Backreaction of accreting matter onto a black hole in the Eddington-Finkelstein coordinates

We study backreaction of accreting matter onto a spherically symmetric black hole in a perturbative way, when accretion is in a quasi-steady state. General expressions for corrections to the metric coefficients are found in the Eddington-Finkelstein coordinates. It is shown that near the horizon of a black hole, independently of the form of the energy-momentum tensor, the leading corrections to the metric are of the Vaidya form. The relation to other solutions is discussed and particular examples are presented.