Multipole expansions of gravitational radiation

摘要: This paper brings together, into a single unified notation, the multipole formalisms for gravitational radiation which various people have constructed. It also extends results of previous workers. More specifically: Part One this reviews scalar, vector, and tensor spherical harmonics used in general relativity literature—including Regge-Wheeler harmonics, symmetric, trace-free ("STF") tensors Sachs Pirani, Newman-Penrose spin-weighted Mathews-Zerilli Clebsch-Gordan-coupled harmonics—which include "pure-orbital" "pure-spin" harmonics. The relationships between are presented. Part then turns attention to radiation. concept "local wave zone" is introduced facilitate clean separation "wave generation" from propagation." generic field local zone decomposed components. energy, linear momentum, angular momentum waves expressed as infinite sums contributions. Attention restricted sources that admit nonsingular, spacetime-covering de Donder coordinate system. (This excludes black holes.) In such system moments volume integrals over source. For slow-motion systems, these source re-expressed power series L / λ≡(size ) (reduced wavelength ). specialized systems with weak internal gravity yield (i) standard Newtonian formulas moments, (ii) post-Newtonian Epstein Wagoner, (iii) post-post-Newtonian formulas. Two derives multipole-moment wave-generation formalism arbitrarily strong gravity, including cannot be covered by coordinates. one calculates, any means, source's instantaneous, near-zone, external solution time-independent Einstein equations. reads off near-zone instantaneous moments; plugs those time-evolving formulae given paper. As building blocks formalism, does following things: (1) linearized theory vacuum exterior an isolated system, it equations (a result due Sachs, Bergmann, Pirani). (2) full nonlinear relativity, structure That sum products matches onto outgoing-wave field. (3) stationary (a) presents definition meshes naturally gravitational-wave theory; (b) introduces "asymptotically Cartesian mass centered" (ACMC) systems; (c) shows how deduce form its metric ACMC As example, lowest few (l ≤ 3) Kerr computed.