Running the Spectrum Backwards: Recovering Remnant Mass and Spin From a Frequency and a Damping Time

Most ringdown work runs forwards, from remnant parameters to mode properties. This paper runs the machinery backwards: given a measured fundamental frequency and damping time, it recovers the mass and spin that a Kerr remnant would need in order to produce them, and it maps how measurement errors in the two observables inflate into parameter errors in the inversion. The adopted measurement pair, 247 Hz and 4.20 ms, implies a quality factor of 3.259, which inverts to a spin of 0.693. At that spin the real dimensionless frequency is 0.5322, and the frequency measurement then gives a remnant mass of 69.6 solar masses; running the recovered pair forwards returns 247.0 Hz and 4.20 ms, closing the round trip. A five per cent frequency error moves the recovered mass by 1.9 per cent and the recovered spin by only 0.036, confirming that frequency is the mass observable. A measured pair of 247 Hz and 4.20 ms inverts to a spin of 0.693 and a mass of 69.6 solar masses, with frequency errors passing into mass at 1.9 per cent for five per cent and damping errors passing into spin at 0.065 for ten per cent. This study is computed from stated assumptions and published catalogue values; it is simulated, not measured.