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Radiation reaction and gravitational waves in the effective field theory approach

Chad R. Galley, Manuel Tiglio

TL;DR

The paper demonstrates that to robustly model radiation reaction and gravitational-wave emission in NRGR, one must employ the in-in (Schwinger–Keldysh) formalism to enforce retarded boundary conditions, ensuring real and causal real-time evolution. It derives the leading $2.5$PN radiation-reaction force and the causal quadrupole radiation within this framework, showing consistency with the Burke–Thorne result and the quadrupole power formula, while highlighting the shortcomings of the traditional in-out approach for dissipative dynamics. By developing explicit NRGR Feynman rules in the in-in formalism and connecting to Kol–Smolkin’s ClEFT, the work provides a practical, self-consistent EFT toolkit for dissipative gravitational physics and hereditary terms, with potential applications to self-force calculations and waveform modeling.

Abstract

We compute the contribution to the Lagrangian from the leading order (2.5 post-Newtonian) radiation reaction and the quadrupolar gravitational waves emitted from a binary system using the effective field theory (EFT) approach of Goldberger and Rothstein. We use an initial value formulation of the underlying (quantum) framework to implement retarded boundary conditions and describe these real-time dissipative processes. We also demonstrate why the usual scattering formalism of quantum field theory inadequately accounts for these. The methods discussed here should be useful for deriving real-time quantities (including radiation reaction forces and gravitational wave emission) and hereditary terms in the post-Newtonian approximation (including memory, tail and other causal, history-dependent integrals) within the EFT approach. We also provide a consistent formulation of the radiation sector in the equivalent effective field theory approach of Kol and Smolkin.

Radiation reaction and gravitational waves in the effective field theory approach

TL;DR

The paper demonstrates that to robustly model radiation reaction and gravitational-wave emission in NRGR, one must employ the in-in (Schwinger–Keldysh) formalism to enforce retarded boundary conditions, ensuring real and causal real-time evolution. It derives the leading PN radiation-reaction force and the causal quadrupole radiation within this framework, showing consistency with the Burke–Thorne result and the quadrupole power formula, while highlighting the shortcomings of the traditional in-out approach for dissipative dynamics. By developing explicit NRGR Feynman rules in the in-in formalism and connecting to Kol–Smolkin’s ClEFT, the work provides a practical, self-consistent EFT toolkit for dissipative gravitational physics and hereditary terms, with potential applications to self-force calculations and waveform modeling.

Abstract

We compute the contribution to the Lagrangian from the leading order (2.5 post-Newtonian) radiation reaction and the quadrupolar gravitational waves emitted from a binary system using the effective field theory (EFT) approach of Goldberger and Rothstein. We use an initial value formulation of the underlying (quantum) framework to implement retarded boundary conditions and describe these real-time dissipative processes. We also demonstrate why the usual scattering formalism of quantum field theory inadequately accounts for these. The methods discussed here should be useful for deriving real-time quantities (including radiation reaction forces and gravitational wave emission) and hereditary terms in the post-Newtonian approximation (including memory, tail and other causal, history-dependent integrals) within the EFT approach. We also provide a consistent formulation of the radiation sector in the equivalent effective field theory approach of Kol and Smolkin.

Paper Structure

This paper contains 18 sections, 127 equations, 3 figures.

Table of Contents

  1. Introduction
  2. The in-out formulation of NRGR
  3. Radiation reaction
  4. Quadrupole radiation
  5. Radiation reaction, self-consistency and hereditary effects
  6. Comparing the In-out and in-in formalisms
  7. One-point functions in the in-out formalism
  8. In-in formalism for expectation values and causality
  9. The Keldysh representation
  10. The in-in formulation of NRGR
  11. NRGR Feynman rules in the in-in formalism
  12. Radiation reaction
  13. Quadrupole radiation
  14. Conclusions
  15. Coupling to conserved quantities
  16. ...and 3 more sections

Figures (3)

  • Figure 1: The diagrams that potentially contribute to radiation reaction at (a) 0.5PN, (b) 1.5PN and (c) 2.5PN orders.
  • Figure 2: The non-zero diagram contributing to the leading order (2.5PN) radiation reaction in NRGR using the in-in formalism. Note the CTP indices $a,b$ that relate to the forward and backward branches of time in the closed-time-path contour of the CTP path integral.
  • Figure 3: The coupling of a radiation graviton to a time-independent vertex $C^{\alpha \beta}$ and a general, time-dependent vertex $V^{\alpha \beta}(t)$, represented here as a gray circle.