From boundary data to bound states
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Springer
Received: October 17, Revised: December 2, Accepted: December 16, Published: January 14,
2019 2019 2019 2020
From boundary data to bound states
a
Department of Physics and Astronomy, Uppsala University, Box 516, 751 20 Uppsala, Sweden b Deutsches Elektronen-Synchrotron DESY, Notkestrasse 85, 22607 Hamburg, Germany c The Abdus Salam International Center for Theoretical Physics, Strada Costiera 11, Trieste 34151, Italy
E-mail: [email protected], [email protected] Abstract: We introduce a — somewhat holographic — dictionary between gravitational observables for scattering processes (measured at the boundary) and adiabatic invariants for bound orbits (in the bulk ), to all orders in the Post-Minkowskian (PM) expansion. Our map relies on remarkable connections between the relative momentum of the twobody problem, the classical limit of the scattering amplitude and the deflection angle in hyperbolic motion. These relationships allow us to compute observables for generic orbits (such as the periastron advance ∆Φ) through analytic continuation, via a radial action depending only on boundary data. A simplified (more geometrical) map can be obtained for circular orbits, enabling us to extract the orbital frequency as a function of the (conserved) binding energy, Ω(E), directly from scattering information. As an example, using the results in Bern et al. [36, 37], we readily derive Ω(E) and ∆Φ(J, E) to two-loop orders. We also provide closed-form expressions for the orbital frequency and periastron advance at tree-level and one-loop order, respectively, which capture a series of exact terms in the Post-Newtonian expansion. We then perform a partial PM resummation, using a no-recoil approximation for the amplitude. This limit is behind the map between the scattering angle for a test-particle and the two-body dynamics to 2PM. We show that it also captures a subset of higher order terms beyond the test-particle limit. While a (rather lengthy) Hamiltonian may be derived as an intermediate step, our map applies directly between gauge invariant quantities. Our findings provide a starting point for an alternative approach to the binary problem. We conclude with future directions and some speculations on the classical double copy. Keywords: Classical Theories of Gravity, Scattering Amplitudes ArXiv ePrint: 1910.03008
c The Authors. Open Access, Article funded by SCOAP3 .
https://doi.org/10.1007/JHEP01(2020)072
JHEP01(2020)072
Gregor K¨ alina and Rafael A. Portob,c
Contents 1
2 From dynamics to scattering angles 2.1 Hamiltonian approach 2.2 Post-Minkowskian expansion
5 5 6
3 From scattering angles to dynamics 3.1 Inversion formula 3.2 Gravitational potential
7 7 8
4 From amplitudes to scattering angles 4.1 Amplitude → impetus. . . 4.2 . . . → deflection angle
9 10 13
5 From scattering data to adiabatic invariants 5.1 Radial action 5.2 Periastron advance to two-loops 5.3 From hyperbolas to ellipses. . . 5.4 . . . to circular orbits 5.5 Orbital frequency to 3PM
16 16 18 20 24 27
6 No-recoil resu
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