Vaidya Spacetime in the Diagonal Coordinates

摘要

We have analyzed the transformation from initial coordinates $(v,r)$ of theVaidya metric with light coordinate $v$ to the most physical diagonalcoordinates $(t,r)$. An exact solution has been obtained for the correspondingmetric tensor in the case of a linear dependence of the mass function of theVaidya metric on light coordinate $v$. In the diagonal coordinates, a narrowregion (with a width proportional to the mass growth rate of a black hole) hasbeen detected near the visibility horizon of the Vaidya accreting black hole,in which the metric differs qualitatively from the Schwarzschild metric andcannot be represented as a small perturbation. It has been shown that, in thiscase, a single set of diagonal coordinates $(t,r)$ is insufficient to cover theentire range of initial coordinates $(v,r)$ outside the visibility horizon; atleast three sets of diagonal coordinates are required, the domains of which areseparated by singular surfaces on which the metric components havesingularities (either $g_{00}=0$ or $g_{00}=\infty$.). The energy-momentumtensor diverges on these surfaces; however, the tidal forces turn out to befinite, which follows from an analysis of the deviation equations forgeodesics. Therefore, these singular surfaces are exclusively coordinatesingularities that can be referred to as false firewalls because there are nophysical singularities on them. We have also considered the transformation fromthe initial coordinates to other diagonal coordinates $(\eta,y)$, in which thesolution is obtained in explicit form, and there is no energy-momentum tensordivergence.