Ingoing Eddington-Finkelstein Metric of an Evaporating Black Hole
  We present an approximate time-dependent metric in ingoing Eddington-Finkelstein coordinates for an evaporating nonrotating black hole as a first-order perturbation of the Schwarzschild metric, using the linearized back reaction from a realistic approximation to the stress-energy tensor for the Hawking radiation in the Unruh quantum state. 
Shohreh Abdolrahimi, Don N. Page and Christos Tzounis
hep-th gr-qc