  We examine the real-time dynamics of a system of one or more black holes
interacting with long wavelength gravitational fields. We find that the
(classical) renormalizability of the effective field theory that describes this
system necessitates the introduction of a time dependent mass counterterm, and
consequently the mass parameter must be promoted to a dynamical degree of
freedom. To track the time evolution of this dynamical mass, we compute the
expectation value of the energy-momentum tensor within the in-in formalism, and
fix the time dependence by imposing energy-momentum conservation. Mass
renormalization induces logarithmic ultraviolet divergences at quadratic order
in the gravitational coupling, leading to a new time-dependent renormalization
group (RG) equation for the mass parameter. We solve this RG equation and use
the result to predict heretofore unknown high order logarithms in the energy
distribution of gravitational radiation emitted from the system.
