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Published November 15, 1989 | public
Journal Article Open

Gravitational-wave measurements of the mass and angular momentum of a black hole


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 ΔM/M≃2.2ρ^-1(1-a)^0.45, Δa≃5.9ρ^-1(1-a)^1.06, where ρ≃hs(πSh)^-1/2 (1-a)^-0.22. Here ρ is the amplitude signal-to-noise ratio at the output of the optimal filter, hs is the wave's amplitude at the beginning of the vibrations, f is the wave's frequency (the angular frequency ω divided by 2π), and Sh is the frequency-independent spectral density of the detectors' noise. These formulas for ΔM and Δa are valid only for ρ≳10. Corrections to these approximate formulas are given in Table II.

Additional Information

©1989 The American Physical Society Received 27 December 1988 I wish to thank Dr. Kip S. Thorne, for suggesting this problem to me and for many suggestions that helped me with the research and in writing this paper. I am also indebted to Dr. E.W. Leaver, who supplied me with his unpublished numerical results on the frequencies and damping times of the fundamental l = m = 2 mode of Kerr black holes, which were necessary for my calculations. This work was supported in part by the National Science Foundation under Grant No. AST85-14911.


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