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Hereditary Terms at Next-To-Leading Order in Two-Body Gravitational Dynamics

Stefano Foffa, Riccardo Sturani

TL;DR

This work advances high-precision modeling of two-body gravitational dynamics by computing next-to-leading order hereditary terms at $5$PN within an EFT framework, including both tail and memory effects. It provides new conservative contributions from octupole and magnetic quadrupole tails, a logarithmic energy term for circular orbits, and local-in-time corrections from angular-momentum memory and GW self-interaction, all while confirming universal relations between tail flux and non-tail emissions. The analysis clarifies how far-zone dynamics feed into near-zone conservative physics, and it outlines the remaining steps to achieve a complete $5$PN description, with significant implications for accurate gravitational-wave templates and tests of GR in the strong-field regime.

Abstract

In the context of the two-body problem in General Relativity, hereditary terms in the long range gravitational field depend on the history rather than the instantaneous state of the source at retarded time. We compute the next-to leading effects of such hereditary terms, that comprise tail and memory, on the two-body dynamics, within effective field theory methods, including both dissipative and conservative effects. The former confirm known results at 2.5 post-Newtonian order with respect to the leading order in the luminosity function; the conservative part is a new result and is an unavoidable ingredient for a derivation of the conservative two-body dynamics at fifth post-Newtonian order.

Hereditary Terms at Next-To-Leading Order in Two-Body Gravitational Dynamics

TL;DR

This work advances high-precision modeling of two-body gravitational dynamics by computing next-to-leading order hereditary terms at PN within an EFT framework, including both tail and memory effects. It provides new conservative contributions from octupole and magnetic quadrupole tails, a logarithmic energy term for circular orbits, and local-in-time corrections from angular-momentum memory and GW self-interaction, all while confirming universal relations between tail flux and non-tail emissions. The analysis clarifies how far-zone dynamics feed into near-zone conservative physics, and it outlines the remaining steps to achieve a complete PN description, with significant implications for accurate gravitational-wave templates and tests of GR in the strong-field regime.

Abstract

In the context of the two-body problem in General Relativity, hereditary terms in the long range gravitational field depend on the history rather than the instantaneous state of the source at retarded time. We compute the next-to leading effects of such hereditary terms, that comprise tail and memory, on the two-body dynamics, within effective field theory methods, including both dissipative and conservative effects. The former confirm known results at 2.5 post-Newtonian order with respect to the leading order in the luminosity function; the conservative part is a new result and is an unavoidable ingredient for a derivation of the conservative two-body dynamics at fifth post-Newtonian order.

Paper Structure

This paper contains 13 sections, 44 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Method
  3. Results
  4. General properties of tails
  5. Next-to-leading order hereditary terms
  6. Octupole tail
  7. Magnetic quadrupole tail
  8. Logarithmic contribution to energy of circular orbit
  9. Angular momentum "failed" tail
  10. GW self interaction
  11. Conclusions
  12. Amplitude construction
  13. Relevant integrals

Figures (1)

  • Figure 1: Hereditary diagrams contributing from leading order (top one), and from next-to leading order (the remaining four). Wiggled lines represent on-shell gravitons (i.e. gravitational waves), straight dashed and dotted ones stand for instantaneous propagators, i.e. interaction with static fields. A relativistic correction to the instantaneous propagator of the top diagram has not been considered because it gives a physically irrelevant term (see discussion in the text). The horizontal continuous black double line represent the external source given by the binary system.