Condições de energia de hawking e ellis e a equação de raychaudhuri

In the Einstein s theory of General Relativity the field equations relate the geometry of space-time with the content of matter and energy, sources of the gravitational field. This content is described by a second order tensor, known as energy-momentum tensor. On the other hand, the energy-momentum tensors that have physical meaning are not specified by this theory. In the 700s, Hawking and Ellis set a couple of conditions, considered feasible from a physical point of view, in order to limit the arbitrariness of these tensors. These conditions, which became known as Hawking-Ellis energy conditions, play important roles in the gravitation scenario. They are widely used as powerful tools for analysis; from the demonstration of important theorems concerning to the behavior of gravitational fields and geometries associated, the gravity quantum behavior, to the analysis of cosmological models. In this dissertation we present a rigorous deduction of the several energy conditions currently in vogue in the scientific literature, such as: the Null Energy Condition (NEC), Weak Energy Condition (WEC), the Strong Energy Condition (SEC), the Dominant Energy Condition (DEC) and Null Dominant Energy Condition (NDEC). Bearing in mind the most trivial applications in Cosmology and Gravitation, the deductions were initially made for an energy-momentum tensor of a generalized perfect fluid and then extended to scalar fields with minimal and non-minimal coupling to the gravitational field. We also present a study about the possible violations of some of these energy conditions. Aiming the study of the single nature of some exact solutions of Einstein s General Relativity, in 1955 the Indian physicist Raychaudhuri derived an equation that is today considered fundamental to the study of the gravitational attraction of matter, which became known as the Raychaudhuri equation. This famous equation is fundamental for to understanding of gravitational attraction in Astrophysics and Cosmology and for the comprehension of the singularity theorems, such as, the Hawking and Penrose theorem about the singularity of the gravitational collapse. In this dissertation we derive the Raychaudhuri equation, the Frobenius theorem and the Focusing theorem for congruences time-like and null congruences of a pseudo-riemannian manifold. We discuss the geometric and physical meaning of this equation, its connections with the energy conditions, and some of its several aplications.

Evolução top-hat hidrodinâmica em campos de velocidades peculiares primordiais

We investigate the cosmology of the vacuum energy decaying into cold dark matter according to thermodynamics description of Alcaniz & Lima. We apply this model to analyze the evolution of primordial density perturbations in the matter that gave rise to the first generation of structures bounded by gravity in the Universe, called Population III Objects. The analysis of the dynamics of those systems will involve the calculation of a differential equation system governing the evolution of perturbations to the case of two coupled fluids (dark matter and baryonic matter), modeled with a Top-Hat profile based in the perturbation of the hydrodynamics equations, an efficient analytical tool to study the properties of dark energy models such as the behavior of the linear growth factor and the linear growth index, physical quantities closely related to the fields of peculiar velocities at any time, for different models of dark energy. The properties and the dynamics of current Universe are analyzed through the exact analytical form of the linear growth factor of density fluctuations, taking into account the influence of several physical cooling mechanisms acting on the density fluctuations of the baryonic component of matter during the evolution of the clouds of matter, studied from the primordial hydrogen recombination. This study is naturally extended to more general models of dark energy with constant equation of state parameter in a flat Universe

Teorias f(R) de gravidade na formulação de Palatini

In this dissertation, after a brief review on the Einstein s General Relativity Theory and its application to the Friedmann-Lemaitre-Robertson-Walker (FLRW) cosmological models, we present and discuss the alternative theories of gravity dubbed f(R) gravity. These theories come about when one substitute in the Einstein-Hilbert action the Ricci 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 neither a dark energy component, nor the existence of extra spatial dimensions. In dealing with f(R) gravity, two different variational approaches may be followed, namely the metric and the Palatini formalisms, which lead to very different equations of motion. We briefly describe the metric formalism and then concentrate on the Palatini variational approach to the gravity action. We make a systematic and detailed derivation of the field equations for Palatini f(R) gravity, which generalize the Einsteins equations of General Relativity, and obtain also the generalized Friedmann equations, which can be used for cosmological tests. As an example, using recent compilations of type Ia Supernovae observations, we show how the f(R) = R − fi/Rn class of gravity theories explain the recent observed acceleration of the universe by placing reasonable constraints on the free parameters fi and n. We also examine the question as to whether Palatini f(R) gravity theories permit space-times in which causality, a fundamental issue in any physical theory [22], is violated. As is well known, in General Relativity there are solutions to the viii 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 every perfect-fluid Gödel-type solution of Palatini f(R) gravity with density and pressure p that satisfy the weak energy condition + p 0 is necessarily isometric to the Gödel geometry, demonstrating, therefore, that these theories present causal anomalies in the form of closed time-like curves. This result extends a theorem on Gödel-type models to the framework of Palatini f(R) gravity theory. We derive an expression for a critical radius rc (beyond which causality is violated) for an arbitrary Palatini f(R) theory. The expression makes apparent that the violation of causality depends on the form of f(R) and on the matter content components. We concretely examine the Gödel-type perfect-fluid solutions in the f(R) = R−fi/Rn class of Palatini gravity theories, and show that for positive matter density and for fi and n in the range permitted by the observations, these theories do not admit the Gödel geometry as a perfect-fluid solution of its field equations. In this sense, f(R) gravity theory remedies the causal pathology in the form of closed timelike curves which is allowed in General Relativity. 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 Gödeltype solution with no violation of causality. Finally, we show that by combining a perfect fluid plus a scalar field as sources of Gödel-type geometries, we obtain both solutions in the form of closed time-like curves, as well as solutions with no violation of causality

Gravitomagnetismo e o teste da sonda gravidade B

The so-called gravitomagnetic field arised as an old conjecture that currents of matter (no charges) would produce gravitational effects similar to those produced by electric currents in electromagnetism. Hans Thirring in 1918, using the weak field approximation to the Einsteins field equations, deduced that a slowly rotating massive shell drags the inertial frames in the direction of its rotation. In the same year, Joseph Lense applied to astronomy the calculations of Thirring. Later, that effect came to be known as the Lense- Thirring effect. Along with the de Sitter effect, those phenomena were recently tested by a gyroscope in orbit around the Earth, as proposed by George E. Pugh in 1959 and Leonard I. Schiff in 1960. In this dissertation, we study the gravitational effects associated with the rotation of massive bodies in the light of the Einsteins General Theory of Relativity. With that finality, we develop the weak field approximation to General Relativity and obtain the various associated gravitational effects: gravitomagnetic time-delay, de Sitter effect (geodesic precession) and the Lense-Thirring effect (drag of inertial frames). We discus the measures of the Lense-Thirring effect done by LAGEOS Satellite (Laser Geodynamics Satellite) and the Gravity Probe B - GPB - mission. The GPB satellite was launched into orbit around the Earth at an altitude of 642 km by NASA in 2004. Results presented in May 2011 clearly show the existence of the Lense-Thirring effect- a drag of inertial frames of 37:2 7:2 mas/year (mas = milliarcsec)- and de Sitter effect - a geodesic precession of 6; 601:8 18:3 mas/year- measured with an accuracy of 19 % and of 0.28 % respectively (1 mas = 4:84810��9 radian). These results are in a good agreement with the General Relativity predictions of 41 mas/year for the Lense-Thirring effect and 6,606.1 mas/year for the de Sitter effect.