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So-called "dirty" black holes are those surrounded by non-zero stress-energy,rather than vacuum. The presence of the non-zero stress-energy modifies keyfeatures of the black hole, such as the surface gravity, Regge-Wheelerequation, linear stability, and greybody factors in a rather nontrivial way.Working within the inverse-Cowling approximation, (effectively the test-fieldlimit), we shall present general forms for the Regge-Wheeler equation forlinearized spin 0, spin 1, and axial spin 2 perturbations on an arbitrarystatic spherically symmetric background spacetime. Using very general featuresof the background spacetime, (in particular the classical energy conditions forthe stress-energy surrounding the black hole), we extract several interestingand robust bounds on the behaviour of such systems, including rigorous boundson the greybody factors for dirty black holes.

Regge-Wheeler equation, linear stability, and greybody factors for dirty black holes