On the local form of static plane symmetric spacetimes in the presence of matter
For any configuration of a static plane symmetric distribution of matter along a spacetime, there are coordinates where the metric can be set explicitly as a function of the energy density and pressures. It satisfies Einstein equations as long as we require conservation of the energy-momentum tensor, which is a single ordinary differential equation for self-gravitating hydrostatic equilibrium. As a direct application, a general solution is given when the pressures are linearly related to the energy density, recovering, in special cases, most of the known solutions of static plane symmetric Einstein equations.