Kerr spin-weighted angular projection

Specification ID: kerr-angular-projection-scalar/v1.

Scope and status

This specification checks the spin-weight \(s=-2\) spherical and spheroidal angular factors used to project positive-\(m\) Kerr ringdown modes into complex plus and cross coefficients. Its independent scalar reference covers six spherical modes and two spheroidal modes at four stated oblateness values. It does not qualify QNM frequencies, excitation amplitudes, a full ringdown waveform, all supported modes or the angular-series stopping rule.

Inputs, conventions, and options

Let \(c=a\omega\) be the complex angular oblateness, \(\theta\) the inclination, and \(\phi\) the azimuth. The spherical basis has unit sphere norm and \(e^{im\phi}\) dependence. A spheroidal eigenfunction has unit Hermitian norm in that basis; its coefficient at the requested \(\ell\) is real and positive. The reference truncates at \(\ell_{\max}=10\) and integrates matrix elements with 32-point Gauss--Legendre quadrature; the large-oblateness regression checks use \(\ell_{\max}=12\). The production solvers use their own spectral matrix and default \(\ell_{\max}=18\).

Mathematics and components

Cook and Zalutskiy, arXiv:1410.7698v2, Eqs. (4), (48), and (49)--(57), specify the spin-weighted angular equation, spherical expansion, and reflected negative-\(m\) partner. For \(s=-2\), write \(x=\cos\theta\) and \({}_{-2}Y_{jm}\) for the spherical basis. The scalar reference obtains each basis function from SymPy's Wigner-\(d\) matrix, then integrates the multiplication operators directly:

\[ X_{jk}=\int {}_{-2}Y_{jm}^{*}\,x\,{}_{-2}Y_{km}\,d\Omega, \qquad X^{(2)}_{jk}=\int {}_{-2}Y_{jm}^{*}\,x^2\,{}_{-2}Y_{km}\,d\Omega. \]

It diagonalizes the finite angular operator

\[ H_{jk}(c)=[j(j+1)-2]\delta_{jk}-4cX_{jk}-c^2X^{(2)}_{jk}, \]

continuing the eigenvalue from \(A_{\ell m}(0)=\ell(\ell+1)-2\) along a straight path to \(c\). If \(S_{\ell m}(\theta,\phi;c)\) is the resulting normalized eigenfunction, the projected coefficients are

\[ P=\tfrac12[S_{\ell m}(\theta,\phi;c) +S_{\ell m}(\pi-\theta,\phi;c)],\qquad C=-\tfrac{i}{2}[S_{\ell m}(\theta,\phi;c) -S_{\ell m}(\pi-\theta,\phi;c)]. \]

At \(c=0\), the spherical check independently constructs the partner as \((-1)^\ell{}_{-2}Y_{\ell,-m}^{*}(\theta,\phi)\). The reflected form above follows from the equatorial partner symmetry in the cited equations.

Implementation mapping

diffgw_reference.kerr_angular uses SymPy Wigner-\(d\), mpmath quadrature and mpmath eigensolutions. It imports no production angular helper or matrix. tests/test_kerr_angular_math_reference.py compares its complex \(P,C\) and their \(\theta,\phi\) derivatives with the Torch and JAX _ringdown_projection paths. It separately checks angular eigenvalues and the Torch derivative with respect to \(\operatorname{Re}c\).

Evidence and limitations

The CPU FP64 gates were declared before evaluating the reference: absolute errors below \(3\times10^{-13}\) for spherical values, \(2\times10^{-12}\) for their angle derivatives, \(3\times10^{-10}\) for spheroidal values and eigenvalues, \(2\times10^{-9}\) for their angle derivatives, and \(3\times10^{-6}\) for the Torch real-\(c\) derivative against a scalar centered difference. The spherical grid covers \((\ell,m)=(2,2),(2,1),(3,3),(3,2),(4,4),(4,3)\) at three off-axis \((\theta,\phi)\) pairs. Spheroidal cases are \((2,2)\) at \(c=0.42-0.10i\) and \((3,2)\) at \(c=0.58-0.12i\). Two additional JAX regression cases cover \((2,2)\) at \(c=1.5\) and \(c=1.9-0.25i\), comparing eigenvalues, projected values, batched JIT evaluation and reverse-mode real-\(c\) derivatives against the independent reference and its centered differences. They retain the \(3\times10^{-10}\) value and \(3\times10^{-6}\) real-\(c\) derivative gates.

Earlier gate evidence records the original 42 checks passing on len with Torch 2.14.0+cpu, JAX 0.11.2, SymPy 1.14.0, and mpmath 1.3.0. All 44 checks, including the continuation regressions, pass on local CPU FP64 after the fix. The JAX spheroidal eigenvector is computed via a pure differentiable inverse-iteration solver that carries the requested eigenpair along eight continuation steps from \(c=0\), without pure_callback, enabling native reverse-mode autodiff with respect to \(c\) and Kerr spin \(a\) via jax.grad (verified in tests/test_differentiable_solvers_and_nsbh.py). Neither this component gate nor the separate public-binary waveform gate establishes a full parameter-domain result.

Reproduce the checks

Run PYTHONPATH=src:tests python -m pytest -q tests/test_kerr_angular_math_reference.py --reference-device=cpu in an environment with the required scalar and frontend packages. The full strict math gate collects all test_*math_reference*.py modules and uses --require-math-reference.