Global geometry of the Vaidya spacetime

Abstract The transformation from initial coordinates (v,r) of the Vaidya metric, with light coordinate v, to the more physical diagonal coordinates (t,r) is obtained and analyzed from the point of view of global geometry. The exact solutions have been obtained in the case of a linear form of the mass function m(v). In the diagonal coordinates under consideration, a narrow region has been revealed near the apparent horizon of the Vaidya accreting black hole, where the metric differs qualitatively from the Schwarzschild one and cannot be represented as a small perturbation of the Schwarzschild solution. The global geometry of the thick photon shell between the two (inner and outer) Schwarzschild regions is constructed and the corresponding matching conditions are discussed. The propagation of light beams in the Vaidya metric in the case of accretion is investigated and the time of the apparent horizon’s crossings is calculated.