  In this paper we present a method to study the frequency shift of signals
sent from near a Schwarzschild black hole that grows or shrinks through
accretion. We construct the numerical solution of Einstein's equations sourced
by a spherical shell of scalar field, with positive energy density to simulate
the growth and with negative energy density to simulate the shrink of the black
hole horizon. We launch a distribution of null rays at various time slices
during the accretion and estimate their energy along their own trajectories.
Spatially the bundles of photons are distributed according to the distribution
of dust, whose dynamics obeys Euler equations in the test field limit during
the evolution of the black hole. With these elements, we construct the
frequency shift of photons during the accretion process of growth or
contraction of the hole, which shows a variability that depends on the
thickness of the scalar field shell or equivalently the time scale of the
accretion.
