  Spherically symmetric spacetimes are ambient spaces for models of stellar
collapse and inhomogeneous cosmology. We obtain results for the Weyl tensor and
the covariant form of the Ricci tensor on general doubly warped (DW)
spacetimes. In a spherically symmetric metric, the Ricci and electric tensors
become rank-2, built with a velocity vector field and its acceleration. Their
structure dictates the general form of the energy-momentum tensor in the
Einstein equations in DW spherical metrics. The anisotropic pressure and the
heat current of an imperfect fluid descend from the gradient of the
acceleration and the electric part of the Weyl tensor. For radiating stellar
collapse with heat flow, the junction conditions of the doubly warped metric
with the Vaidya metric are reviewed, with the boundary condition for the radial
pressure. The conditions for isotropy simply accomodate various models in the
literature. The anisotropy of the Ricci tensor in the special case of spherical
GRW space-times (geodesic velocity), gives Friedmann equations deviating from
standard FRW cosmology by terms due to the electric tensor. We introduce
"perfect 2-scalars" to discuss f(R) gravity with anisotropic fluid source in a
doubly warped spacetime, and show that the new geometric terms in the field
equations do not change the tensor structure of the fluid energy-momentum
tensor.
