The ringdown of a perturbed black hole contains fundamental information about space-time in the form of Quasi Normal Modes (QNM). Modifications to general relativity, or extended profiles of other fields surrounding the black hole, so called black hole hair, can perturb the QNM frequencies. Previous works have examined the QNM frequencies of spherically symmetric dirty black holes - that is black holes surrounded by arbitrary matter fields. Such analyses were restricted to static systems, making the assumption that the metric perturbation was independent of time. However, in most physical cases such black holes will actually be growing dynamically due to accretion of the surrounding matter. Here we develop a perturbative analytic method that allows us to compute for the first time the time dependent QNM deviations of such growing dirty black holes. Whilst both are small, we show that the change in QNM frequency due to the accretion can be of the same order or larger than the change due to the static matter distribution itself, and therefore should not be neglected in such calculations. We present the case of spherically symmetric accretion of a complex scalar field as an illustrative example, but the method has the potential to be extended to more complicated cases.
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