Slowly rotating stars in Bondi-Sachs form; extension to tri-axial bodies

A modern approach to describing the stationary exterior gravitational field of slowly rotating stars has been previously developed separately by both the Author (2006), and Vera et al (2006), building on the original framework by Hartle-Thorne (1968). However, all these formulations implicitly assume that the configuration has always been stationary, and continues to be forever stationary, and are therefore not suitable for describing the start or end point of a process involving the emission of gravitational waves. Suitable invariant radiation coordinates are required, and the Bondi-Sachs system can satisfy this requirement. Based on a method by Bishop and Venter (2006) for the Kerr metric, here we demonstrate that it is possible to convert both the Kerr and non-Kerr parts of the stationary vacuum metric into the Bondi-Sachs representation, such that closed-form expressions can be obtained. This quasi-spherical conversion can also be used for radiating processes, and using an approach similar to the Regge-Wheeler and Zerilli frameworks, we briefly discuss how the exterior of a radiating tri-axial rotating star could be constructed.