Lorentz transformation of three dimensional gravitational wave tensor

TL;DR

This paper derives the explicit Lorentz transformation for three-dimensional gravitational wave tensors, enabling the construction of waveforms from moving sources based on rest frame data, with applications to binary black hole kicks.

Contribution

It provides the first explicit Lorentz transformation formula for 3D gravitational wave tensors using a traditional tensor approach.

Findings

Derived the Lorentz transformation for gravitational wave tensors.

Enabled waveform construction for moving sources from rest waveforms.

Applied the method to binary black hole kick velocities.

Abstract

Recently there are more and more interest on the gravitational wave of moving sources. This introduces a Lorentz transformation problem of gravitational wave. Although Bondi-Metzner-Sachs (BMS) theory has in principle already included the Lorentz transformation of gravitational wave, the transformation of the three dimensional gravitational wave tensor has not been explicitly calculated before. Within four dimensional spacetime, gravitational wave have property of `boost weight zero' and `spin weight 2'. This fact makes the Lorentz transformation of gravitational wave difficult to understand. In the current paper we adopt the traditional three dimensional tensor description of gravitational wave. Such a transverse-traceless tensor describes the gravitational wave freedom directly. We derive the explicit Lorentz transformation of the gravitational wave tensor. The transformation is similar to the Lorentz transformation for electric field vector and magnetic field vector which are three dimensional vectors. Based on the deduced Lorentz transformation of the gravitational wave three dimensional tensor, we can construct the gravitational waveform of moving source with any speed if only the waveform of the corresponding rest waveform is given. As an example, we apply our method to the effect of kick velocity of binary black hole. The adjusted waveform by the kick velocity is presented.