In Vaidya's metric for a radiating sphere, $d{s}^2=-(1-2m{r}^{-1\mathrm{}})d{u}^2-2dudr+r^{2}d{\Omega }^2,$ where $m\left(u\right)\mathrm{}$ is a nonincreasing function of the retarded time $u=t-r$, we verify that $-\frac{\mathrm{dm}}{\mathrm{du}}$ is the total power output as given by the Landau-Lifshitz stress-energy pseudotensor, and relate it through red-shift and Doppler-shift factors to the apparent luminosity $L$ for an observer moving radially in this gravitational field. We argue that the hypersurface $r=2m\left(u\right)\mathrm{}$ cannot be realized physically, but see that a hypersurface $r=2m\left(\infty \right)$ at $u=\infty$ (which is not adequately represented in presently available coordinate systems) shows the total red-shift characteristic of the Schwarzschild "singularity." The geodesic equations are written out to display a gravitational "induction field" $-\frac{\mathrm{GL}}{c^{3}r}$ associated with a changing mass in the Newtonian $-\frac{\mathrm{Gm}}{r^2}$ field.

• Received 9 September 1964

DOI:https://doi.org/10.1103/PhysRev.137.B1364