The Bondi–Sachs 4-momentum in non-axisymmetric Robinson–Trautman spacetimes

We derive the Bondi–Sachs 4-momentum conservation laws that regulate the emission of gravitational waves in non-axisymmetric Robinson–Trautman spacetimes. Although Robinson–Trautman spacetimes describe the exterior vacuum gravitational field of a bounded system radiating gravitational waves, the Robinson–Trautman metrics do not satisfy the appropriate Bondi–Sachs boundary conditions formulated for radiating systems. We therefore derive the transformations between Robinson–Trautman and Bondi–Sachs coordinates which allow us to obtain some of the basic Bondi–Sachs physical quantities, namely the news functions and the Bondi–Sachs energy–momentum fluxes of the gravitational waves. These quantities are fundamental to the description of the radiative transfer processes involved in the generation and emission of gravitational waves by the system. The Bondi–Sachs 4-momentum conservation laws are covariant under the transformations of the Bondi–Metzner–Sachs group. The section of this paper containing the derivation of the coordinate transformations extends the work of von der Gönna and Kramer done for the case of axisymmetric Robinson–Trautman spacetimes. We discuss some possible applications of the conservation laws in numerical simulations of black hole systems emitting gravitational waves, and also future research of Bondi–Sachs conservation laws for twisting Robinson–Trautman-type spacetimes. (paper)