-Minkowskian scattering angles {amplitudes} Scattering angle χ (encodes the full dynamics...) Impact parameter b ; γ = p 1 p 2 /m 1 m 2 ; M = m 1 + m 2, ν = m 1 m 2 /M 2 Schw.-Schw. 1PM/tree: χ = 2 GM b 1 + 2ν(γ 1) γ 2 (2γ 2 1) + O(G 2 ). 1 Kerr-Kerr (aligned-spins) 1PM/tree: {O(a ) Guevara 17} [O(a 1 ) Bini, Damour 17] [O(a ) JV 17] χ = 2 GM b 1 + 2ν(γ 1) γ 2 1 2γ 2 1 2γ γ 2 1 a 1 + a 2 b ( ) 2 + O(G 2 ). a1 + a 2 1 b

49 Post-Minkowskian scattering angles {amplitudes} Kerr-Kerr (aligned-spins) 1PM/tree: {O(a ) Guevara 17} [O(a 1 ) Bini, Damour 17] [O(a ) JV 17] χ = 2 GM b 1 + 2ν(γ 1) γ 2 1 2γ 2 1 2γ γ 2 1 a 1 + a 2 b ( ) 2 + O(G 2 ). a1 + a 2 1 b Schw.-Schw. 2PM/one-loop: [Westpfahl 85] {Damour 17} χ = 2 GM [ 1 + 2ν(γ 1) b γ 2 2γ π ] GM (5γ 2 1) + O(G 3 ). 1 8 b Kerr-Kerr 2PM/one-loop: {O(a 4 ) Guevara 17} 3PM/two-loop???

50 Post-Minkowskian scattering angles {amplitudes} Kerr-Kerr (aligned-spins) 1PM/tree: {O(a ) Guevara 17} [O(a 1 ) Bini, Damour 17] [O(a ) JV 17] χ = 2 GM b 1 + 2ν(γ 1) γ 2 1 2γ 2 1 2γ γ 2 1 a 1 + a 2 b ( ) 2 + O(G 2 ). a1 + a 2 1 b Schw.-Schw. 2PM/one-loop: [Westpfahl 85] {Damour 17} χ = 2 GM [ 1 + 2ν(γ 1) b γ 2 2γ π ] GM (5γ 2 1) + O(G 3 ). 1 8 b Kerr-Kerr 2PM/one-loop: {O(a 4 ) Guevara 17} 3PM/two-loop???

51 Post-Minkowskian scattering angles {amplitudes} Kerr-Kerr (aligned-spins) 1PM/tree: {O(a ) Guevara 17} [O(a 1 ) Bini, Damour 17] [O(a ) JV 17] χ = 2 GM b 1 + 2ν(γ 1) γ 2 1 2γ 2 1 2γ γ 2 1 a 1 + a 2 b ( ) 2 + O(G 2 ). a1 + a 2 1 b Schw.-Schw. 2PM/one-loop: [Westpfahl 85] {Damour 17} χ = 2 GM [ 1 + 2ν(γ 1) b γ 2 2γ π ] GM (5γ 2 1) + O(G 3 ). 1 8 b Kerr-Kerr 2PM/one-loop: {O(a 4 ) Guevara 17} 3PM/two-loop???

52 Post-Minkowskian scattering angles {amplitudes} Kerr-Kerr (aligned-spins) 1PM/tree: {O(a ) Guevara 17} [O(a 1 ) Bini, Damour 17] [O(a ) JV 17] χ = 2 GM b 1 + 2ν(γ 1) γ 2 1 2γ 2 1 2γ γ 2 1 a 1 + a 2 b ( ) 2 + O(G 2 ). a1 + a 2 1 b Schw.-Schw. 2PM/one-loop: [Westpfahl 85] {Damour 17} χ = 2 GM [ 1 + 2ν(γ 1) b γ 2 2γ π ] GM (5γ 2 1) + O(G 3 ). 1 8 b Kerr-Kerr 2PM/one-loop: {O(a 4 ) Guevara 17} 3PM/two-loop???