Shear and vorticity of perfect-fluid spacetimes and the shear-free conjecture

TL;DR

This paper derives expressions for shear and vorticity in perfect-fluid spacetimes, proving a key condition linking shear, expansion, and vorticity, and introduces criteria for identifying generalized Robinson-Walker spacetimes.

Contribution

It provides new formulas for shear and vorticity tensors and proves a significant condition related to the shear-free conjecture in perfect-fluid spacetimes.

Findings

Derived expressions for shear and vorticity tensors in terms of Weyl tensor divergence.

Proved that parallel energy density gradient implies zero expansion or vorticity.

Established a new condition characterizing Generalized Robinson-Walker spacetimes.

Abstract

We obtain expressions for the shear and the vorticity tensors of perfect-fluid spacetimes, in terms of the divergence of the Weyl tensor. For such spacetimes, we prove that if the gradient of the energy density is parallel to the velocity, then either the expansion rate is zero, or the vorticity vanishes. This statement recalls the "shear-free conjecture" for a perfect barotropic fluid: vanishing shear implies either vanishing expansion rate or vanishing vorticity. Finally, we give a new condition for a perfect fluid to be a Generalized Robinson-Walker spacetime.