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The shear-free perfect fluid conjecture

There is mounting evidence that general relativistic shear-free perfect fluids, obeying a barotropic equation of state with , are necessarily irrotational or non-expanding. This conjecture has been demonstrated in a number of particular cases, but a general proof is still lacking. In the tetrad-based approach two particular cases require a special treatment, namely = constant and = constant. A proof is given that the conjecture holds for each of these. In addition, a formalism is presented enabling one to deal with the more general case of a -law equation of state + constant.