Spin-Weighted Harmonics for Waveforms

Gravitational-wave strain is not a scalar on the sky. It depends on the orientation of the complex polarization dyad and acquires two units of phase under a dyad rotation. Spin-weighted spherical harmonics are therefore the natural angular basis: they encode that local frame dependence while leaving the mode coefficients as a finite, systematically improvable description of the radiation. This page fixes the phase, normalization, conjugation, and ladder conventions used throughout the volume. The general harmonic construction and coupling algebra remain the province of black-hole perturbation theory.

Required background. Spacetime Symmetries and Killing Fields supplies rotations and their generators on the sphere.

Helpful background. Einstein Equations supplies the tensorial radiative field; Waves, Green Functions, and Geometric Optics supplies angular mode expansions; and Gravitational Radiation and Waveforms supplies the complex strain $h_+-ih_\times$.

Why the strain has spin weight minus two

Choose an oriented orthonormal screen basis and its complex unit dyad $m^A$, normalized by $m^A\bar m_A=1$. At future null infinity define the rescaled strain

$$H=C_{AB}\bar m^A\bar m^B =\lim_{r\to\infty}r(h_+-ih_\times).$$

Under a local dyad rotation $m^A\mapsto e^{i\chi}m^A$,

$$H\mapsto e^{-2i\chi}H.$$

A spin-$s$ field transforms as $\eta\mapsto e^{is\chi}\eta$, so $H$ has $s=-2$. This transformation is a change of polarization basis, not a new physical wave. A scalar spherical-harmonic expansion would hide the dyad phase in coordinate-dependent components; the spin-weighted basis carries it explicitly.

The declared harmonic convention

Use the Condon–Shortley associated Legendre functions and the Wigner small-$d$ convention

$$d^\ell_{m0}(\theta) =\sqrt{\frac{(\ell-m)!}{(\ell+m)!}}\, P_\ell^m(\cos\theta).$$

The spin-weighted harmonics are

$${}_sY_{\ell m}(\theta,\phi) =(-1)^s\sqrt{\frac{2\ell+1}{4\pi}} d^\ell_{m,-s}(\theta)e^{im\phi},$$

for integer $s$, $\ell\ge\lvert s\rvert$, and $-\ell\le m\le\ell$. This declaration fixes the Condon–Shortley phase, the dyad phase, and the sign of $m$. At $s=0$ it reduces to the ordinary unit-normalized $Y_{\ell m}$. Goldberg and collaborators give the definitions, rotations, orthogonality, and differential ladders in this convention family Goldberg et al. 1967, pp. 2155–2161.

The waveform expansion and inverse projection are

$$H(u,\theta,\phi) =\sum_{\ell=2}^{\infty}\sum_{m=-\ell}^{\ell} H_{\ell m}(u)\,{}_{-2}Y_{\ell m}(\theta,\phi),$$ $$H_{\ell m}(u) =\int_{S^2}H(u,\theta,\phi) \,{}_{-2}\bar Y_{\ell m}(\theta,\phi)d\Omega.$$

These formulas use the positively oriented unit-sphere measure and

$$\int_{S^2}{}_sY_{\ell m}\,{}_s\bar Y_{\ell' m'}d\Omega =\delta_{\ell\ell'}\delta_{mm'}.$$

Conjugation and angular ladders

Complex conjugation reverses spin weight:

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

It does not turn a spin-$-2$ harmonic into another spin-$-2$ harmonic. Relations such as $H_{\ell,-m}=(-1)^\ell\bar H_{\ell m}$ require additional physical reflection symmetry and a specified source frame; they are not identities of an arbitrary waveform.

Define the eth operators on a spin-$s$ field by

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

Then

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

The ladder coefficients vanish at the allowed endpoints, a useful check on both $s$ and the sign assigned to $\bar\eth$.

Pole and power checks

Because $d^\ell_{m,-s}(0)=\delta_{m,-s}$, only $m=-s$ can be nonzero at the north pole in this coordinate dyad. For gravitational strain, that is $m=2$. The residual $e^{2i\phi}$ there reflects that the coordinate dyad itself is singular at the pole; the tensor field is regular.

Orthonormality also makes the sky-integrated strain power diagonal:

$$\int_{S^2}\lvert H\rvert^2d\Omega =\sum_{\ell,m}\lvert H_{\ell m}\rvert^2.$$

A rotation mixes $m$ values at fixed $\ell$ through a unitary Wigner matrix, so this total is preserved. Truncating the mode sum, changing harmonic phases, or mixing active and passive rotations can spoil an apparent mode-by-mode comparison without changing the underlying field.

Reading mode coefficients correctly

The complex field $H$ packages two real polarizations, so it is not required to be a real function on the sphere. The usual scalar reality condition $a_{\ell,-m}=(-1)^m\bar a_{\ell m}$ therefore does not apply. A relation between positive and negative $m$ strain modes comes from a physical reflection symmetry of the source and the chosen frame, and precession or asymmetry can break it.

Mode labels also refer to a basis, not separately measurable pieces of radiation. A rigid rotation preserves each fixed-$\ell$ norm but redistributes power among its $m$ components. A boost or supertranslation is not merely such a rotation and can couple different $\ell$ values or retarded times. Accordingly, a claim that a particular $(\ell,m)$ mode “grew” is meaningful only after the Bondi frame and rotation convention have been held fixed.

Four errors are especially easy to diagnose. A nonzero coefficient with $\ell<\lvert s\rvert$ violates the representation range. Failure of the orthogonality integral signals an inconsistent sphere measure or normalization. A conjugation formula that leaves $s$ unchanged has confused the complex field with its tensor origin. Finally, swapping $e^{im\phi}$ for $e^{-im\phi}$ while keeping every other formula fixed reverses the interpretation of $m$ and invalidates later Wigner-$D$ and phase relations. Declaring the full convention once is safer than repairing these signs formula by formula.

Exercise

In the declared convention,

$${}_{-2}Y_{22} =\sqrt{\frac{5}{64\pi}}(1+\cos\theta)^2e^{2i\phi}.$$

Verify its unit norm and its north- and south-pole behavior.

Solution

With $x=\cos\theta$,

$$\int\lvert{}_{-2}Y_{22}\rvert^2d\Omega =\frac{5}{64\pi}(2\pi) \int_{-1}^{1}(1+x)^4dx =\frac{10}{64}\frac{32}{5}=1.$$

At the south pole, $x=-1$ and the harmonic vanishes. At the north pole, $x=1$ and its magnitude is $\sqrt{5/(4\pi)}$. The phase $e^{2i\phi}$ is the expected spin-frame phase of the singular polar coordinate dyad, not a failure of regularity.

References

• Goldberg, Joshua N., Alan 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.