Metric Reconstruction for Generic Black-Hole Perturbations

Standard (radiation-gauge) metric reconstruction excludes generic sources because it requires a tracefree metric perturbation. We remove this obstruction for perturbations of Petrov type D spacetimes by introducing a traceful radiation gauge. Two first-order transport equations determine the metric trace from the stress-energy tensor, and the remaining metric components follow hierarchically from the Newman-Penrose equations. We illustrate the method for a Schwarzschild black hole with a thin static shell, including a source-supported static completion sector.