What the paper is about: The authors present a fully explicit analytic formula for the conservative gravitational Hamiltonian of N point particles at second post-Newtonian (2PN) order. In plain terms, they give a closed-form expression that describes the relativistic corrections to Newtonian gravity up to the next-to-next-to-leading order for many interacting bodies. The work is in the Arnowitt–Deser–Misner (ADM) canonical framework, a standard way to write general relativity in terms of positions and momenta.

What the researchers did: Earlier work had reduced the general 2PN N-body Hamiltonian to an expression that contained a single unresolved spatial integral. That unresolved piece prevented writing the whole result in closed form. In this paper the authors evaluate that remaining integral analytically. They show how the troublesome term — arising from the transverse-traceless part of the gravitational field and contributing four-body interactions — can be written in closed form. They also check the new analytic expression by comparing it with the previous numerical evaluations and find agreement.

How it works at a high level: The post-Newtonian (PN) expansion is a way to approximate Einstein’s equations when velocities are small compared with the speed of light and fields are not too strong. A Hamiltonian is a function of particle positions and canonical momenta that generates the conservative equations of motion. At 2PN order the Hamiltonian contains several pieces: Newtonian terms, first-order (1PN) corrections, and second-order (2PN) corrections. The difficult part came from a four-point contribution in the so-called transverse-traceless potential, which was expressible as a single spatial integral over a combination of particle distances. The authors evaluate that integral exactly and thereby complete the closed-form ADM Hamiltonian at 2PN order.