Slowly Rotating Radiating Sphere and a Kerr-Vaidya Metric

The problem of a spherically symmetric radiating body was first considered by Vaidya, who obtained what is often referred to as the "radiating Schwarzschild metric." It is well known that this metric, if expressed in radiation coordinates, differs from the Schwarzschild metric only in that the parameter $m$ has been replaced by a function of retarded time. In this paper, the parameter $m$ of the Kerr metric is considered to be a function of the retarded time and an exact expression for the energy tensor is obtained. It is shown that if $\frac{a}{m}$ is small then this energy tensor is appropriate for directed radiation. To this approximation [i.e., terms of order ${\left(\frac{a}{m}\right)}^2$ are neglected] the Landau-Lifschitz pseudotensor is used to show that the angular momentum radiated is $-m^{\prime }a$. The metric is also used to describe the physical properties of a slowly rotating radiating spherical shell, and it is shown that (provided $\frac{2m}{R}\ll 1$) the radiation gives rise to surface pressures pro-portional to the momentum radiated.