Levi-Cività solutions with a cosmological constant

Abstract

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.

Cylindrical systems in general relativity

“…Since in the latter case the curvatures of the two spacelike surfaces t, r = Constant are identically zero, it is difficult to consider this spacetime as having cylindrical symmetry. Instead, one may extend the ϕ coordinate from the range [0, 2π] to the range (−∞, +∞), so the resulted spacetime has a plane symmetry, as in the vacuum case [124,332].…”

Cylindrically symmetric static solutions of the Einstein field equations for elastic matter

“…The solution has in general five free parameters, namely the cosmological constant Λ and the real constant parameters A, B, C = 0 and σ ≥ 0. The parameter A can be interpreted as a time rescaling, B and C are related to the conicity and σ is interpreted as the mass per unit length (see [27,26]). The case Λ > 0 is obtained by replacing the hyperbolic functions by trigonometric ones (see [23], [24]) and, for that case, the calculations in this section remain also valid.…”

Editorial note to: T. Levi-Civita, The physical reality of some normal spaces of Bianchi and to: Einsteinian ds 2 in Newtonian fields. IX: The analog of the logarithmic potential

“…The analogous transformations for the cosmological Kasner case were considered by McIntosh [37]. The generalization of (3) to add a cosmological constant has been discussed by several authors [38][39][40], and electromagnetic generalizations have also been considered [41,42].…”