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 GN/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 [1] and compare to expressions obtained in the effective one body formalism.
Highlights
The use of a non-relativistic effective field theory [2,3,4,5,6,7,8,9], provides one way to derive the equations of motion of a binary mass system within the post-Newtonian (PN) approach
The general principle is to expand in the ratio G N /r, with G N Newton’s constant and r denoting the distance between the two point masses m1 and m2, and to retain all velocity corrections up to the order implied by the virial theorem
We have shown that one may determine the expansion coefficients of the potential in the post-Minkowskian approach from the velocity expansion in the effective field theory approach by finite terms, using very general mathematical algorithms
Summary
The use of a non-relativistic effective field theory [2,3,4,5,6,7,8,9], provides one way to derive the equations of motion of a binary mass system within the post-Newtonian (PN) approach. The results up to the 4th post-Newtonian order have already been derived using different methods, see Refs. The question arises, whether these corrections can be obtained using the effective field theory approach, in which one usually can only expand up to finite terms in the velocity. All the master integrals in the case one expands in the momentum are already known up to five-loop order from the post-Newtonian approach. In an appendix we discuss how some special sums occurring can be carried out
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