Reviving The Shear-Free Perfect Fluid Conjecture In General Relativity

TL;DR

This paper uses computer algebra to provide covariant proofs of the shear-free perfect fluid conjecture in general relativity, identifying conditions involving Killing vectors and basic vector fields that support the conjecture.

Contribution

It offers new covariant proofs for specific cases of the shear-free perfect fluid conjecture, employing symbolic computation and revealing geometric conditions necessary for the conjecture's validity.

Findings

Existence of a Killing vector along vorticity under certain conditions

Conditions involving basic vector fields are necessary for the conjecture

Covariant proofs for cases with constant pressure and specific acceleration-vorticity relations

Abstract

Employing a Mathematica symbolic computer algebra package called xTensor, we present $(1+3)$-covariant special case proofs of the shear-free perfect fluid conjecture in General Relativity. We first present the case where the pressure is constant, and where the acceleration is parallel to the vorticity vector. These cases were first presented in their covariant form by Senovilla et. al. We then provide a covariant proof for the case where the acceleration and vorticity vectors are orthogonal, which leads to the existence of a Killing vector along the vorticity. This Killing vector satisfies the new constraint equations resulting from the vanishing of the shear. Furthermore, it is shown that in order for the conjecture to be true, this Killing vector must have a vanishing spatially projected directional covariant derivative along the velocity vector field. This in turn implies the existence of another \textit{basic} vector field along the direction of the vorticity for the conjecture to hold. Finally, we show that in general, there exist a \textit{basic} vector field parallel to the acceleration for which the conjecture is true.