From momentum expansions to post-Minkowskian Hamiltonians by computer algebra algorithms

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 [1].1 It has been shown in [15] that the results of [1] are in accordance with the energy-and angular momentumorbital frequency relation, E(ω) and L(ω), obtained in the post-Newtonian approach up to O((G N M ω) 2 ), with M = m 1 + m 2 . We have verified that the results in [1], when compared to the post-Newtonian results in ADM coordinates in [10], agree up to terms of O(G 3 N (p 2 ) 2 ).

Second-order post-Minkowskian scattering in arbitrary dimensions

“…In the case of classical General Relativity and to second Post-Minkwoskian order the scattering angle can be read off from of eq. (5.15) and (5.13), 16) since sin θ θ at this level of approximation. To this order the scattering angle arises from two contributions, one linear and one quadratic in the involved scattering amplitudes.…”

“…Given the expected improvements in detector sensitivity, it will be extremely important in the future to have high-precision theoretical predictions from General Relativity. To this aim the use of quantum field theory amplitudes to extract the post-Minkowskian (PM) expansion of General Relativity has recently gained considerable momentum [6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21], and progress is now also being made on extensions to spinning objects [22][23][24][25][26][27][28][29][30][31]. The underlying physical motivation for this approach lies in the observation that, during the early stages of a merger event, when the two compact objects are still far apart, gravitational interactions are weak and can be conveniently treated JHEP07(2020)122 in a weak-coupling approximation.…”

Extremal black hole scattering at $$ \mathcal{O} $$(G3): graviton dominance, eikonal exponentiation, and differential equations

“…[19,20] in the case of Einstein gravity. Since then, the correctness of the latter result has been verified at high orders in the velocity expansion [48,49], an alternative method for resummation of the velocity series has been used with identical results [50], and the unitarity cut construction of the loop integrand has been checked against direct Feynman diagram computations [51]. Still, a direct relativistic calculation that bypasses velocity resummation will be a valuable additional confirmation of the result, and will provide a way to streamline future calculations at O(G 3 ) and beyond.…”

Sixth post-Newtonian local-in-time dynamics of binary systems

“…The latter equation shows that one can formally consider that j = O c G , so that a term or order 1 j n is of order G n c n . In the following, we shall often use the shorthand notations 39) and…”