Basic Vaidya White Hole Evaporation Process

We developed a self-consistent double-null description of an evaporating white-hole spacetime by embedding the outgoing Vaidya solution in a coordinate system that remains regular across the future horizon. Starting from the radiation-coordinate form, we specialize in retarded time so that a monotonically decreasing mass function M(u) encodes outgoing positive-energy flux. Expressing the metric in null coordinates (u,v), Einstein’s equations for a single-directional null-dust stress–energy tensor, Tuu=ρ(u), then reduce to one first-order PDE for the areal radius: ∂vr=B(u)1−2M(u)/r. Its integral, r+2M(u)ln|r−2M(u)|=v−C(u), defines an implicit foliation of outgoing null cones. The metric coefficient follows algebraically as f(u,v)=1−2M(u)/r. Residual gauge freedom in B(u) and C(u) is fixed so that u matches the Bondi retarded time at null infinity, while v remains analytic at the apparent horizon, generalizing the Kruskal prescription to dynamical mass loss. In the limit M(u)→M, the construction reduces to the familiar Eddington–Finkelstein and Kruskal forms. Our solution, therefore, provides a compact analytic framework for studying white-hole evaporation, Hawking-like energy fluxes, and back-reaction in spherically symmetric settings without encountering coordinate singularities.