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. 
E. Babichev, V. Dokuchaev, Yu. Eroshenko
gr-qc