The Vaidya metric: expected and unexpected traits of evaporating black holes
Julius Piesnack, Klaus Kassner
备注：
V3, condit. accepted by Am. J. Phys.; publ. variant may differ in minor corrections. Material in publ. vers. essentially the same, so readers without subscript. to AJP may consult archive vers. to know what content referring to in citing AJP paper. V3 consid. shortened comp. to V2. Better focus, but V2 has appendix explaining how eqns prepared for numerics in order to avoid divergences
The ingoing Vaidya metric is introduced as a model for a non-rotating uncharged black hole emitting Hawking radiation. This metric is expected to capture the physics of the spacetime for radial coordinates up to a small multiple $(>1)$ of the Schwarzschild radius. For larger radii, it will give an excellent approximation to the spacetime geometry in the case of astrophysical black holes $(M\ge M_{\astrosun})$, except at extremely large distances from the horizon (exceeding the cosmic particle horizon). In the classroom, the model may serve as a first exploration of non-stationary gravitational fields. Several interesting predictions are developed. First, particles dropped early enough before complete evaporation of the black hole cross its horizon as easily as with an eternal black hole. Second, the Schwarzschild radius takes on the properties of an apparent horizon, and the true event horizon of the black hole is \emph{inside} of it, because light can escape from the shrinking apparent horizon. Third, a particle released from rest close enough to the apparent horizon is strongly repelled and may escape to infinity. An interpretation is given, demonstrating that such a particle would be able to compete, for a short time, in a race with a photon.
