(New) perspectives on the relativistic binary problem Jan Steinhoff Max-Planck-Institute for Gravitational Physics (Albert-Einstein-Institute), Potsdam-Golm, Germany QCD Meets Gravity 2019, UCLA, December 10th Jan Steinhoff (AEI) (New) perspectives on the relativistic binary problem UCLA, December 10th, 2019 0 / 13
The relativistic binary problem Jan Steinhoff (AEI) (New) perspectives on the relativistic binary problem UCLA, December 10th, 2019 1 / 13
’Old’ perspective on the relativistic binary problem J. A. Wheeler nytimes.com “The Hamilton-Jacobi description of motion: natural because ratified by the quantum principle” Box 25.3 in [ Gravitation , Misner, Thorne, Wheeler (MTW)] Jan Steinhoff (AEI) (New) perspectives on the relativistic binary problem UCLA, December 10th, 2019 2 / 13
Hamilton-Jacobi is natural! MTW, Box 25.3 “quantum mechanical” amplitude ∼ wavefunction ψ ∼ e iS S satisfies Hamilton-Jacobi equation: p µ = ∂ S ∂ x µ classical case: build wavepackets! interference of waves with different frequency/energy E + ∆ E constructive interference: 0 = S E +∆ E − S E ∂ S E classical − → ∂ E = 0 ∆ E Jan Steinhoff (AEI) (New) perspectives on the relativistic binary problem UCLA, December 10th, 2019 3 / 13
Hamilton-Jacobi is natural! MTW, Box 25.3 “quantum mechanical” amplitude ∼ wavefunction ψ ∼ e iS S satisfies Hamilton-Jacobi equation: p µ = ∂ S ∂ x µ classical case: build wavepackets! interference of waves with different frequency/energy E + ∆ E constructive interference: 0 = S E +∆ E − S E ∂ S E classical − → ∂ E = 0 ∆ E MTW, Box 25.3 Jan Steinhoff (AEI) (New) perspectives on the relativistic binary problem UCLA, December 10th, 2019 3 / 13
Hamilton-Jacobi is natural! MTW, Box 25.3 “quantum mechanical” amplitude ∼ wavefunction ψ ∼ e iS S satisfies Hamilton-Jacobi equation: p µ = ∂ S ∂ x µ classical case: build wavepackets! interference of waves with different frequency/energy E + ∆ E constructive interference: 0 = S E +∆ E − S E ∂ S E classical − → ∂ E = 0 ∆ E MTW, Box 25.3 Jan Steinhoff (AEI) (New) perspectives on the relativistic binary problem UCLA, December 10th, 2019 3 / 13
New perspective: scattering black holes are natural! χ Classical scattering: scattering angle χ (more for spinning black holes) Quantum scattering: probability amplitude M M talks by Solon, Zeng, Damgaard, Guevara, Kosower, Bjerrum-Bohr, K¨ alin, Shen, Luna black holes ∼ higher-spin massive particles ? Vaidya (2015); Guevara, Ochirov, Vines (2018); Chung, Huang, Kim, Lee (2019); Guevara, Ochirov, Vines (2019); Siemonsen, Vines (2019); Arkani-Hamed, Huang, O’Connell (2019); Bautista, Guevara (2019); Guevara (2019); Arkani-Hamed, Huang, Huang (2017); talks by Vines, Ochirov Jan Steinhoff (AEI) (New) perspectives on the relativistic binary problem UCLA, December 10th, 2019 4 / 13
New perspective: scattering black holes are natural! χ black holes ( ∼ higher-spin + massive particles) G graviton Classical scattering: scattering angle χ (more for spinning black holes) Quantum scattering: probability amplitude M M talks by Solon, Zeng, Damgaard, Guevara, Kosower, Bjerrum-Bohr, K¨ alin, Shen, Luna black holes ∼ higher-spin massive particles ? Vaidya (2015); Guevara, Ochirov, Vines (2018); Chung, Huang, Kim, Lee (2019); Guevara, Ochirov, Vines (2019); Siemonsen, Vines (2019); Arkani-Hamed, Huang, O’Connell (2019); Bautista, Guevara (2019); Guevara (2019); Arkani-Hamed, Huang, Huang (2017); talks by Vines, Ochirov Jan Steinhoff (AEI) (New) perspectives on the relativistic binary problem UCLA, December 10th, 2019 4 / 13
Compact binaries from effective field theory (EFT) A modern approach to the relativistic binary problem: EFT! [Goldberger, Rothstein, PRD 73 (2006) 104029], talks by Levi, Maia, Maier, Yang, . . . A i A j leading-order spin(1)-spin(2) S 1 S 2 interaction as graviton exchange L S 1 S 2 = 1 1 � ∂ k A i ∂ ℓ A j � 1 2 S ki 2 S ℓ j [ignoring time integrals and δ ( t 1 − t 2 ) factors] 2 e i � k ( � x 1 − � x 2 ) = 1 1 ∂ � dk 2 S ℓ j 2 S ki 2 δ ij ( − 16 π G ) 1 � ∂ x k 1 ∂ x ℓ ( 2 π ) 3 k 2 2 � 1 � ∂ = − GS ki 1 S ℓ i (where r 12 = | � x 1 − � x 2 | ) 2 ∂ x k 1 ∂ x ℓ r 12 2 Jan Steinhoff (AEI) (New) perspectives on the relativistic binary problem UCLA, December 10th, 2019 5 / 13
Results for the post-Newtonian potential conservative part of the motion of the binary post-Newtonian (PN) approximation: expansion around 1 c → 0 (Newton) c 0 c − 1 c − 2 c − 3 c − 4 c − 5 c − 6 c − 7 c − 8 order N 1PN 2PN 3PN 4PN � � � � � non spin � � � spin-orbit � � � Spin 2 � Spin 3 � Spin 4 . ... . . Work by many people (“just” for the spin sector): Barker, Blanchet, Boh´ e, Buonanno, O’Connell, Damour, D’Eath, Faye, Hartle, Hartung, Hergt, Jaranowski, Marsat, Levi, Ohashi, Owen, Perrodin, Poisson, Porter, Porto, Rothstein, Sch¨ afer, Steinhoff, Tagoshi, Thorne, Tulczyjew, Vaidya Code for the spin part using EFT: M. Levi, JS, CQG 34 (2017), 244001 Jan Steinhoff (AEI) (New) perspectives on the relativistic binary problem UCLA, December 10th, 2019 6 / 13
Results for the post-Newtonian potential conservative part of the motion of the binary post-Newtonian (PN) approximation: expansion around 1 c → 0 (Newton) c 0 c − 1 c − 2 c − 3 c − 4 c − 5 c − 6 c − 7 c − 8 order N 1PN 2PN 3PN 4PN � � � � � non spin � � � spin-orbit � � � Spin 2 � Spin 3 Possible resummation: along diagonal � Spin 4 ∼ naked (st)ring singularities . ... . . Work by many people (“just” for the spin sector): Barker, Blanchet, Boh´ e, Buonanno, O’Connell, Damour, D’Eath, Faye, Hartle, Hartung, Hergt, Jaranowski, Marsat, Levi, Ohashi, Owen, Perrodin, Poisson, Porter, Porto, Rothstein, Sch¨ afer, Steinhoff, Tagoshi, Thorne, Tulczyjew, Vaidya Code for the spin part using EFT: M. Levi, JS, CQG 34 (2017), 244001 Jan Steinhoff (AEI) (New) perspectives on the relativistic binary problem UCLA, December 10th, 2019 6 / 13
Summing spin to infinity (leading PN order) J. Vines, JS, PRD 97 (2018), 064010 Start from an effective point-particle action: � ∞ � � 1 � ℓ !( I L E L − J L B L ) + ... S = d τ − m + · · · + ℓ = 2 Infinite number of higher dimensional couplings, one for each multipole I L , J L . Here: E L / B L are the electric/magnetic parts of (derivatives of) the curvature. For black-holes (BHs): a = spin / m = radius of ring singularity (mass ℓ -pole I L ) + i (current ℓ -pole J L ) = m i ℓ a L � � � STF Jan Steinhoff (AEI) (New) perspectives on the relativistic binary problem UCLA, December 10th, 2019 7 / 13
Summing spin to infinity (leading PN order) J. Vines, JS, PRD 97 (2018), 064010 Start from an effective point-particle action: � ∞ � � 1 � ℓ !( I L E L − J L B L ) + ... S = d τ − m + · · · + ℓ = 2 Infinite number of higher dimensional couplings, one for each multipole I L , J L . Here: E L / B L are the electric/magnetic parts of (derivatives of) the curvature. For black-holes (BHs): a = spin / m = radius of ring singularity (mass ℓ -pole I L ) + i (current ℓ -pole J L ) = m i ℓ a L � � � STF Jan Steinhoff (AEI) (New) perspectives on the relativistic binary problem UCLA, December 10th, 2019 7 / 13
Summing spin to infinity (result) J. Vines, JS, PRD 97 (2018), 064010 Infinite sum of higher dimensional couplings, one for each multipole. . . Double-infinite sum of multipole-multipole interaction. . . Still, the S ∞ series can be resummed! (in the leading-order Hamiltonian H ) � � � � � P 2 A + 1 S 1 S 2 2 µ − µ U + 4 � P · � � · � H = P × + ∇ µ U 2 m 2 m 2 1 2 where M = m 1 + m 2 , µ = M 1 m 2 / M , a i = � � a 0 = � a 1 + � � a 2 , S i / m i � R × � Mr A = − U a 0 � U = 0 cos 2 θ , r 2 + a 2 r 2 + a 2 2 0 Linearized Kerr metric! ∼ Test-mass motion in Kerr metric Jan Steinhoff (AEI) (New) perspectives on the relativistic binary problem UCLA, December 10th, 2019 8 / 13
Summing spin to infinity (result) J. Vines, JS, PRD 97 (2018), 064010 Infinite sum of higher dimensional couplings, one for each multipole. . . Double-infinite sum of multipole-multipole interaction. . . Still, the S ∞ series can be resummed! (in the leading-order Hamiltonian H ) � � � � � P 2 A + 1 S 1 S 2 2 µ − µ U + 4 � P · � � · � H = P × + ∇ µ U 2 m 2 m 2 1 2 where M = m 1 + m 2 , µ = M 1 m 2 / M , a i = � � a 0 = � a 1 + � � a 2 , S i / m i � R × � Mr A = − U a 0 � U = 0 cos 2 θ , r 2 + a 2 r 2 + a 2 2 0 Linearized Kerr metric! ∼ Test-mass motion in Kerr metric oblate-spheroidal coord. r , θ Jan Steinhoff (AEI) (New) perspectives on the relativistic binary problem UCLA, December 10th, 2019 8 / 13
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