Canonical formalism for compact sources

This thesis aims to describe the ADM formalism of General Relativity and to use the latter to describe a spherical compact source consisting of a perfect fluid. With two different choices for three-dimensional metric on hypersurfaces, we analyze the constraints of the system in the non-static case and the resulting equations of motion, both for canonical gravitational variables and those of matter. After examining some special cases, we also show that it is possible, in the case of static nature, to obtain the value of the Misner-Sharp mass from the Hamiltonian constraint, while near the trapping surfaces we obtain a relationship between the density of matter and the dynamic variables of the metric. Finally we propose a possible method for quantizing the constraints using the procedure that in the vacuum leads to the Wheeler-DeWitt equations.