Dimensional regularization of the gravitational interaction of point masses

Abstract

We show how to use dimensional regularization to determine, within the Arnowitt-Deser-Misner canonical formalism, the reduced Hamiltonian describing the dynamics of two gravitationally interacting point masses. Implementing, at the third post-Newtonian (3PN) accuracy, our procedure we find that dimensional continuation yields a finite, unambiguous (no pole part) 3PN Hamiltonian which uniquely determines the heretofore ambiguous "static" parameter: namely, ωs = 0. Our work also provides a remarkable check of the perturbative consistency (compatibility with gauge symmetry) of dimensional continuation through a direct calculation of the "kinetic" parameter ω k , giving the unique answer compatible with global Poincaré invariance (ω k = 41 24 ) by summing ∼ 50 different dimensionally continued contributions.

Sources of Gravitational Waves: Theory and Observations

“…Thus, it is necessary to introduce a regularisation. Until the early 2000's the Hadamard regularisation was employed [110], but it does not provide unambiguous results at 3PN order, and so dimensional regularisation, which is a well-known regularisation scheme in particle physics, was adopted [111].…”

“…This allowed the computation of spin-orbit and spin-spin couplings at quite high PN orders. So far, the ADM Hamiltonian approach has been developed mostly for the conservative dynamics through high PN orders [32,50,51,101,102,111,[120][121][122][123][124][125][126].…”

Measuring orbital eccentricity and periastron advance in quasicircular black hole simulations

“…static was computed by Ref. [53] to be zero, and the ambiguity parameter ! kinetic was shown to be 41=24 by Ref.…”

Dimensional regularization of the third post-Newtonian gravitational wave generation from two point masses

“…Thus, we simply have We shall compute DH in the limit where ε → 0, keeping the pole part ∝ ε −1 (at 3PN order only simple poles will occur) and the finite term ∝ ε 0 , but neglecting O(ε). Using the same method as in [39,40] …”

“…The ambiguity parameter λ entering the 3PN equations of motion has been computed in Refs. [39,40], with result λ = −1987/3080. (This result has also been obtained with an alternative approach in Refs.…”