Reviving the shear-free perfect fluid conjecture in general relativity

Abstract

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 basic vector field along the direction of the vorticity for the conjecture to hold. Finally, we show that in general, there exist a basic vector field parallel to the acceleration for which the conjecture is true. IntroductionThis paper presents a method of investigating shear-free perfect fluid solutions in General Relativity. This method is used to examine a number of special cases. The shear-free perfect fluid conjecture in General Relativity states that, a shear-free velocity vector field of a barotropic perfect fluid with µ + p = 0 and p = p(µ), is either expansion or rotation free [1], where µ and p are the energy density and pressure of the fluid respectively. The first indication that the vanishing of the shear can have restrictive properties between the rotation and the expansion of a cosmological model was given by Gödel [2,3]. Some of the well known examples of these cosmological models include: the Einstein (static)Reviving The Shear-Free Perfect Fluid Conjecture In General Relativity 2 universe, FLRW universes (expanding models) and the Gödel (purely rotating) universe.The requirement that the matter equation of state be of a barotropic perfect fluid produces new additional constraints on the full Einstein field equations, for instance all expanding and non-rotating shear-free perfect fluids with a barotropic equation of state are known to exist [4]. In contrast, not all non-expanding and rotating shear-free perfect fluids with a barotropic equation of state are known [5], but a fair amount of these models are known to exist especially all stiff rotating axisymmetric and stationary perfect fluids belong in this class. A simultaneously expanding and rotating model (Bianchi IX) was found to have an equation of state of the form p = −µ = constant, which is of the cosmological constant type and is excluded in this discussion [6].The validity of the above mentioned conjecture would be a striking feature of the full Einstein field equations, while in Newtonian cosmology there exist shear-free perfect fluids with a barotropic equation of state which are both expanding and rotating simultaneously [1,[7...

Shear-free perfect fluids with a barotropic equation of state in general relativity: the present status

“…Schücking's result was generalized in 1967 by Ellis [12], who used the orthonormal tetrad formalism to show that the restriction of spatial homogeneity was redundant for dust spacetimes, while in [52] it was observed that Ellis' result remained valid in the presence of a cosmological constant. A fully covariant proof of both results was presented more recently in [33]. In [40] Treciokas and Ellis proved, again using a combination of an orthonormal tetrad formalism and an adapted choice of coordinates, that the conjecture held true also for the EOS p = 1 3 µ, a result which was generalised by Coley [10] to allow for a possible non-zero cosmological constant.…”

“…A 'covariant' proof of this same result was given by Sopuerta in [38]. While Treciokas and Ellis already questioned the possible existence of rotating and expanding perfect fluids with p = p(µ), their nonexistence was explicitly conjectured by Collins [9], following a series of papers in which the conjecture was proved successively for the cases where the vorticity vector is parallel to the acceleration (see [52], or [32,33,34] for a fully covariant proof), or in which the Weyl tensor is purely magnetic [8] or purely electric [11,24].…”

“…with the vorticity. A fully covariant proof of this result was presented recently in [34], generalizing it furthermore to the property that an expanding and rotating barotropic shear-free perfect fluid admits a Killing vector along the vorticity, if the projection of the acceleration in the plane orthogonal to the vorticity has components u1 = F B 1 and u2 = F B 2 , with F a function of the matter density µ and B 1 , B 2 basic functions.…”

“…A special case is of course the situation where U1 = U2 = 0, i.e. the well known case in which the acceleration and vorticity vectors are parallel, see [52], or [32,33,34] for a covariant proof.…”

Shear free barotropic perfect fluids cannot rotate and expand simultaneously

“…As both Ω and ξ are non-zero in this case, we can immediately see that A = 0 and Θ = 0. Therefore all the shear free spacetimes where A = 0 (for example dust [6], or when the acceleration vector is perpendicular to vorticity [11]) fall under Case 1 and the theorem is proven for these. Now, taking the hat derivative of (101), and using equations (90), ( 81) and (83) we can solve for  to get…”