  We have analyzed the transformation from initial coordinates $(v,r)$ of the
Vaidya metric with light coordinate $v$ to the most physical diagonal
coordinates $(t,r)$. An exact solution has been obtained for the corresponding
metric tensor in the case of a linear dependence of the mass function of the
Vaidya metric on light coordinate $v$. In the diagonal coordinates, a narrow
region (with a width proportional to the mass growth rate of a black hole) has
been detected near the visibility horizon of the Vaidya accreting black hole,
in which the metric differs qualitatively from the Schwarzschild metric and
cannot be represented as a small perturbation. It has been shown that, in this
case, a single set of diagonal coordinates $(t,r)$ is insufficient to cover the
entire range of initial coordinates $(v,r)$ outside the visibility horizon; at
least three sets of diagonal coordinates are required, the domains of which are
separated by singular surfaces on which the metric components have
singularities (either $g_{00}=0$ or $g_{00}=\infty$.). The energy-momentum
tensor diverges on these surfaces; however, the tidal forces turn out to be
finite, which follows from an analysis of the deviation equations for
geodesics. Therefore, these singular surfaces are exclusively coordinate
singularities that can be referred to as false firewalls because there are no
physical singularities on them. We have also considered the transformation from
the initial coordinates to other diagonal coordinates $(\eta,y)$, in which the
solution is obtained in explicit form, and there is no energy-momentum tensor
divergence.
