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

TL;DR

This paper derives the conservative 4th post-Newtonian Hamiltonian for two-point-mass systems, including logarithmic terms and test-mass limits, advancing the precision of gravitational dynamics modeling.

Contribution

It provides the first derivation of 45 coefficients of the 4PN Hamiltonian, including new coefficients for circular orbit energy expressions.

Findings

Derived 45 of 57 Hamiltonian coefficients at 4PN order.

Presented the first 4PN-accurate energy formula for circular orbits.

Included all logarithmic terms and test-mass limit contributions.

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 Poincar\'e 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.