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Gravitational Radiation from Post-Newtonian Sources and Inspiralling Compact Binaries

Luc Blanchet

TL;DR

This work develops a comprehensive analytical framework for gravitational-wave generation from general-relativistic, weak-field sources, with a focus on inspiralling compact binaries. It blends post-Minkowskian and post-Newtonian methods to derive a complete multipolar description: the exterior field is constructed from source moments via a careful matching to the interior PN solution, and radiative moments encode tails, tails-of-tails, and memory effects. The paper delivers explicit 3PN conservative dynamics and up to 3.5PN radiative flux for binaries, resolves regularization ambiguities through dimensional regularization, and provides detailed expressions for orbital phase and gravitational-wave polarizations critical for LIGO/VIRGO data analysis. These results enable high-precision waveform templates, enabling stringent tests of general relativity and improving binary black-hole and neutron-star merger predictions. The methods and findings significantly advance the theoretical modeling necessary for detector-scale gravitational-wave astronomy.

Abstract

The article reviews the current status of a theoretical approach to the problem of the emission of gravitational waves by isolated systems in the context of general relativity. Part A of the article deals with general post-Newtonian sources. The exterior field of the source is investigated by means of a combination of analytic post-Minkowskian and multipolar approximations. The physical observables in the far-zone of the source are described by a specific set of radiative multipole moments. By matching the exterior solution to the metric of the post-Newtonian source in the near-zone we obtain the explicit expressions of the source multipole moments. The relationships between the radiative and source moments involve many non-linear multipole interactions, among them those associated with the tails (and tails-of-tails) of gravitational waves. Part B of the article is devoted to the application to compact binary systems. We present the equations of binary motion, and the associated Lagrangian and Hamiltonian, at the third post-Newtonian (3PN) order beyond the Newtonian acceleration. The gravitational-wave energy flux, taking consistently into account the relativistic corrections in the binary moments as well as the various tail effects, is derived through 3.5PN order with respect to the quadrupole formalism. The binary's orbital phase, whose prior knowledge is crucial for searching and analyzing the signals from inspiralling compact binaries, is deduced from an energy balance argument.

Gravitational Radiation from Post-Newtonian Sources and Inspiralling Compact Binaries

TL;DR

This work develops a comprehensive analytical framework for gravitational-wave generation from general-relativistic, weak-field sources, with a focus on inspiralling compact binaries. It blends post-Minkowskian and post-Newtonian methods to derive a complete multipolar description: the exterior field is constructed from source moments via a careful matching to the interior PN solution, and radiative moments encode tails, tails-of-tails, and memory effects. The paper delivers explicit 3PN conservative dynamics and up to 3.5PN radiative flux for binaries, resolves regularization ambiguities through dimensional regularization, and provides detailed expressions for orbital phase and gravitational-wave polarizations critical for LIGO/VIRGO data analysis. These results enable high-precision waveform templates, enabling stringent tests of general relativity and improving binary black-hole and neutron-star merger predictions. The methods and findings significantly advance the theoretical modeling necessary for detector-scale gravitational-wave astronomy.

Abstract

The article reviews the current status of a theoretical approach to the problem of the emission of gravitational waves by isolated systems in the context of general relativity. Part A of the article deals with general post-Newtonian sources. The exterior field of the source is investigated by means of a combination of analytic post-Minkowskian and multipolar approximations. The physical observables in the far-zone of the source are described by a specific set of radiative multipole moments. By matching the exterior solution to the metric of the post-Newtonian source in the near-zone we obtain the explicit expressions of the source multipole moments. The relationships between the radiative and source moments involve many non-linear multipole interactions, among them those associated with the tails (and tails-of-tails) of gravitational waves. Part B of the article is devoted to the application to compact binary systems. We present the equations of binary motion, and the associated Lagrangian and Hamiltonian, at the third post-Newtonian (3PN) order beyond the Newtonian acceleration. The gravitational-wave energy flux, taking consistently into account the relativistic corrections in the binary moments as well as the various tail effects, is derived through 3.5PN order with respect to the quadrupole formalism. The binary's orbital phase, whose prior knowledge is crucial for searching and analyzing the signals from inspiralling compact binaries, is deduced from an energy balance argument.

Paper Structure

This paper contains 39 sections, 231 equations, 1 figure, 2 tables.

Table of Contents

  1. Introduction
  2. Gravitational-wave generation formalisms
  3. Problem posed by compact binary systems
  4. Post-Newtonian equations of motion and radiation
  5. Post-Newtonian Sources
  6. Einstein's Field Equations
  7. Linearized Vacuum Equations
  8. Non-linear Iteration of the Field Equations
  9. The post-Minkowskian solution
  10. Generality of the solution
  11. Near-zone and far-zone structures
  12. The radiative multipole moments
  13. Exterior Field of a Post-Newtonian Source
  14. The matching equation
  15. General expression of the multipole expansion
  16. ...and 24 more sections

Figures (1)

  • Figure 1: Results for the binding energy $E_\mathrm{ICO}$ versus $\omega_\mathrm{ICO}$ in the equal-mass case ($\nu=1/4$). The asterisk marks the result calculated by numerical relativity. The points indicated by $\mathrm{1PN}$, $\mathrm{2PN}$, and $\mathrm{3PN}$ are computed from the minimum of Equation (\ref{['133']}), and correspond to irrotational binaries. The points denoted by $\mathrm{1PN}^\mathrm{corot}$, $\mathrm{2PN}^\mathrm{corot}$, and $\mathrm{3PN}^\mathrm{corot}$ come from the minimum of the sum of Equations (\ref{['133']}) and (\ref{['Ecorot']}), and describe corotational binaries.