Post-Newtonian methods: Analytic results on the binary problem

TL;DR

This work provides a comprehensive analytic treatment of Post-Newtonian methods for the binary problem in general relativity, contrasting ADM-based canonical formalisms with Lorentz-covariant (harmonic) approaches and detailing near- and far-zone structures. It develops and analyzes conservative and dissipative dynamics for point masses and spinning black holes, introducing the skeleton Hamiltonian and BRILL–Lindquist initial data, and treats delta-function sources with dimensional regularization to high PN orders. The near-zone PN expansion is connected to the far-zone multipole expansion, yielding energy flux and gravitational-wave luminosity with tail terms, and the paper supplies explicit waveform models through 1.5PN for GW data analysis, alongside ISCO analyses and spin-coupling Hamiltonians relevant for effective-one-body and waveform-generation frameworks. Collectively, these results provide a robust analytic foundation for accurate gravitational-wave templates and deepen understanding of regularization, spin dynamics, and radiation reaction in binary GR systems.

Abstract

A detailed account is given on approximation schemes to the Einstein theory of general relativity where the iteration starts from the Newton theory of gravity. Two different coordinate conditions are used to represent the Einstein field equations, the generalized isotropic ones of the canonical formalism of Arnowitt, Deser, and Misner and the harmonic ones of the Lorentz-covariant Fock-de Donder approach. Conserved quantities of isolated systems are identified and the Poincaré algebra is introduced. Post-Newtonian expansions are performed in the near and far (radiation) zones. The natural fitting of multipole expansions to post-Newtonian schemes is emphasized. The treated matter models are ideal fluids, pure point masses, and point masses with spin and mass-quadrupole moments modelling rotating black holes. Various Hamiltonians of spinning binaries are presented in explicit forms to higher post-Newtonian orders. The delicate use of black holes in post-Newtonian expansion calculations and of the Dirac delta function in general relativity find discussions.