On the Limitations of a Generalized Vaidya Metric

Abstract

We prove that there can not be a smooth matching of the Generalized Vaidya metric with an exterior Schwarzschild/Vaidya patch across a finite boundary hypersurface unless the mass function is a function of the null coordinate alone. By explicitly deriving the extrinsic curvature components, we show that for $\partial m / \partial r \neq 0$ one has a discontinuity in the curvature and induces a surface stress-energy tensor, corresponding to a thin shell of matter. This discontinuity also appears in the geometric invariant $\mathcal{K} = K_{ab}K^{ab}$ and in the Kodama current, indicating a mismatch in quasi-local energy flux across the boundary. The analysis of timelike geodesics leads to the same condition, reinforcing that the generalized Vaidya geometry with $\partial m / \partial r \neq 0$ cannot represent a consistent stellar interior bounded by a regular surface. We therefore note that the generalized Vaidya spacetime should be interpreted as an unbounded geometry with intrinsic heat flux rather than a viable bounded source.