Plane symmetric relativistic fluids with Taub geometry

Abstract

We investigate the problem of generating perfect fluid models by performing a conformal transformation on a non-conformally flat but conformally Ricci-flat (vacuum) seed metric such as the Taub (Ann. Math. 53, 472 (1951)) spacetime. The Taub metric is a static plane symmetric matter-free solution of the Einstein field equations possessing three Killing vectors. It turns out that, assuming a conformal factor depending on the temporal coordinate and one space variable, the resultant metrics are necessarily static. We are able to solve completely the field equations and obtain the geometric and dynamical variables explicitly. A study of the elementary properties required of realistic fluids is made and it is found that the fluid constructed displays necessary qualitative features desirable in realistic cosmological distributions. In particular the energy density and pressure profiles are positive definite and the adiabatic sound-speed index is found to be causal (subluminal) in a region excluding the central axis. Importantly, the weak, strong and dominant energy conditions are all satisfied. It is not possible to obtain a barotropic equation of state in this model.