Cosmological exact solutions of Petrov type D of a non-lineal fluid asymptotic to a fluid of dark energy in real and complex geometries with double singularity. First case

Abstract

In this paper, exact solutions to the Einstein’s equations are obtained for an anisotropic and homogeneous symmetry of Petrov Type D, of a non-lineal fluid that responds to the equation of state R+R− = 0, where R± = P + Λ − Λ4 ( Λ μ−Λ ± √ 4 + Λ 2 (μ−Λ) ) where μ, P , and Λ are the volumetric energy density, the pressure, and a constant linked to the concept of dark energy. Two general solutions that are different between them because of the initial expansion degree that a coordinate can have in relation to a plane perpendicular to it are obtained. For each one of the solutions, two cases are presented: one represents a space-time with real geometry (R) for all the values of t, and asymptotically in time this case becomes a isotropic space-time of FLRW, of a fluid of dark energy and the other case presents a double singularity, so that since the second singularity the space-time becomes a complex one (C), with the increase of time the complex part of the interval tends to be very small in relation to the real part, and the real part tends to an isotropic space-time of FLRW, of a fluid of dark energy. The implications of the complex solution (of geometry of the complex space-time) are discussed in quantum processes, such as the quantum entanglement, and the behavior of the temperature in relation to the time is obtained.