Kerr 220 energy-normalized carrier

Specification ID: kerr-220-energy-carrier-conditional/v1.

Scope and status

This specification covers the causal, physical-energy fundamental 220 polarizations and their Fourier-minus transform in kerr220_td/kerr220_fd. It conditions on supplied QNM frequency/damping and normalized angular factors. The separate angular projection specification now checks selected spherical and spheroidal factors independently. This carrier check itself does not verify the QNM eigenvalue problem; a separate scalar 220 QNM input check covers five selected spin values and angular eigenvalues. Neither check qualifies excitation, the inferred public-compatible angular-series stopping rule, or other modes and overtones.

Inputs, conventions, and options

The remnant mass \(M\) and luminosity distance \(D\) enter in geometric seconds, \(M_s=GM/c^3\) and \(D_s=D/c\). Let \(\epsilon\) be the fraction of remnant rest energy emitted by one mode, \(\omega>0\) its angular frequency in rad/s, \(\lambda=1/\tau>0\) its damping rate, \(t_c\) the onset, and \(S_+\) and \(S_-\) the supplied spin-weighted angular values at \(\iota\) and \(\pi-\iota\). The frontend inputs use solar masses, Mpc, dimensionless QNM frequency and damping, and phase \(\phi_c\) included in the angular factors. The two sides of the onset use \(H(t-t_c)\) with \(H(0)=1\). The Fourier convention is \(\widetilde h(f)=\int h(t)e^{-2\pi ift}\,dt\).

Mathematics and components

The finite-quality-factor amplitude for a single complex mode is

\[ A=\frac{\sqrt{32\pi\epsilon M_s\tau}} {D_s\sqrt{1+(\omega\tau)^2}}. \]

This follows by integrating the mode energy flux for \(t>t_c\), retaining both the oscillatory and damping derivatives. The energy-flux convention is from Ruiz et al., arXiv:0707.4654v3, Eq. (2.8), with unit angular norm as in Flanagan and Hughes, arXiv:gr-qc/9701039v2, Eq. (3.22). For comparison, Berti, Cardoso, and Will, arXiv:gr-qc/0512160v2, Eq. (3.18), write a dimensionless coefficient \(A_{lmn}^{\rm FH}\) using \(Q=\omega\tau/2\) and \(f_{lmn}=\omega/(2\pi)\). Identifying their \(\epsilon_{\rm rd}\) with \(\epsilon\) gives the algebraic mapping \(A=(M_s/D_s)A_{lmn}^{\rm FH}\). Their equation assumes the FH doubled-time waveform and equal-polarization convention; this coefficient identity does not establish equivalence of that waveform to the causal projection below.

For \(u=t-t_c\geq0\), \(z=e^{-(\lambda+i\omega)u}\), the selected projection is

\[ h_+(t)=-\frac{A}{\sqrt2}\operatorname{Re}[z(S_++S_-)],\qquad h_\times(t)=\frac{A}{\sqrt2}\operatorname{Im}[z(S_+-S_-)]. \]

Both are zero for \(u<0\). Define \(P=S_++S_-\), \(C=S_+-S_-\), and \(L_\pm=[\lambda+i(2\pi f\pm\omega)]^{-1}\). Direct Laplace integration gives

\[ \widetilde h_+=-\frac{A e^{-2\pi ift_c}}{2\sqrt2} (P L_+ + P^*L_-),\qquad \widetilde h_\times=\frac{A e^{-2\pi ift_c}}{2i\sqrt2} (C L_+ - C^*L_-). \]

Implementation mapping

diffgw_reference.kerr_ringdown evaluates these expressions with scalar mpmath arithmetic and also integrates the time-domain polarizations by quadrature as a separate Fourier check. The Torch and JAX production paths are torchwave.approximants.kerr_ringdown_public_clean and jaxwave.approximants.kerr_ringdown_public_clean. The maintained test_kerr_ringdown_math_reference.py supplies each frontend's QNM and angular values explicitly to the reference and compares both polarizations before, at, and after onset and at positive, zero, and negative frequencies.

Evidence and limitations

The scalar checks isolate energy normalization, causal projection, Fourier signs, mirror contribution, and time shift on two off-canonical parameter sets. Since they share QNM and angular inputs with each tested frontend, they do not independently establish those inputs. A fixed off-canonical full-waveform holdout uses the public SimBlackHoleRingdown TD callable, converged numerical Fourier integration, and the unchanged strict parity gate. It tests one additional point, not a parameter domain.

An independent opt-in public-clean composition audit finds angular and complete-waveform discrepancies after matching the QNM and angular phase gauge. Thus this specification remains conditional; it does not qualify the opt-in public-clean physical-energy 220 path as a whole waveform. The direct KerrRingdown class takes a separate source-backed spectral angular route; its bounded physical-energy 220 composition is versioned separately.

Reproduce the checks

Run PYTHONPATH=src:tests python -m pytest -q tests/test_kerr_ringdown_math_reference.py in a reference-enabled environment. The independent public-binary holdout is the KerrRingdown_holdout_off_axis_energy_onset case in tests/parity_harness.py.