Abstract
The geodesic equation encodes test-particle dynamics at arbitrary gravitational coupling, hence retaining all orders in the post-Minkowskian (PM) expansion. Here we explore what geodesic motion can tell us about dynamical scattering in the presence of perturbatively small effects such as tidal distortion and higher derivative corrections to general relativity. We derive an algebraic map between the perturbed geodesic equation and the leading PM scattering amplitude at arbitrary mass ratio. As examples, we compute formulas for amplitudes and isotropic gauge Hamiltonians for certain infinite classes of tidal operators such as electric or magnetic Weyl to any power, and for higher derivative corrections to gravitationally interacting bodies with or without electric charge. Finally, we present a general method for calculating closed-form expressions for amplitudes and isotropic gauge Hamiltonians in the test-particle limit at all orders in the PM expansion.
Highlights
In recent years, powerful tools from the modern amplitudes program [1,2] and effective field theory have [3,4,5] been unified to derive new results [6,7,8,9] of relevance to the search for gravitational waves at the LIGO/Virgo experiment [10]
Powerful tools from the modern amplitudes program [1,2] and effective field theory have [3,4,5] been unified to derive new results [6,7,8,9] of relevance to the search for gravitational waves at the LIGO/Virgo experiment [10]. These efforts have spurred a resurgence of interest in post-Minkowskian (PM) perturbation theory [11,12], which organizes dynamics in powers of the gravitational constant G while retaining all orders in velocity
Many recent developments have benefited immensely from a genuine cross-pollination of ideas between classical general relativity and quantum field theory [18,19,20,21], culminating in new results pertaining to spinning black holes [22,23,24], orbital mechanics [25,26,27,28], and radiation [29]
Summary
Powerful tools from the modern amplitudes program [1,2] and effective field theory have [3,4,5] been unified to derive new results [6,7,8,9] of relevance to the search for gravitational waves at the LIGO/Virgo experiment [10]. We demonstrate the simplicity of our approach by deriving analytic formulas for scattering amplitudes and Hamiltonians at arbitrary mass ratio for certain infinite classes of tidal operators, including electric Weyl and magnetic Weyl to any power, as shown in Eq (27) and Eq (33) We apply this method to other types of perturbative corrections, e.g., which arise for electrically charged bodies or from higher derivative corrections to Einstein-Maxwell theory. As an application of these general formulas, we derive all orders in PM expressions for the isotropic Hamiltonian and scattering amplitude for a set of tidal operators, as presented in Table III and Table IV These results may provide useful data for checking future higher order PM calculations
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