We investigate the geometrical structure of Vaidya’s spacetime in the case of a white hole with decreasing mass, stabilising to a black hole in finite or infinite time or evaporating completely. Our approach relies on a detailed analysis of the ordinary differential equation describing the incoming principal null geodesics, among which are the generators of the past horizon. We devote special attention to the case of a complete evaporation in infinite time and establish the existence of an asymptotic light-like singularity of the conformal curvature, touching both the past space-like singularity and future time-like infinity. This singularity is present independently of the decay rate of the mass. We derive an explicit formula that relates directly the strength of this null singularity to the asymptotic behaviour of the mass function.
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Note that Fayos and Torres  also exhibit a null singularity but under slightly different assumptions. The evaporation ends in finite time, therefore the singularity is not asymptotic but well present in the spacetime. Moreover the evaporation is allowed to end brutally: the mass vanishes in finite retarded time with a possibly non-zero slope, in which case it is non-differentiable at its vanishing point.
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Coudray, A., Nicolas, JP. Geometry of Vaidya spacetimes. Gen Relativ Gravit 53, 73 (2021). https://doi.org/10.1007/s10714-021-02839-7
- Black hole
- White hole
- Vaidya metric
- Einstein equations with matter
- Null singularity