From Momentum Expansions to Post-Minkowskian Hamiltonians by Computer Algebra Algorithms

TL;DR

This paper develops an algorithmic method to derive complete post-Minkowskian Hamiltonians for binary systems in gravity using computer algebra, advancing the computational techniques in gravitational physics.

Contribution

It introduces a new algorithmic approach to obtain post-Minkowskian Hamiltonians from finite velocity terms, applicable to higher orders.

Findings

Successfully derived third post-Minkowskian order Hamiltonians algorithmically.

Demonstrated the method's general applicability under mathematical conditions.

Enhanced computational efficiency in gravitational two-body problem calculations.

Abstract

The post-Newtonian and post-Minkowskian solutions for the motion of binary mass systems in gravity can be derived in terms of momentum expansions within effective field theory approaches. In the post-Minkowskian approach the expansion is performed in the ratio $G_N/r$, retaining all velocity terms completely, while in the post-Newtonian approach only those velocity terms are accounted for which are of the same order as the potential terms due to the virial theorem. We show that it is possible to obtain the complete post-Minkowskian expressions completely algorithmically, under most general purely mathematical conditions from a finite number of velocity terms and illustrate this up to the third post-Minkowskian order given in \cite{Bern:2019crd}.