There is an intimate connection between multipole moments and the conformal group. While this connection is not emphasized in the usual formulation of moments, it provides the starting point for a consideration of multipole moments in curved space. As a preliminary step in defining multipole moments in general relativity (a program which will be carried out in a subsequent paper), the moments of a solution of Laplace's equation in flat 3‐space are studied from the standpoint of the conformal group. The moments emerge as certain multilinear mappings on the space of conformal Killing vectors. These mappings are re‐expressed as a collection of tensor fields, which then turn out to be conformal Killing tensors (first integrals of the equation for null geodesics). The standard properties of multipole moments are seen to arise naturally from the algebraic structure of the conformal group.

1.
Strictly speaking, it is ∇2ψ−18Rψ0 which is conformally invariant, provided ψ is assigned conformal weight −12. In flat space, however, this equation reduces to Laplace’s equation.
2.
In particular, V will be topological S2×R.
3.
See, for example, T. J. Willmore, Introduction to Differential Geometry (Oxford U.P. London, 1959).
4.
See, for example, C. Chevalley, Theory of Lie Groups (Princeton U.P., Princeton, N.J., 1946).
5.
It is shown in Appendix B that the constancy of either of these tensors implies the constancy of the other.
6.
No restrictions on the dimension of the manifold or on the signature of the metric will be required in this appendix.
7.
That is, la is so scaled that lm∇mla = 0.
8.
Conformal Killing tensors are a generalization of Killing tensors, which represent first integrals of the equations for arbitrary geodesies. Killing tensors are discussed, for example, in L. P. Eisenhart, Riemannian Geometry (Princeton U.P., Princeton, N.J., 1964), pp. 128.
9.
If the metric hab is positive definite, there are, of course, no null geodesies. In this case, (A1) is taken as the definition of a conformal Killing tensor.
10.
The difference between the actions of two derivative operators, Dm and D̃m, may be expressed in terms of a tensor field Kabc = K(ab)c. [See S. Kobayashi and K. Nomizu, Foundations of Differential Geometry (Interscience, New York, 1963), Vol. I.] The coefficients on the right in (A3) have been so chosen that, when Dm is eliminated in favor of D̃m, all K terms vanish identically.
11.
A. Nijenhuis, in Proceedings of the International Congress of Mathematicians (Cambridge U.P., Cambridge, England, 1960), p. 463.
12.
There is a similar algebra for totally antisymmetric contravariant tensors (Ref. 11). If P, Q, and R are skew, of ranks p, q, and r, respectively, then Eqs. (A2)–(A5) must be replaced by
P∩AQ = P[a1⋯apQap+1⋯ap+q] = (−1)pqQ∩AP,[P,Q] = p(−1)p+1Qm[a1⋯ap−1∇mQap⋯ap+q−1]−q(−1)q+pqQm[a1⋯aq−1∇mPap⋯ap+q−1][P,Q∩AR] = [P,Q]∩AR+(−1)q[P,R]∩AQ,(−1)pr+p[P,[Q,R]]+(−1)qp+q[Q,[R,P]]+(−1)rq+τ[R,[P,Q]] = 0
, respectively.
13.
Evidently, the Killing tensors [P’s which satisfy (A6) with the right side set equal to zero] form a subalgebra of the algebra of conformal Killing tensors.
14.
R.
Geroch
,
Commun. Math. Phys.
13
,
180
(
1969
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15.
The position vector of p′ relative to p is defined as follows. Let γ be any curve from p to p′, with parameter s and tangent vector η(s)m, such that ηmm∇s = 1. Then xm = ∫pp′η̃m(s)ds, where η̃m(s) is the s‐dependent vector at p′ obtained by parallel transport of ηm(s) along γ to p′. This xm is certainly invariant under continuous deformations of γ: that it is totally independent of γ is a consequence of the global assumption on V mentioned earlier. Note that in certain spaces distinct points may be related by the zero position vector.
16.
This number is obtained by counting data for ξb:n dimensions for ξa,12n(n−1) for Fab, 1 for φ, and n for ka.
17.
The first term in (B5) contributes 12(p2+q2−p−q) plus’s and pq minus’s, the second term, 0 plus’s and 1 minus, and the third term (p+q) plus’s and (p+q) minus’s.
18.
The converse is false. The conformal Killing tensor gab, for example, is not trace free, yet it certainly comes from some tensor over 𝒞.
19.
This proof requires a modification for the case in which the metric is positive definite: one must permit ka,⋯,la to be complex.
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© 1970 The American Institute of Physics.
1970
The American Institute of Physics
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