Theorems on shear-free perfect fluids with their Newtonian analogues

TL;DR

This paper provides covariant proofs of theorems on shear-free perfect fluids in relativity, compares them with Newtonian analogues, and highlights key differences in their behavior regarding expansion and rotation.

Contribution

It offers new covariant proofs of shear-free perfect fluid theorems and compares relativistic and Newtonian cases to clarify their differences.

Findings

Shear-free perfect fluids with acceleration proportional to vorticity are either non-expanding or non-rotating.

Relativistic and Newtonian shear-free dust behave differently, especially regarding expansion and rotation.

Explicit comparison reveals where and why the Newtonian and relativistic results diverge.

Abstract

In this paper we provide fully covariant proofs of some theorems on shear-free perfect fluids. In particular, we explicitly show that any shear-free perfect fluid with the acceleration proportional to the vorticity vector (including the simpler case of vanishing acceleration) must be either non-expanding or non-rotating. We also show that these results are not necessarily true in the Newtonian case, and present an explicit comparison of shear-free dust in Newtonian and relativistic theories in order to see where and why the differences appear.