CONICET | Buscador de Institutos y Recursos Humanos

We present the exact solution of Einstein´s equationcorresponding to a static and plane symmetric distribution ofmatter with constant positive density located below $z=0$ matchedto vacuum solutions. The internal solution depends essentially ontwo constants: the density $ ho$ and a parameter $kappa$. Weshow that these space-times finish down below at an innersingularity at finite depth $dleqsqrt{ rac{pi}{24 ho}}$. Weshow that for $kappageq0.3513dots$, the dominant energycondition is satisfied all over the space-time. We match these singular solutions to the vacuum one and computethe external gravitational field in terms of slab´s parameters.Depending on the value of $kappa$, these slabs are eitherattractive, repulsive or neutral. The external solution turns outto be a Rindler´s st. Repulsive slabs explicitly show hownegative, but finite pressure can dominate the attraction of thematter. In this case, the presence of horizons in the vacuum showsthat there are null geodesics which never reach the surface of theslab. We also consider a static and plane symmetric non-singulardistribution of matter with constant positive density $ ho$ andthickness $d$ ($0<d<sqrt{ rac{pi}{24 ho}}$) surrounded by twoexternal vacuums. We explicitly write down the pressure and theexternal gravitational fields in terms of $ ho$ and $d$. Thesolution turns out to be attractive and remarkably asymmetric:the ``upper" solution is Rindler´s vacuum, whereas the ``lower"one is the singular part of Taub´s plane symmetric solution.Inside the slab, the pressure is positive and bounded, presentinga maximum at an asymmetrical position between the boundaries. Weshow that if $0<sqrt{6pi ho},d<1.527dots$, the dominantenergy condition is satisfied all over the space-time. We alsoshow how the mirror symmetry is restored at the Newtonian limit. We also find thinner repulsive slabs by matching a singular sliceof the inner solution to the vacuum. We also discuss solutions in which an attractive slab and arepulsive one, and two neutral ones are joined. We also discusshow to assemble a ``gravitational capacitor" by inserting a sliceof vacuum between two such slabs.