Gravitational waves in Brans-Dicke theory: Analysis by test particles around a Kerr black hole

Abstract

Analyzing test particles falling into a Kerr black hole, we study gravitational waves in Brans-Dicke theory of gravity. First we consider a test particle plunging with a constant polar angle into a rotating black hole and calculate the waveform and emitted energy of both scalar and tensor modes of gravitational radiation. We find that the waveform as well as the energy of the scalar gravitational waves weakly depends on the rotation parameter of a black hole $a$ and on the polar angle. Second, using a model of a nonspherical dust shell of test particles falling into a Kerr black hole, we study when the scalar modes dominate. When a black hole is rotating, the tensor modes do not vanish even for a ``spherically symmetric'' shell; instead a slightly oblate shell minimizes their energy but with a nonzero finite value, which depends on the Kerr parameter $a$. As a result, we find that the scalar modes dominate only for highly spherical collapse, but they never exceed the tensor modes unless the Brans-Dicke parameter ${\ensuremath{\omega}}_{\mathrm{BD}}\ensuremath{\lesssim}750$ for $a/M=0.99$ or unless ${\ensuremath{\omega}}_{\mathrm{BD}}\ensuremath{\lesssim}20000$ for $a/M=0.5$, where $M$ is the mass of a black hole. We conclude that the scalar gravitational waves with ${\ensuremath{\omega}}_{\mathrm{BD}}\ensuremath{\lesssim}$ several thousands do not dominate except for very limited situations (observation from the face-on direction of a test particle falling into a Schwarzschild black hole or highly spherical dust shell collapse into a Kerr black hole). Therefore, observation of polarization is also required when we determine the theory of gravity by the observation of gravitational waves.