Calculating the Gravitational Waves Emitted from High-speed Sources

The possibility of forming gravitational-wave sources with high center-of-mass (c.m.) velocities in the vicinity of supermassive black holes requires us to develop a method of deriving the waveform in the observer's frame. Here we show that in the limit where the c.m. velocity is high but the relative velocities of the components of the source are small, we can solve the problem by directly integrating the relaxed Einstein field equation. In particular, we expand the result into multipole components which can be conveniently calculated given the orbit of the source in the observer's frame. Our numerical calculations using arbitrary c.m. velocities show that the result is consistent with the Lorentz transformation of GWs to the leading order of the radiation field. Moreover, we show an example of using this method to calculate the waveform of a scattering event between the high-speed ($\sim 0.1c$) stellar objects embedded in the accretion disk of an active galactic nucleus. Our multipole-expansion method not only has advantages in analyzing the results from stellar-dynamical models but also provides new insight into the multipole properties of the GWs emitted from a high-speed source.