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

摘要

We present the exact solution of Einstein's equation corresponding to astatic and plane symmetric distribution of matter with constant positivedensity located below $z=0$. This solution depends essentially on twoconstants: the density $\rho$ and a parameter $\kappa$. We show that thisspace-time finishes down below at an inner singularity at finite depth. Wematch this solution to the vacuum one and compute the external gravitationalfield in terms of slab's parameters. Depending on the value of $\kappa$, theseslabs can be attractive, repulsive or neutral. In the first case, thespace-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 arepulsive one, and two neutral ones. We also discuss how to assemble a"gravitational capacitor" by inserting a slice of vacuum between two suchslabs.