Multipole expansions of gravitational radiation

This paper brings together, into a single unified notation, the multipole formalisms for gravitational radiation which various people have constructed. It also extends the results of previous workers. More specifically:

Part One of this paper reviews the various scalar, vector, and tensor spherical harmonics used in the general relativity literature—including the Regge-Wheeler harmonics, the symmetric, trace-free ("STF") tensors of Sachs and Pirani, the Newman-Penrose spin-weighted harmonics, and the Mathews-Zerilli Clebsch-Gordan-coupled harmonics—which include "pure-orbital" harmonics and "pure-spin" harmonics. The relationships between the various harmonics are presented. Part One then turns attention to gravitational radiation. The concept of "local wave zone" is introduced to facilitate a clean separation of "wave generation" from "wave propagation." The generic radiation field in the local wave zone is decomposed into multipole components. The energy, linear momentum, and angular momentum in the waves are expressed as infinite sums of multipole contributions. Attention is then restricted to sources that admit a nonsingular, spacetime-covering de Donder coordinate system. (This excludes black holes.) In such a coordinate system the multipole moments of the radiation field are expressed as volume integrals over the source. For slow-motion systems, these source integrals are re-expressed as infinite power series in $\frac{L}{\lambda }\equiv \frac{\left(\mathrm{size}\mathrm{of}\mathrm{source}\right)}{\left(\mathrm{reduced}\mathrm{wavelength}\mathrm{of}\mathrm{waves}\right)}$. The slow-motion source integrals are then specialized to systems with weak internal gravity to yield (i) the standard Newtonian formulas for the multipole moments, (ii) the post-Newtonian formulas of Epstein and Wagoner, and (iii) post-post-Newtonian formulas.

Part Two of this paper derives a multipole-moment wave-generation formalism for slow-motion systems with arbitrarily strong internal gravity, including systems that cannot be covered by de Donder coordinates. In this formalism one calculates, by any means, the source's instantaneous, near-zone, external gravitational field as a solution of the time-independent Einstein field equations. One then reads off of this near-zone field the source's instantaneous multipole moments; and one plugs those time-evolving moments into the standard radiation formulae given in Part One of this paper.

As building blocks for this formalism, Part Two also does the following things: (1) In the linearized theory of gravity, for the vacuum exterior of an isolated system, it derives the general solution of the field equations (a result due to Sachs, Bergmann, and Pirani). (2) In full nonlinear general relativity, for the vacuum near-zone exterior of an isolated system, it derives the structure of the general solution of the Einstein field equations. That structure is expressed as a sum of products of multipole contributions. It also matches this near-zone field onto an outgoing-wave radiation field. (3) In full nonlinear general relativity, for the vacuum exterior of a stationary isolated system, (a) it presents a definition of multipole moments which meshes naturally with gravitational-wave theory; (b) it introduces the concept of "asymptotically Cartesian and mass centered" (ACMC) coordinate systems; and (c) it shows how to deduce the multipole moments of a source from the form of its metric in an ACMC coordinate system. As an example, the lowest few ($l\le 3$) multipole moments of the Kerr metric are computed.