Gravitational-wave measurements of the mass and angular momentum of a black hole
Abstract
A deformed black hole produced in a cataclysmic astrophysical event should undergo damped vibrations which emit gravitational radiation. By fitting the observed gravitational waveform h(t) to the waveform predicted for black-hole vibrations, it should be possible to deduce the hole's mass M and dimensionless rotation parameter a=(c/G)(angular momentum)/${M}^{2}$. This paper estimates the accuracy with which M and a can be determined by optimal signal processing of data from laser-interferometer gravitational-wave detectors. It is assumed that the detector noise has a white spectrum and has been made Gaussian by cross correlation of detectors at different sites. Assuming, also, that only the most slowly damped mode (which has spheroidal harmonic indices l=m=2) is significantly excited---as probably will be the case for a hole formed by the coalescence of a neutron-star binary or a black-hole binary---it is found that the one-sigma uncertainties in M and a are \ensuremath{\Delta}M/M\ensuremath{\simeq}2.2${\ensuremath{\rho}}^{\mathrm{\ensuremath{-}}1}$(1-a${)}^{0.45}$, \ensuremath{\Delta}a\ensuremath{\simeq}5.9${\ensuremath{\rho}}^{\mathrm{\ensuremath{-}}1}$(1-a${)}^{1.06}$, where \ensuremath{\rho}\ensuremath{\simeq}${h}_{s}$(\ensuremath{\pi}${S}_{h}$${)}^{\mathrm{\ensuremath{-}}1/2}$ (1-a${)}^{\mathrm{\ensuremath{-}}0.22}$. Here \ensuremath{\rho} is the amplitude signal-to-noise ratio at the output of the optimal filter, ${h}_{s}$ is the wave's amplitude at the beginning of the vibrations, f is the wave's frequency (the angular frequency \ensuremath{\omega} divided by 2\ensuremath{\pi}), and ${S}_{h}$ is the frequency-independent spectral density of the detectors' noise. These formulas for \ensuremath{\Delta}M and \ensuremath{\Delta}a are valid only for \ensuremath{\rho}\ensuremath{\gtrsim}10. Corrections to these approximate formulas are given in Table II.