Infinite slabs and other weird plane symmetric space-times with constant positive density

TL;DR

This paper derives exact solutions for static, plane symmetric spacetimes with constant positive density, exploring their properties, singularities, and potential configurations like gravitational capacitors.

Contribution

It provides new exact solutions for plane symmetric matter distributions with positive density, analyzing their global structure and possible physical configurations.

Findings

Solutions depend on density and a parameter, showing different gravitational behaviors.

Space-times can have singularities or be asymptotically flat depending on parameters.

Configurations include slabs, joined slabs, and vacuum slices forming a gravitational capacitor.

Abstract

We present the exact solution of Einstein's equation corresponding to a static and plane symmetric distribution of matter with constant positive density located below $z=0$. This solution depends essentially on two constants: the density $\rho$ and a parameter $\kappa$. We show that this space-time finishes down below at an inner singularity at finite depth. We match this solution to the vacuum one and compute the external gravitational field in terms of slab's parameters. Depending on the value of $\kappa$, these slabs can be attractive, repulsive or neutral. In the first case, the space-time also finishes up above at another singularity. In the other cases, they turn out to be semi-infinite and asymptotically flat when $z\to\infty$. We also find solutions consisting of joining an attractive slab and a repulsive one, and two neutral ones. We also discuss how to assemble a "gravitational capacitor" by inserting a slice of vacuum between two such slabs.