Uniqueness properties of purely magnetic LRS perfect fluid spacetimes

We show that all purely magnetic Petrov type D, irrotationall aligned perfect fluid spacetimes are locally rotationally symmetric class III by the Ellis classification. This result is combined with a similar theorem for purely magnetic Petrov type D, shear-free, aligned perfect fluid spacetimes. For all purely magnetic locally rotationally symmetric class III perfect fluid spacetimes the pressure and energy density are both expressed in terms of a single function (of the timelike coordinate only). Therefore, these expressions define, implicitly, a nonlinear barotropic equation of state on intervals of this single function where the energy conditions are satisfied. However, a general expression is obtained for which the pressure is explicitly written as a function of the energy density. It is Shown that in the asymptotic limit of large pressure and energy density the ratio between the pressure and energy density approaches the value 1/5. The only member of these spacetimes with a linear barotropic equation of state and a homothetic vector field is the purely magnetic Collins-Stewart spacetime.