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Self-force effects in post-Minkowskian scattering

Samuel E. Gralla, Kunal Lobo

TL;DR

This work develops a comprehensive treatment of self-force effects in weak-field gravitational scattering within the post-Minkowskian framework. By computing scalar, electromagnetic, and gravitational self-forces at leading PM order and including the matter-mediated force for gravity, the authors derive explicit, parameterized particle trajectories and conserved quantities, culminating in a full 2PM two-body gravitational scattering description in the center-of-energy frame. The analysis reproduces Westpfahl’s momentum transfer, aligns with radiative angular-momentum flux results, and introduces the mass moment (and its “scoot”) as a new diagnostic, thereby clarifying the role of self-force in PM scattering and laying groundwork for higher-order investigations. The approach combines Green-function techniques, tail integrals, and a frame-transformation symmetry to connect single-body self-force calculations with full two-body dynamics at 2PM order.

Abstract

We revisit the old problem of the self-force on a particle moving in a weak-field spacetime in the context of renewed interest in two-body gravitational scattering. We analytically calculate the scalar, electromagnetic, and gravitational self-force on a particle moving on a straight-line trajectory at a large distance from a Newtonian star, and use these results to find the associated correction to its motion. In the gravitational case we must also include the matter-mediated force, which acts at the same perturbative order as the gravitational self-force. We further augment the gravitational results with geodesic calculations at second order in the central body mass to determine the full, explicit solution to the two-body gravitational scattering problem at second post-Minkowskian order (2PM). We calculate the momentum transfer (which reproduces Westpfahl's old result), the change in mechanical angular momentum (which matches the radiative flux recently computed by Damour), and the change in mechanical mass moment (the time-space components of the angular momentum tensor), which has not previously appeared. Besides the new 2PM results of explicit trajectories and all conserved quantities, this work clarifies the role of gravitational self-force in PM scattering theory and provides a foundation for higher-order calculations.

Self-force effects in post-Minkowskian scattering

TL;DR

This work develops a comprehensive treatment of self-force effects in weak-field gravitational scattering within the post-Minkowskian framework. By computing scalar, electromagnetic, and gravitational self-forces at leading PM order and including the matter-mediated force for gravity, the authors derive explicit, parameterized particle trajectories and conserved quantities, culminating in a full 2PM two-body gravitational scattering description in the center-of-energy frame. The analysis reproduces Westpfahl’s momentum transfer, aligns with radiative angular-momentum flux results, and introduces the mass moment (and its “scoot”) as a new diagnostic, thereby clarifying the role of self-force in PM scattering and laying groundwork for higher-order investigations. The approach combines Green-function techniques, tail integrals, and a frame-transformation symmetry to connect single-body self-force calculations with full two-body dynamics at 2PM order.

Abstract

We revisit the old problem of the self-force on a particle moving in a weak-field spacetime in the context of renewed interest in two-body gravitational scattering. We analytically calculate the scalar, electromagnetic, and gravitational self-force on a particle moving on a straight-line trajectory at a large distance from a Newtonian star, and use these results to find the associated correction to its motion. In the gravitational case we must also include the matter-mediated force, which acts at the same perturbative order as the gravitational self-force. We further augment the gravitational results with geodesic calculations at second order in the central body mass to determine the full, explicit solution to the two-body gravitational scattering problem at second post-Minkowskian order (2PM). We calculate the momentum transfer (which reproduces Westpfahl's old result), the change in mechanical angular momentum (which matches the radiative flux recently computed by Damour), and the change in mechanical mass moment (the time-space components of the angular momentum tensor), which has not previously appeared. Besides the new 2PM results of explicit trajectories and all conserved quantities, this work clarifies the role of gravitational self-force in PM scattering theory and provides a foundation for higher-order calculations.

Paper Structure

This paper contains 17 sections, 104 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Prelude: scattering angle
  3. Green functions in a weakly curved spacetime
  4. Self-forces
  5. Results
  6. Method of computation
  7. Limits
  8. Matter-mediated Force
  9. Motion of the particle
  10. Physical Quantities
  11. 2PM gravitational scattering
  12. Initial conditions
  13. Energy, momentum, and scattering angle
  14. Angular momentum
  15. Mass moment
  16. ...and 2 more sections

Figures (1)

  • Figure 1: Support of the retarded Green function (scalar, electromagnetic, or gravitational) in the spacetime of a Newtonian star, when both points are at large spatial distances from the star. The star can be idealized as occupying a worldline at the spatial origin of coordinates, shown as a vertical line. The direct portion of the Green function has support on the past light cone, which can be represented as delta functions and derivatives on the flat-spacetime past light cone $T=R$ (gold). The tail portion of the Green function has support only on and within a secondary cone $T=r+r'$ emanating from the intersection of the past light cone with the star's worldline (beige).