Exact solutions of a Chaplygin gas in an anisotropic space-time of Petrov D

Abstract

This document obtains two exact solutions to the anisotropic spacetime of Petrov D by using the model of a perfect fluid. These solutions represent a scenario of a universe in which the pressure P and the energetic density µ of the fluid are inversely proportional (Chaplygin's type P = −Q 2 /µ), where Q is a constant of proportionality. It is established that the symmetry of those models, in the proximities when t → 0, is equivalent to the analogues for the dust model, and might tend to behave as the solutions of the flat or vacuum LRS Kasner solution (Local Rotational Symmetry). Although the solutions are not flat or vacuum in any of the cases, in those proximities, the density tends to infinite and with no pressure. When t → ∞, the models tend to behave as the isotropic flat model of the type FRWL. In the analysis of the Hubble and the deceleration parameters obtained that in those solutions, the Hubble's constant and the deceleration parameter, depend on time and manner in which their values, or tendency, significantly evolve. The deceleration parameter q changes its sign as times passes, so that it represents an initial deceleration process that, in continuity, constantly changes to a process of acceleration.

Interactive spinorial and scalar fields congruent to the Chaplygin's gas. Exact cosmologic solutions in a space-time of Petrov D

“…The non-lineal self-consistent interaction of the scalar and spinorial fields of the kind (6), with a non-lineal element of the spinorial field of the type (17) in an anisotropic symmetry of Petrov D, results in a state of the fields that is equivalent to the fluid of Chaplygin, which is studied in [7], and with similar cosmologic consequences. For said state of the fields, it was obtained that the function of the scalar field (ϕ) increases its value with time.…”

“…which defines (22), (20) and (14). The metric function K in (22), has been already determined and studied in [7], and also defined in the appendix. The no-null components of the energy-stress tensor (21), which represent the volumetric density of the energy and the pressure, take the pattern…”

“…and α = 0, 1, the function F (a, c) is the incomplete elliptic integral of the first kind and Π(a, b, c) is the incomplete elliptic integral of the third kind. The functions and constants L, E, A i , a α c, b, e i (the same cited in [7]), σ and ξ, and others are defined in the appendix at the end.…”

“…In the case of the Chaplygin fluid, under an homogeneous anisotropic symmetry of Petrov D, there were obtained two solutions, and their respective analysis, in [7] determines that in a Universe where the pressure P and the energetic density µ of the fluid are inversely proportional (P = −Q 2 /µ), where Q is a constant of proportionality, the symmetry of said models, in the proximities when t → 0 is equivalent to the analogous of the dust model, and they can tend to behave as the solutions of the flat or the vacuum of Kasner LRS(Local Rotational Symmetry), although said solutions are not flat or vacuum in any of the cases, in that proximity, the density tends infinite and the pressure to zero. Moreover, when t → ∞, the models tend to behave as the isotropic flat model of dark energy of FRWL.…”

“…Moreover, when t → ∞, the models tend to behave as the isotropic flat model of dark energy of FRWL. In [7], there were obtained and analyzed the parameter of Hubble and the deceleration model, and the results were, for example, that the parameter of deceleration q changes it sign when time increases, so that it represents a process of initial deceleration through the increase of time; in a continuous manner, its acceleration changes. This study analyzes the interaction of the scalar and the spinorial fields, which results in two solutions of the space-time, equivalent to the ones obtained for the model of the fluid of Chaplygin in [7]; these conclusions are valid for this work, therefore, the emphasis will be about the new aspects related to the interaction among the scalar and spinorial fields.…”