Exact gravitational wave signatures from colliding extreme black holes

The low-energy dynamics of any system admitting a continuum of static configurations is approximated by slow motion in moduli (configuration) space. Here, following Ferrell and Eardley, this moduli space approximation is utilized to study collisions of two maximally charged Reissner-Nordström black holes of arbitrary masses, and to compute analytically the gravitational radiation generated by their scattering or coalescence. The motion remains slow even though the fields are strong, and the leading radiation is quadrupolar. A simple expression for the gravitational waveform is derived and compared at early and late times to expectations.

Figure 1

Relative trajectories ${\mathbf{x}}_{*}(t)$, and (north pole, $+$ polarization) gravitational waveforms $h_{+}^{TT}$, for two equal mass BHs: a head-on coalescence, in dotted green (impact parameter $b=0$); a near-critical coalescence, in solid blue ($b=2.3658M$); and a scattering, in dashed red ($b=2.4M$). For this mass ratio, $b_{\mathrm{crit}}\simeq 2.3660M$.