  The construction of Dirac observables, that is gauge invariant objects, in
General Relativity is technically more complicated than in other gauge theories
such as the standard model due to its more complicated gauge group which is
closely related to the group of spacetime diffeomorphisms. However, the
explicit and usually cumbersome expression of Dirac observables in terms of
gauge non invariant quantities is irrelevant if their Poisson algebra is
sufficiently simple. Precisely that can be achieved by employing the relational
formalism and a specific type of matter proposed originally by Brown and
Kucha{\v r}, namely pressureless dust fields. Moreover one is able to derive a
compact expression for a physical Hamiltonian that drives their physical time
evolution. The resulting gauge invariant Hamiltonian system is obtained by
Higgs -- ing the dust scalar fields and has an infinite number of conserved
charges which force the Goldstone bosons to decouple from the evolution. In
previous publications we have shown that explicitly for cosmological
perturbations. In this article we analyse the spherically symmetric sector of
the theory and it turns out that the solutions are in one--to--one
correspondence with the class of Lemaitre--Tolman--Bondi metrics. Therefore the
theory is capable of properly describing the whole class of gravitational
experiments that rely on the assumption of spherical symmetry.
