Comparing Kerr Coordinates for Parameterizing Fundamental Ringdown Damping in Numerical Relativity

This paper compares candidate Kerr coordinate parameterizations for organizing the fundamental (ℓ=m=2, n=0) quasinormal-mode damping quality factor Q220 of Kerr black holes, using non-circular numerical-relativity simulations from the SXS collaboration catalog. The central result is that dimensionless surface gravity κM = (r+ − r−)/(2(r+2 + a2)) · M emerges as the leading physical coordinate for parameterizing Q220, outperforming dimensionless spin af and horizon angular velocity ΩHM in residual structure, monotonicity, and physical interpretability. Key contributions: • Systematic comparison of three Kerr coordinate parameterizations for Q220 • Non-circular NR benchmark using 18 SXS simulations (mass ratios q = 1–10) • Identification of κM as the coordinate with minimal residual structure • Robustness analysis across mass-ratio subsets and fit-order sensitivity • Reproducible comparison protocol documented for independent verification