Divergent integrals of post-Newtonian gravity: Nonanalytic terms in the near-zone expansion of a gravitationally radiating system found by matching

Journal Article · Thu Apr 15 04:00:00 EST 1982 · Phys. Rev. D; (United States)

DOI:https://doi.org/10.1103/PhysRevD.25.2038· OSTI ID:5381325

Anderson, J L; Kates, R E; Kegeles, L S; Madonna, R G

We study the divergent integrals that occur in all post-Newtonian (PN) treatments of radiation reaction in slow-motion, gravitationally bound systems in general relativity. The PN methods implicitly assume that the near-zone metric has a valid asymptotic expansion in powers of the small velocity parameter epsilon. We first show explicitly (for the gauge to be used here) that a PN-approximation method leads to a divergent integral at 4-PN order. This divergence arises from the second iteration. Matching arguments are then used to calculate a near-zone term of O(log epsilon) larger than 4-PN order. On the basis of this calculation and several previous model problems, we argue that the PN divergences signify the breakdown of the PN power-series assumption, rather than a breakdown of the near and wave zones. Our results suggest that the PN calculations in fact give correct answers at least up to the orders at which divergences appear. The nonanalytic term of O(log epsilon) beyond 4-PN order arises in the near zone via matching to the wave-zone expansion when we include terms of O(epsilon/sup 3/) beyond linearized order. (This order corresponds to the second PN iteration.) We also solve the wave-zone equations at O(epsilon/sup 6/) beyond linearized order and analyze the inner expansion of the solutions. Matching gives rise to a nonanalytic term in the wave zone at O(epsilon/sup 11/log/sup epsilon/), i.e., at O(e/sup 6/log/sup epsilon/) beyond linearized order. A straining technique is used in the wave-zone expansion to give a sufficiently accurate approximation to the null surfaces near past and future null infinity. The lowest-order strained solution at first appears to contribute a large, anomalous, time-odd piece to the reaction potential. However, after analyzing the contribution of higher-order wave-zone terms, we obtain agreement with the Burke reaction potential. Our results thus strongly support the usual quadrupole formula.

• Research Organization: Department of Physics, Stevens Institute of Technology, Hoboken, New Jersey 07030
• OSTI ID: 5381325
• Journal Information: Phys. Rev. D; (United States), Journal Name: Phys. Rev. D; (United States) Vol. 25:8; ISSN PRVDA
• Country of Publication: United States
• Language: English

Related Subjects

657003* -- Theoretical & Mathematical Physics-- Relativity & Gravitation
71 CLASSICAL AND QUANTUM MECHANICS
GENERAL PHYSICS
BASIC INTERACTIONS
FIELD THEORIES
GENERAL RELATIVITY THEORY
GRAVITATIONAL INTERACTIONS
INTEGRALS
INTERACTIONS
MATHEMATICAL SPACE
MINKOWSKI SPACE
PERTURBATION THEORY
SCHWARZSCHILD RADIUS
SPACE