Towards the 4th post-Newtonian Hamiltonian for two-point-mass systems

TL;DR

The paper advances conservative gravitational two-body dynamics to 4PN order (up to quadratic order in G) using the ADM formalism in the ADMTT gauge, deriving an autonomous center-of-mass Hamiltonian by solving the constraint equations to fourth order. It reports a 4PN Hamiltonian comprising 57 terms, with 45 coefficients determined and all logarithmic contributions computed, and it confirms consistency with the test-mass limit and Poincaré invariance. The circular-orbit binding energy E(x;nu) is obtained with two new nu-dependent coefficients at 4PN order, improving the precision of waveform models, while 12 coefficients remain undetermined and require more sophisticated regularization. These results provide essential inputs for high-precision gravitational-wave modeling and effective-one-body approaches.

Abstract

The article presents the conservative dynamics of gravitationally interacting two-point-mass systems up to the eight order in the inverse power of the velocity of light, i.e.\ 4th post-Newtonian (4PN) order, and up to quadratic order in Newton's gravitational constant. Additionally, all logarithmic terms at the 4PN order are given as well as terms describing the test-mass limit. With the aid of the Poincaré algebra additional terms are obtained. The dynamics is presented in form of an autonomous Hamiltonian derived within the formalism of Arnowitt, Deser and Misner. Out of the 57 different terms of the 4PN Hamiltonian in the center-of-mass frame, the coefficients of 45 of them are derived. Reduction of the obtained results to circular orbits is performed resulting in the 4PN-accurate formula for energy expressed in terms of angular frequency in which two coefficients are obtained for the first time.