Explicit relations among the classical definitions of multipole moments for stationary isolated systems in general relativity (Thorne, 1980)

Kip Thorne's 1980 review 'Multipole expansions of gravitational radiation' noted that five earlier definitions of multipole moments for stationary, asymptotically flat systems in general relativity — those of Janis and Newman (1965), van der Burg (1968), Geroch (1970), Hansen (1974, the stationary extension of Geroch's static construction), and Clarke and Sciama (1971) — each fix the exterior gravitational field of the source completely, so their moments must be expressible as combinations of one another; yet, he wrote, 'nobody has yet exhibited those relationships explicitly'. Later literature resolved the load-bearing piece of this question: Gürsel (1983) proved that Thorne's own moments coincide with the Geroch-Hansen moments up to normalization, and subsequent work re-derived and generalized this equivalence (a Noether-charge definition matching both, and extensions to electrovac, asymptotically de Sitter, and general stationary matter spacetimes). However, the combination rules connecting the three pre-1970 sets — Janis-Newman, van der Burg, Clarke-Sciama — to the now-standard moments have not been exhibited explicitly in the literature. The open problem is to complete this dictionary for stationary isolated systems with arbitrarily strong internal gravity, exhibiting the relations on whatever common domain an older definition shares with the standard moments; a claim that some older definition admits no such dictionary must be established rigorously on that common domain, not merely from a domain mismatch.

Background

In Newtonian gravity the exterior field of an isolated source is completely characterized by its multipole moments — the mass, center of mass, angular momentum, quadrupole, and so on, one symmetric trace-free (STF; a Cartesian tensor index-symmetric with all contractions removed) tensor per order $\ell$. In general relativity the nonlinear field equations make 'multipole moments of a source' nontrivial to define, and by 1980 several inequivalent-looking definitions existed for stationary (time-independent, possibly rotating) isolated systems whose exterior is asymptotically flat, meaning the spacetime metric approaches that of flat space far from the source.

Geroch (1970; static case) and Hansen (1974; stationary case) gave the geometric definition now regarded as standard: one passes to the quotient of spacetime by the stationary symmetry flow (the three-dimensional manifold of Killing trajectories, a Killing vector being the generator of the time-translation symmetry) and to its conformal compactification (a mathematical rescaling of the metric that brings the 'points at infinity' to a finite distance), and defines two sequences of STF moments — mass moments and angular-momentum or current moments — by a recursive covariant-derivative construction acting on two scalar potentials. Thorne (1980) gave a coordinate definition: in asymptotically Cartesian and mass-centered (ACMC) coordinates — a preferred far-zone class of coordinates, realized within harmonic (de Donder) gauge (coordinates satisfying the standard gauge condition $\Gamma^{\alpha}=0$ on the contracted Christoffel symbol), in which the remaining coordinate freedom is fixed so that only the mass-centered STF pieces of the $1/r$ expansion survive — the moments are read off as the coefficients of the irreducible $1/r^{\ell+1}$ STF pieces of the metric components $g_{00}$ and $g_{0j}$. Thorne's definition was designed to mesh with gravitational-wave generation theory, which computes emitted radiation from source multipoles.

Three earlier definitions predate these. Janis and Newman (1965) specified a stationary source by data evolved from characteristic initial data on an outgoing null hypersurface (a lightlike surface on which the Einstein equations reduce to a hierarchical evolution system), tying their moments to structures at null infinity. Van der Burg (1968) expanded stationary fields in multipoles directly. Clarke and Sciama (1971), for static fields, connected the field multipoles to integrals over the source using the Sciama-Waylen-Gilman device, which rewrites the field equations formally as a linear wave equation sourced by an effective stress-energy tensor that contains all the nonlinearity.

Each of these constructions was claimed to fix the exterior field completely, including its nonlinearities — the characterization property, proven rigorously for the Geroch-Hansen moments by Beig and Simon (1980) and Kundu (1981). Since all sets characterize the same spacetimes, the moments of any one set must be (generally nonlinear, order-mixing) combinations of the moments of any other; Thorne observed in 1980 that nobody had exhibited those combination rules explicitly. The question is not merely decorative: each older definition encodes a different facet of the theory (null-infinity data, source integrals, direct stationary-field expansions), and explicit dictionaries would tie those facets to the moments used in wave generation and in modern black-hole and compact-source modeling.

Problem Statement

The problem as stated by Thorne (Rev. Mod. Phys. 52, 299 (1980), Section X, p. 332): for stationary isolated systems in general relativity, 'The moments in each set (Janis and Newman's, Van der Burg's, Geroch and Hansen's, Clarke and Sciama's, and mine) must be some combination of the moments in each other set. However, nobody has yet exhibited those relationships explicitly.' The task is to exhibit those relationships explicitly.

Later literature resolved one load-bearing piece: Gürsel (1983) proved that Thorne's moments and the Geroch-Hansen moments are identical up to normalization for stationary asymptotically flat systems, and the modern literature additionally established the equivalence of both with a Noether-charge definition (Compère, Oliveri and Seraj 2018). Geroch's static definition and Hansen's stationary definition are one formalism, not two. The remaining open problem: for stationary, asymptotically flat systems with arbitrarily strong internal gravity, exhibit explicitly the combination rules translating the moments of each of the three remaining sets — Janis-Newman (1965), van der Burg (1968), and Clarke-Sciama (1971) — into the standard Geroch-Hansen/Thorne moments (equivalently, complete the dictionary among all sets). For each such set, derive the explicit combination rules together with the precise domain of validity (which spacetimes both definitions cover) and the order-mixing structure; where an older definition and the standard moments share only a restricted common domain — for example the static overlap in the case of Clarke-Sciama — the relations are to be exhibited on that common domain. Any claim that a definition supports no such dictionary must be established rigorously on the common domain of definition; a mere domain mismatch, such as a definition being static-only or not extending to strong fields, is not by itself a resolution.

References

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