Spin-Weighted Spherical Harmonics

Radiative field components depend on a tangent-frame choice even when the underlying tensor is geometric. Spin-weighted spherical harmonics encode this local phase behavior while retaining ordinary angular-momentum labels. With a fixed dyad, Condon–Shortley phase, and eth convention, they form an orthonormal basis for spin-$s$ fields, turn angular derivatives into ladder coefficients, and make product-coupling rules algebraic.

Required background. Killing Fields and Isometry Groups supplies angular momentum and rotations; Events, Intervals, and Minkowski Geometry supplies local Lorentz transformations and helicity intuition.

Helpful background. Black-Hole Geometry and Mechanics supplies radiative black-hole backgrounds; PDEs and Green Functions for Gravitation supplies spectral expansions; Formulations of General Relativity places frame components within constrained field equations; Frames, Coframes, and Anholonomy develops the local frame freedom used here.

Spin weight is a dyad transformation law

Choose the complex unit-sphere dyad

$$m^A=\frac{1}{\sqrt2} \left((\partial_\theta)^A+i\csc\theta(\partial_\phi)^A\right), \qquad m^A\bar m_A=1.$$

Under $m^A\mapsto e^{i\chi}m^A$, a quantity $\eta$ has spin weight $s$ if $\eta\mapsto e^{is\chi}\eta$. This is not field spin or spherical-harmonic $\ell$. For example, contracting a trace-free tensor with $m^Am^B$ produces a spin-$2$ component, while the conjugate contraction has spin $-2$. A regular spin-weighted field is a section of the corresponding line bundle, so one should not expect a single globally nonsingular dyad.

We use the Condon–Shortley scalar harmonics defined on the preceding page. The spheroidal convention used later is $c=a\omega$, with time dependence $e^{-i\omega t}$, inner product $\langle f,g\rangle=\int_{S^2}\bar f g\,d\Omega$, and $A_{s\ell m}(0)=\ell(\ell+1)-s(s+1)$.

Eth operators and normalized construction

For a spin-$s$ field in the displayed dyad, define

$$ð\eta =-(\sin\theta)^s \left(\partial_\theta+i\csc\theta\,\partial_\phi\right) \left[(\sin\theta)^{-s}\eta\right],$$ $$\barð\eta =-(\sin\theta)^{-s} \left(\partial_\theta-i\csc\theta\,\partial_\phi\right) \left[(\sin\theta)^s\eta\right].$$

The first raises spin weight by one and the second lowers it. For $s\ge0$,

$${}_sY_{\ell m} =\sqrt{\frac{(\ell-s)!}{(\ell+s)!}}\,ð^sY_{\ell m},$$

while for $s\le0$,

$${}_sY_{\ell m} =(-1)^s\sqrt{\frac{(\ell+s)!}{(\ell-s)!}}\, \barð^{-s}Y_{\ell m}.$$

These are defined only for $\ell\ge|s|$. The eth signs and normalized construction agree with Goldberg et al. 1967, § 2, Eqs. (2.3)–(2.8), p. 2156, PDF, but their displayed scalar phase needs translation to the site’s Condon–Shortley phase. Their finite sum gives $Y_{11}^{\mathrm G}=+\sqrt{3/(8\pi)}\sin\theta\,e^{i\phi}$, whereas the site’s $Y_{11}$ has a minus sign. In general,

$${}_sY_{\ell m}^{\mathrm{site}} =(-1)^m{}_sY_{\ell m}^{\mathrm G}.$$

For $m\ge0$, compare the scalar finite sum with Rodrigues’ associated Legendre formula; the difference is its Condon–Shortley factor $(-1)^m$. The negative-$m$ conjugation rule extends the relation, and applying eth does not change $m$, so it holds for every allowed spin weight Goldberg et al. 1967, Eq. (3.1), p. 2157, PDF. This basis-phase translation is distinct from changing the dyad or either eth sign.

The definition is intrinsically patchwise. The displayed dyad is singular at the coordinate poles, but the spin-weighted field is not: on the overlap of north- and south-regular dyads, its two coordinate representatives differ by the transition phase $e^{is\chi}$. This observation separates a genuine lack of regularity from a dyad singularity. It also explains why integrating a spin-$s$ representative as though it were an ordinary scalar can give a phase-dependent answer. A scalar observable must have net spin weight zero, either before integration or after contraction with complementary frame factors.

Ladder, conjugation, and orthogonality identities

Applying the definitions gives

$$ð\,{}_sY_{\ell m} =\sqrt{(\ell-s)(\ell+s+1)}\,{}_{s+1}Y_{\ell m},$$ $$\barð\,{}_sY_{\ell m} =-\sqrt{(\ell+s)(\ell-s+1)}\,{}_{s-1}Y_{\ell m}.$$

At $s=0$, ${}_0Y_{\ell m}=Y_{\ell m}$. Conjugation is

$$\overline{{}_sY_{\ell m}} =(-1)^{s+m}{}_{-s}Y_{\ell,-m}.$$

For fixed $s$, the harmonics obey

$$\int_{S^2}\overline{{}_sY_{\ell m}}\, {}_sY_{\ell' m'}\,d\Omega =\delta_{\ell\ell'}\delta_{mm'},$$

and are complete in the square-integrable spin-$s$ sections. The angular eigenvalue identity

$$\barð ð\,{}_sY_{\ell m} =-(\ell-s)(\ell+s+1){}_sY_{\ell m}$$

is an immediate independent sign check. Reversing the order gives a different coefficient because $[\barð,ð]\eta=2s\eta$ in this convention. The weight used in the second operation is the new weight: explicitly,

$$\barð_{s+1}ð_s-ð_{s-1}\barð_s=2s.$$

Couplings become Wigner selection rules

For integer weights with $s_1+s_2+s_3=0$, three harmonics in the same dyad form a scalar product. Their spin-weighted Gaunt coefficient is

$$\begin{aligned} &\int {}_{s_1}Y_{\ell_1m_1} {}_{s_2}Y_{\ell_2m_2} {}_{s_3}Y_{\ell_3m_3}\,d\Omega\\ &=\sqrt{\frac{(2\ell_1+1)(2\ell_2+1)(2\ell_3+1)}{4\pi}} \begin{pmatrix}\ell_1&\ell_2&\ell_3\\-s_1&-s_2&-s_3\end{pmatrix} \begin{pmatrix}\ell_1&\ell_2&\ell_3\\m_1&m_2&m_3\end{pmatrix}. \end{aligned}$$

Under the stated total-spin-zero hypothesis, the result vanishes unless $m_1+m_2+m_3=0$ and the three $\ell$ values satisfy the triangle inequalities. Additional zeros can occur. For three scalar harmonics an odd sum of $\ell$ vanishes, but that is not a universal rule for nonzero individual spin weights. The hypothesis and the caution about integrating nonzero-spin coordinate representatives are explicit in Allen 2024, Appendix A, Eq. (A7) and following paragraph, p. 043043-20. The Goldberg-to-site phases do not alter this two-symbol formula: when the azimuthal sum is zero, their product is $(-1)^{m_1+m_2+m_3}=1$.

What fails when the total spin is nonzero

In the displayed dyad the normalized dipoles are

$${}_1Y_{10}=+\sqrt{\frac3{8\pi}}\sin\theta, \qquad {}_{-1}Y_{10}=-\sqrt{\frac3{8\pi}}\sin\theta.$$

Since $Y_{00}=1/\sqrt{4\pi}$, direct integration gives

$$\begin{aligned} \int_{S^2}{}_1Y_{10}Y_{00}^2\,d\Omega &=\frac12\sqrt{\frac3{8\pi}}\int_0^\pi\sin^2\theta\,d\theta\\ &=\sqrt{\frac{3\pi}{128}}\ne0. \end{aligned}$$

The two-Wigner-symbol expression would be zero, but its hypothesis fails. This is an integral of a chosen component, not an invariant scalar observable. A constant dyad rotation by $\chi=\pi/2$ multiplies it by $i$. The coordinate poles have measure zero and do not prevent this finite integral; they do not make its answer dyad-independent either.

Allowed and forbidden coefficients

Set $x=\cos\theta$. The spin-zero product of the two dipoles is $-\frac{3}{8\pi}(1-x^2)$. Consequently

$$\int_{S^2}{}_1Y_{10}{}_{-1}Y_{10}Y_{00}\,d\Omega =-\frac1{\sqrt{4\pi}}.$$

For $Y_{20}=\sqrt{5/(16\pi)}(3x^2-1)$, the calculation is

$$\begin{aligned} \int_{S^2}{}_1Y_{10}{}_{-1}Y_{10}Y_{20}\,d\Omega &=-\frac34\sqrt{\frac5{16\pi}} \int_{-1}^1(1-x^2)(3x^2-1)\,dx\\ &=\frac1{\sqrt{20\pi}}, \end{aligned}$$

using the polynomial integral $-8/15$. These opposite signs agree with the two Wigner symbols and test the relative phase of the spin-one bases. The zero cases test different restrictions:

• $\int{}_1Y_{11}{}_{-1}Y_{10}Y_{00}\,d\Omega=0$ because the azimuthal factor is $e^{i\phi}$; total spin and the triangle condition are satisfied.
• $\int{}_1Y_{10}{}_{-1}Y_{10}Y_{40}\,d\Omega=0$ because $4\gt1+1$. Equivalently, the Legendre polynomial $P_4(x)$ is orthogonal to $1-x^2$. The azimuthal sum and even-$\ell$-sum condition hold here.
• $\int{}_1Y_{10}{}_{-1}Y_{10}Y_{10}\,d\Omega=0$ because the integrand is odd in $x$, although the triangle and azimuthal rules permit it.

The last zero must not be promoted to a general odd-$\ell$-sum rule. For example, direct integration and the Wigner expression both give

$$\int_{S^2}{}_1Y_{11}{}_{-1}Y_{10}Y_{1,-1}\,d\Omega =\frac{\sqrt3}{4\sqrt\pi}\ne0.$$

These checks distinguish a required selection rule from an additional zero of a particular coupling.

Why rotations predict the couplings

The rotational definition gives a second normalization check. In the adopted phase convention, ${}_sY_{\ell m}$ is a fixed matrix element of the $\ell$ representation of $SO(3)$, multiplied by $\sqrt{(2\ell+1)/(4\pi)}$. Orthogonality of the rotation-matrix elements under the rotationally invariant measure gives harmonic orthonormality, while the Clebsch–Gordan product law yields the two Wigner symbols above. This is why the triangle and $m$-sum rules are representation-theoretic necessities rather than empirical features of particular integrals. The coordinate and rotation constructions must agree after one phase is fixed; disagreement diagnoses a convention translation, not new physics.

The harmonics organize angular dependence; they do not by themselves prove that a radial–angular field equation separates, nor that a mode sum is complete under quasinormal boundary conditions. General representation theory remains external mathematical preparation, and the exact hidden geometry that may produce separability remains in Exact Solutions.

Exercise: test both ladder signs

Starting with ${}_0Y_{10}=\sqrt{3/(4\pi)}\cos\theta$, apply $ð$ and $\barð$ directly. Compare the results with the normalized ${}_{\pm1}Y_{10}$ and verify $\barð ðY_{10}=-2Y_{10}$.

Solution

Because $m=0$, the azimuthal derivative vanishes. At spin weight zero,

$$ðY_{10}=-\partial_\theta Y_{10} =\sqrt{\frac{3}{4\pi}}\sin\theta,$$

and the ladder rule says this equals $\sqrt2\,{}_1Y_{10}$. Likewise $\barðY_{10}=-\partial_\theta Y_{10}=-\sqrt2\,{}_{-1}Y_{10}$ in the adopted phase convention. Thus the two coordinate expressions are equal, while the normalized spin-$+1$ and spin-$-1$ basis functions differ by a sign. Acting with $\barð$ on the spin-one result gives

$$\barð ðY_{10} =-(1-0)(1+0+1)Y_{10}=-2Y_{10}.$$

The equality of the first coordinate expressions does not mean the resulting sections have the same spin weight: they transform oppositely when the dyad is rotated. Missing that transformation law is the common diagnostic failure.

References

• Allen, Bruce. “Pulsar Timing Array Harmonic Analysis and Source Angular Correlations.” Physical Review D 110 (2024): 043043. DOI.
• Goldberg, Joshua N., A. J. Macfarlane, Ezra T. Newman, Fritz Rohrlich, and E. C. G. Sudarshan. “Spin-s Spherical Harmonics and ð.” Journal of Mathematical Physics 8 (1967): 2155–2161. DOI.