Abstract
The connection between spherical harmonics and symmetric tensors is explored. For each spherical harmonic, a corresponding traceless symmetric tensor is constructed. These tensors are then extended to include nonzero traces, providing an orthonormal angular-momentum eigenbasis for symmetric tensors of any rank. The relationship between the spherical-harmonic tensors and spin-weighted spherical harmonics is derived. The results facilitate the spherical-harmonic expansion of a large class of tensor-valued functions. Several simple illustrative examples are discussed, and the formalism is used to derive the leading-order effects of violations of Lorentz invariance in Newtonian gravity.
Highlights
Spherical harmonics Yjm provide an orthonormal basis for scalar functions on the 2-sphere and have numerous applications in physics and related fields
While they are commonly written in terms of the spherical-coordinate polar angle θ and azimuthal angle φ, spherical harmonics can be expressed in terms of Cartesian coordinates, which is convenient in certain applications
We first develop a method for calculating the scalar spherical harmonics Yjm in terms of components of the direction unit vector n = sin θ cos φ ex + sin θ sin φ ey + cos θ ez. (1)
Summary
Spherical harmonics Yjm provide an orthonormal basis for scalar functions on the 2-sphere and have numerous applications in physics and related fields. Contracting Y jm with a single n vector gives a traceless symmetric rank( j − 1) tensor function of n This decreases the spin by 1 and increases the orbital angular momentum by 1, while leaving the total angular momentum unchanged. The tensor Y30 generates four different angular-momentum eigenfunctions: the spin-3 constant (Y30 )abc, the spin-2 (Y30 )abcnc, the spin-1 (Y30 )abcnbnc, and the scalar (Y30)abcnanbnc. This procedure yields a natural set of tensor spherical harmonics of different ranks and spins.
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