Lorentz transformation of three dimensional gravitational wave tensor

Abstract

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