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

TL;DR

This paper develops an inverse method to determine fluid flows that generate specific spacetimes via Einstein's equations, focusing on fluids with zero anisotropic stress, applicable to compact objects and cosmology.

Contribution

It introduces an algorithmic framework for deriving fluid flows from spacetime metrics with vanishing anisotropic stress in warped product spacetimes of class B1.

Findings

Derived conditions for fluid flows in various coordinate systems.

Provided criteria for perfect fluid sources.

Developed a computationally implementable approach.

Abstract

We expand previous work on an inverse approach to Einstein Field Equations where we include fluids with energy flux and consider the vanishing of the anisotropic stress tensor. We consider the approach using warped product spacetimes of class $B_1$. Although restricted, these spacetimes include many exact solutions of interest to compact object studies and to cosmological models studies. The question explored here is as follows: given a spacetime metric, what fluid flow (timelike congruence), if any, could generate the spacetime via Einstein's equations. We calculate the flow from the condition of a vanishing anisotropic stress tensor and give results in terms of the metric functions in the three canonical types of coordinates. A condition for perfect fluid sources is also provided. The framework developed is algorithmic and suited for the study and validation of exact solutions using computer algebra systems. The framework can be applied to solutions in comoving and non-comoving frames of reference, and examples in different types of coordinates are worked out.