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We expand previous work on an inverse approach to Einstein Field Equationswhere we include fluids with energy flux and consider the vanishing of theanisotropic stress tensor. We consider the approach using warped productspacetimes of class $B_1$. Although restricted, these spacetimes include manyexact solutions of interest to compact object studies and to cosmologicalmodels studies. The question explored here is as follows: given a spacetimemetric, what fluid flow (timelike congruence), if any, could generate thespacetime via Einstein's equations. We calculate the flow from the condition ofa vanishing anisotropic stress tensor and give results in terms of the metricfunctions in the three canonical types of coordinates. A condition for perfectfluid sources is also provided. The framework developed is algorithmic andsuited for the study and validation of exact solutions using computer algebrasystems. The framework can be applied to solutions in comoving and non-comovingframes of reference, and examples in different types of coordinates are workedout.

Inverse approach to Einstein's equations for fluids with vanishing anisotropic stress tensor