Structure of the post-Newtonian expansion in general relativity

Abstract

In the continuation of a preceding work, we derive a new expression for the metric in the near zone of an isolated matter system in post-Newtonian approximations of general relativity. The post-Newtonian metric, a solution of the field equations in harmonic coordinates, is formally valid up to any order, and is cast in the form of a particular solution of the wave equation, plus a specific homogeneous solution which ensures the asymptotic matching to the multipolar expansion of the gravitational field in the exterior of the system. The new form is suitable for practical computations of the gravitational radiation reaction forces in harmonic coordinates. It also provides some insights on the structure of the post-Newtonian expansion in general relativity and the radiation reaction terms therein.

Sources of Gravitational Waves: Theory and Observations

“…The multipolar expansions employ a multipole decomposition in irreducible representations of the rotation group. The gravitational field all over space can be obtained by matching the near-zone to the wave-zone fields and the matching can be accomplished at all orders, as shown by Blanchet and collaborators [86][87][88][89]. We shall denote this approach the MPM-PN formalism.…”

“…The most general solution is obtained by adding the homogeneous solution (n) v αβ that is regular at the origin r = 0, to the inhomogeneous solution, that is (n) h αβ = (n) u αβ + (n) v αβ . As derived in [86][87][88][89][90][91], the solution in the near zone that matches the external field and satisfies correct boundary conditions at infinity involves a specific homogenous solution which can be expressed in terms of STF tensors as…”

“…They were overcome by applying a more sophisticated mathematical method to invert the Laplacian and find the correct solution. This method is a variant of the analytic continuation technique developed by Blanchet and Damour in the wave zone, and it was carried out in [86][87][88][89][90]. Basically, one multiplies the function (n) f αβ by r B , where B is a negative real number whose modu-lus is sufficiently large that the integral is regular at spatial infinity.…”

The Newtonian Limit for Perfect Fluids

“…The majority of results are based on formal expansions in the parameter ǫ which are used to calculate the (approximate) values of physical quantities and also to investigate the behavior of the gravitational and matter fields in the limit ǫ ց 0. For some classic and recent results of this type see [2,3,6,9,13,[20][21][22]31,41] and reference cited therein. The main difficulty with the formal expansions is that they leave completely unanswered the question of convergence.…”

The Newtonian Limit on Cosmological Scales

“…Most interest in this subject has focused on understanding the relationship in the setting of isolated systems, and the investigations have almost exclusively involved formal calculations, see [3,2,6,9,10,11,14,26,27,28] and references therein, with a few exceptions [32,33,39] where rigorous results were obtained. More recently, interest has shifted to understanding the relationship between Newtonian gravity and General Relativity on cosmological scales [5,12,8,17,18,20,19,23,24,31,37].…”