Um estudo sobre a violação de causalidade em teorias f (R) de gravidade

Abstract

The currently observed accelerated expansion of the Universe, as well as the so called dark mater problem in astrophysics, has raised many discussions and some doubts about the very well tested Einstein’s theory of gravitation, known as General Relativity. Several modified, as well as extended gravity theories, have been formulated in the last 15 years, and some others been resuscitated in new aspect. In this thesis, we present and discuss, in the class of extended gravity, the alternative theories known as f(R) gravity. These theories come about when one substitute in the Einstein-Hilbert action the Ricci scalar curvature R by some well behaved nonlinear function f(R). They provide an alternative way to explain the current cosmic acceleration with no need of invoking either a dark energy component or the existence of an extra spatial dimension. In dealing with f(R) gravity, two different variational approaches may be followed, namely the metric and the Palatini formalisms. In the metric approach the connections are assumed, ab initio, to be the Levi-Civita connections and variation of the action is taken with respect to the metric only, whereas in the Palatini approach the metric and the connections are treated as independent fields and the variation of the action is taken with respect to both. Although they give the same equations for the Einstein-Hilbert action, for a general f(R) nonlinear term in the action they give rise to very different equations of motion. For both formalisms, we make a systematic and detailed derivation of the field equations, which generalize the Einstein’s equations of General Relativity and examine the covariant conservation of this equations. In this regard, we detect and call attention for the covariant conservation of the equations in Palatini f(R) gravity, which, in our view, deserves some more debate on the physical relevance of conformal aspects of the Palatini approach. In order to shed some light on the debate about the role of f(R) gravity, we also examine the question as to whether this theories permit space-times in which the causality, a fundamental issue in any physical theory, is violated. In the framework of f(R) gravity, the causal structure of four-dimensional space-time has locally the same qualitative nature as the flat space-time of special relativity: causality holds locally. The nonlocal question, however, is left open, and violation of causality can occur. As is well known, in General Relativity there are solutions to the field equations that have causal anomalies in the form of closed time-like curves, the renowned Gödel model being the best known example of such a solution. Here we show that for f(R) gravity satisfying the condition df/dR > 0, independently of being formulated in metric or Palatini formalism, every perfect type-fluid Gödel-type solution with density ρ and pressure p that satisfy the weak energy condition (ρ + p ≥ 0) is necessarily isometric to the Gödel geometry. This demonstrate that these theories present causal anomalies in the form of closed time-like curves. We also derive expressions for critical radius rc , beyond which the causality is violated, for an arbitrary Palatini, as well as metric f(R) theory of gravity. The expressions make apparent that the violation of causality depends on the form of f(R) and on the matter content components. As an example, we examine the Gödel-type perfect type-fluid solutions in the f(R) = R − β/Rn class of Palatini gravity theories, and show that for positive matter density and for β and n in the range permitted by the observations, these theories does not admit the Gödel geometry as a perfect type-fluid solution of its field equations. We also examine the violation of causality of Gödel-type by considering a single scalar field as the matter content. For this source we show that Palatini f(R) gravity gives rise to a unique Gödel-type solution with no violation of causality. Finally we show that by combining a perfect type-fluid plus a scalar field as source of Gödel-type geometries, we obtain either solutions in the form of closed time-like curves as well as solutions with no violation of causality. In the metric formalism we take another example, the f(R) = R − αR∗ ln(1 + R/R∗) gravity, which is free from singularities of the Ricci scalar and is cosmologically viable. Here we also show that combining perfect type-fluid with a scalar field as source of the Gödel geometry, this class of theories accommodate both causal and noncausal solutions for the range of cosmologically allowed parameters. Our conclusion is that f(R) gravity theory may remedies the causal pathology in the form of closed time-like curves which is allowed in General Relativity.