http://arxiv.org/abs/1708.02462
Employing a Mathematica symbolic computer algebra package called xTensor, we present $(1+3)$-covariant special case proofs of the shear-free conjecture for perfect fluids in General Relativity. We first present the case where the pressure is constant and then where the acceleration is parallel to the vorticity vector, which 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, and it is shown that for the conjecture to be true, this Killing vector must have a vanishing spatially projected directional covariant derivative along the velocity vector field, which in turn implies the existence of another \textit{basic} vector field along the direction of the vorticity for the theorem to hold. Finally, we show that in general if the acceleration is non-zero, there exist a \textit{basic} vector field parallel to the acceleration for the conjecture to be true.
M. Sikhonde and P. Dunsby
Wed, 9 Aug 17
18/32
Comments: 38 pages, Submitted to CQG