When we solve the Einstein Field Equations

$$G_{\mu\nu} = 8\pi T_{\mu\nu}$$

one way of doing it is by specifying a symmetry (and thus a general form of the metric) and then specifying the stress-energy-tensor $$T_{\mu\nu}$$. Most examples found in text are for vacuum solutions (vanishing $$T_{\mu\nu}$$) or homogeneous isotropic $$T_{\mu\nu}=\textrm{diag}(\rho,p,p,p)$$.

Is it physically meaningful to solve the field equations for a radially-dependent stress-energy tensor? For example

$$T_{\mu\nu}=\textrm{diag}(\rho(r),p(r),p(r),p(r))$$

where there is a radially-dependent mass-energy density $$\rho(r)$$. For example, a mass-energy density that resembles a Gaussian or Poisson distribution, where the highest concentration is at the center. And if the mass-energy density is radially-dependent, is the above expression for $$T_{\mu\nu}$$ correct? Or will the momentum flux and shear now be non-zero so that $$T_{\mu\nu}$$ is no longer a diagonal matrix?