Métricas tipo-Gödel na gravitação modificada f(R, Q, P)

Abstract

In this dissertation, we review the most relevant properties of Einstein’s theory of gravity and modified theories of gravity. We address the fundamental principles and mathematical tools on which general relativity (GR) is based. In particular, we study the Schwarzschild solution and Gödel-type universes and Friedmann-Robertson-Walker metric in the GR framework. Next we discuss the motivations that gave rise to Einstein’s modified theories of GR. We focus our study on verifying the consistency of spacetime-homogeneous Gödel-type metrics within the particular of gravity, f(R, Q, P), for well-motivated matter sources. It is worth stressing that f is an arbitrary function of curvature invariants: R is the Ricci scalar, Q is the contraction of two Ricci tensors and P is the contraction of two Riemann tensors. As it is well known, such geometries allow for global causality violation. We compare this theory with standard GR, we check the consistency of Gödel-type solutions within the f(R, Q, P) gravity and we discuss issues related to causality. Explicitly, we find new completely causal Godel-type solutions without any analogue in GR. In particular, a remarkable Gödel-type solution corresponding to the conformally flat space and maximally symmetric space, with no need for cosmological constant Λ, has been found. We provide general conditions to engender completely causal solutions in a manner utterly different from general relativity. We take some specific models, for instance, f(R, Q, P) = R− μ4n+2 (aR2 + bQ + cP )n , to illustrate the general results. Notably, we also find an unusual completely causal vacuum solution in the presence of a non-trivial cosmological constant which corresponds to the case m2 = 4ω2.