We show rigorously that the coefficient for spontaneous emission of an atom placed in a dielectric is proportional to the local radiative density of states-that is only a part of the local density of the eigenmodes of the Maxwell equations. Spontaneous emission is inhibited if the atom is located at a position where this local radiative density is small, even if the total density of states is not vanishing. This radiative density of states can be obtained without having to perform a full quantum calculation of the radiation-matter system. We demonstrate this principle by solving numerically a scalar model for a dielectric that consists of a lattice of resonating dipoles.