A duality relation for fluid spacetime

We consider the electromagnetic resolution of the gravitational field. We show that under the duality transformation, in which active and passive electric parts of the Riemann curvature are interchanged, a fluid spacetime in comoving coordinates remains invariant in character with density and pressure transforming, while energy flux and anisotropic pressure remain unaltered. Furthermore, if the fluid admits a barotropic equation of state, p = (gamma - 1)rho where 1 less than or equal to gamma less than or equal to 2, which will transform to p = (2 gamma/(3 gamma - 2)-1)rho. Clearly the stiff fluid and dust are dual to each other, while rho + 3p = 0 will go to flat spacetime. However, the radiation (rho - 3p = 0) and the de Sitter (rho + p = 0) universes are self-dual.