Relativistic Closed-Form Hamiltonian for Many-Body Gravitating Systems in the Post-Minkowskian Approximation

TL;DR

This paper derives a simple, closed-form Hamiltonian for many-body relativistic gravitational systems in the post-Minkowskian approximation, including all linear gravitational effects, and applies it to relativistic scattering scenarios.

Contribution

It presents a novel, compact Hamiltonian formulation for relativistic many-body gravitational systems in the post-Minkowskian approximation, extending classical results.

Findings

Hamiltonian expressed in canonical variables for relativistic bodies

Includes all effects linear in gravitational constant

Applied to relativistic scattering of objects

Abstract

The Hamiltonian for a system of relativistic bodies interacting by their gravitational field is found in the post-Minkowskian approximation, including all terms linear in the gravitational constant. It is given in a surprisingly simple closed form as a function of canonical variables describing the bodies only. The field is eliminated by solving inhomogeneous wave equations, applying transverse-traceless projections, and using the Routh functional. By including all special relativistic effects our Hamiltonian extends the results described in classical textbooks of theoretical physics. As an application, the scattering of relativistic objects is considered.