The main properties of the Levi-Civita solutions with the cosmological constant are studied. In particular, it is found that some of the solutions need to be extended beyond certain hypersurfaces in order to have geodesically complete spacetimes. Some extensions are considered and found to give rise to black hole structure but with plane symmetry. All the spacetimes that are not geodesically complete are Petrov type D, while in general the spacetimes are Petrov type I.
A class of solutions to Einstein field equations is studied, which represents gravitational collapse of thick spherical shells made of self-similar and shear-free fluid with heat flow. It is shown that such shells satisfy all the energy conditions, and the corresponding collapse always forms naked singularities.
We have constructed star models consisting of four parts: (i) a homogeneous inner core with anisotropic pressure (ii) an infinitesimal thin shell separating the core and the envelope; (iii) an envelope of inhomogeneous density and isotropic pressure; (iv) an infinitesimal thin shell matching the envelope boundary and the exterior Schwarzschild spacetime. We have analyzed all the energy conditions for the core, envelope and the two thin shells. We have found that, in order to have static solutions, at least one of the regions must be constituted by dark energy. The results show that there is no physical reason to have a superior limit for the mass of these objects but for the ratio of mass and radius.
The static vacuum plane spacetimes are considered, which have two non-trivial
solutions: The Taub solution and the Rindler solution. Imposed reflection
symmetry, we find that the source for the Taub solution does not satisfy any
energy conditions, which is consistent with previous studies, while the source
for the Rindler solution satisfies the weak and strong energy conditions (but
not the dominant one). It is argued that the counterpart of the Einstein theory
to the gravitational field of a massive Newtonian plane should be described by
the Rindler solution, which represents also a uniform gravitational field
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