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Mining the Geodesic Equation for Scattering Data

Clifford Cheung, Nabha Shah, Mikhail P. Solon

TL;DR

This work develops a robust, algebraic bridge between geodesic motion in a perturbed Schwarzschild background and conservative scattering data in the post-Minkowskian regime. It provides two complementary routes: (i) a leading-$PM$ map that yields closed-form isotropic Hamiltonians and amplitudes for tidal operators and certain higher-derivative/charge perturbations, and (ii) an all-orders, diffeomorphism-based method to transform to isotropic coordinates and extract the corresponding amplitudes at arbitrary PM order in the test-particle limit. The authors derive explicit, closed-form expressions for infinite classes of tidal operators such as $[\mathcal E^n]$ and $[\mathcal B^n]$, as well as for higher-derivative and electric-charge corrections, and they validate these results against known PM/PN limits and consistency checks. A key outcome is a practical toolkit to bootstrap higher-PM computations directly from geodesic dynamics, with potential applications to neutron-star tides and beyond‑GR scenarios, including checks via an all-orders test-particle construction. The methods unify gauge choices through an isotropic mapping and offer explicit formulas for scattering amplitudes and Hamiltonians that can aid future theoretical and phenomenological analyses of gravitational scattering.

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.

Mining the Geodesic Equation for Scattering Data

TL;DR

This work develops a robust, algebraic bridge between geodesic motion in a perturbed Schwarzschild background and conservative scattering data in the post-Minkowskian regime. It provides two complementary routes: (i) a leading- map that yields closed-form isotropic Hamiltonians and amplitudes for tidal operators and certain higher-derivative/charge perturbations, and (ii) an all-orders, diffeomorphism-based method to transform to isotropic coordinates and extract the corresponding amplitudes at arbitrary PM order in the test-particle limit. The authors derive explicit, closed-form expressions for infinite classes of tidal operators such as and , as well as for higher-derivative and electric-charge corrections, and they validate these results against known PM/PN limits and consistency checks. A key outcome is a practical toolkit to bootstrap higher-PM computations directly from geodesic dynamics, with potential applications to neutron-star tides and beyond‑GR scenarios, including checks via an all-orders test-particle construction. The methods unify gauge choices through an isotropic mapping and offer explicit formulas for scattering amplitudes and Hamiltonians that can aid future theoretical and phenomenological analyses of gravitational scattering.

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.

Paper Structure

This paper contains 21 sections, 73 equations, 2 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Test-Particle Dynamics
  3. Geodesic to Hamiltonian
  4. Hamiltonian to Amplitude
  5. Leading Order in $G$
  6. Lifting to Arbitrary Mass Ratio
  7. Fourier Transform
  8. Tidal Operators
  9. Kretschmann Scalar to a Power: ${\cal O} = [C^2]^n$
  10. Electric Weyl to a Power: ${\cal O} = [{\cal E}^n]$
  11. Magnetic Weyl to a Power: ${\cal O} = [{\cal B}^{n}]$
  12. Beyond Schwarzschild
  13. Higher Derivative Corrections
  14. Electric Charge
  15. Electric Charge and Higher Derivative Corrections
  16. ...and 6 more sections

Figures (2)

  • Figure 1: At leading PM order, tidal corrections to the scattering amplitude arise from fan diagrams. For an operator composed of $n$ curvature tensors, this contribution corresponds to an $(n-1)$-loop diagram. Thick and thin lines denote matter and graviton lines while the black dots denote tidal operator insertions.
  • Figure 2: To all orders in PM, contributions to the scattering amplitude of a tidally distorted test particle are generated by fan diagrams. Shown here are examples at one-, two-, and three-loop orders.