  Shape Dynamics is a 3D conformally invariant theory of gravity which
possesses a large set of solutions in common with General Relativity. When
looked closely, these solutions are found to behave in surprising ways, so in
order to probe the fitness of Shape Dynamics as a viable alternative to General
Relativity one must find and understand increasingly more complex, less
symmetrical exact solutions, on which to base perturbative studies and
numerical analyses in order to compare them with data. Spherically symmetric
exact solutions have been studied, but only in a static vacuum setup. In this
work we construct a class of time-dependent exact solutions of Shape Dynamics
from first principles, representing a central inhomogeneity in an evolving
cosmological environment. By assuming only a perfect fluid source in a
spherically symmetric geometry we show that this fully dynamic non-vacuum
solution satisfies in all generality the Hamiltonian structure of Shape
Dynamics. The simplest choice of solutions is shown to be a member of the
McVittie family.
