Academic literature on the topic 'Raychaudhuri equation'

The effects on Raychaudhuri’s equation of an intrinsically-discrete or particle nature of spacetime are investigated. This is done through the consideration of null congruences emerging from, or converging to, a generic point of spacetime, i.e., in geometric circumstances somehow prototypical of singularity issues. We do this from an effective point of view, that is through a (continuous) description of spacetime modified to embody the existence of an intrinsic discreteness on the small scale, this adding to previous results for non-null congruences. Various expressions for the effective rate of change of expansion are derived. They in particular provide finite values for the limiting effective expansion and its rate of variation when approaching the focal point. Further, this results in a non-vanishing of the limiting cross-sectional area itself of the congruence.

We know that in general relativity the Raychaudhuri equation gives the congruence and evolution of the geodesics. In this work, it is proposed that in brane cosmology the Raychaudhuri equation, apart from giving the collapse of the brane, it gives the evolution of the brane. Therefore, the Raychaudhuri equation gives the same information as the one of Friedmann’s, in the context of evolution.

Raychaudhuri equation is generalized in the parametrized absolute parallelism geometry. This version of absolute parallelism is more general than the conventional one. The generalization takes into account the suggested interaction between the quantum spin of the moving particle and the torsion of the background gravitational field. The generalized Raychaudhuri equation obtained contains some extra terms, depending on the torsion of space–time, that would have some effects on the singularity theorems of general relativity. Under a certain condition, this equation could be reduced to the original Raychaudhuri equation without any need for a vanishing torsion.

More than twenty years ago, the first of the two authors of this paper has deduced the generalized Landau–Raychaudhuri equation and demonstrated its numerous applications. Now we present some new interesting applications of the generalized Landau–Raychaudhuri equation.

In this work, we obtain the Raychaudhuri equations for various types of Finsler spaces as the Finsler–Randers (FR) space-time and in a generalized geometrical structure of the space-time manifold which contains two fibers that represent two scalar fields [Formula: see text]. We also derive the Klein–Gordon equation for this model. In addition, the energy conditions are studied in a FR cosmology and are correlated with FRW model. Finally, we apply the Raychaudhuri equation for the model [Formula: see text], where M is a FRW-space-time.

Thesis (Ph.D.)--Tufts University, 2004.
Adviser: Larry H. Ford. Submitted to the Dept. of Physics. Includes bibliographical references (leaves 117-120). Access restricted to members of the Tufts University community. Also available via the World Wide Web;

The work presented in this thesis is devoted to the study of geodesic motion in the context of General Relativity. The motion of a single test particle is governed by the geodesic equations of the given space-time, nevertheless one can be interested in the collective behavior of a family (congruence) of test particles, whose dynamics is controlled by the Raychaudhuri equations. In this thesis, both the aspects have been considered, with great interest in the latter issue. Geometric quantities appear in these evolution equations, therefore, it goes without saying that the features of a given space-time must necessarily arise. In this way, through the study of these quantities, one is able to analyze the given space-time. In the first part of this dissertation, we study the relation between geodesic motion and gravity. In fact, the geodesic equations are a useful tool for detecting a gravitational field. While, in the second part, after the derivation of Raychaudhuri equations, we focus on their applications to cosmology. Using these equations, as we mentioned above, one can show how geometric quantities linked to the given space-time, like expansion, shear and twist parameters govern the focusing or de-focusing of geodesic congruences. Physical requirements on matter stress-energy (i.e., positivity of energy density in any frame of reference), lead to the various energy conditions, which must hold, at least in a classical context. Therefore, under these suitable conditions, the focusing of a geodesics "bundle", in the FLRW metric, bring us to the idea of an initial (big bang) singularity in the model of a homogeneous isotropic universe. The geodesic focusing theorem derived from both, the Raychaudhuri equations and the energy conditions acts as an important tool in understanding the Hawking-Penrose singularity theorems.

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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.
Na teoria da Relatividade Geral de Einstein as equa??es de campo relacionam a geometria do espa?o-tempo com o conte?do de mat?ria e de energia, fontes do campo gravitacional. Esse conte?do ? descrito por um tensor de segunda ordem, conhecido como tensor energia-momento. Por outro lado, os tensores energia-momento que possuem significado f?sico n?o s?o especificados por essa teoria. Na d?cada de 70, Hawking e Ellis estabeleceram algumas condi??es, consideradas plaus?veis do ponto de vista f?sico, com o intuito de limitar as arbitrariedades desses tensores. Essas condi??es ficaram conhecidas como condi??es de energia de Hawking-Ellis, desempenham pap?is importantes no cen?rio da gravita??o. Elas s?o largamente usadas como poderosas ferramentas de an?lise, desde a demonstra??o de importantes teoremas relativos ao comportamento de campos gravitacionais e geometrias associadas, comportamento qu?ntico da gravita??o, at? as an?lises de modelos cosmol?gicos. Nesta disserta??o apresentamos uma dedu??o rigorosa das v?rias condi??es de energia em voga atualmente na literatura cient?fica, tais como: Condi??o de Energia Nula (NEC), Condi??o de Energia Fraca (WEC), Condi??o de Energia Forte (SEC), Condi??o de Energia Dominante (DEC) e Condi??o de Energia Dominante Nula (NDEC). Tendo em mente as aplica??es mais corriqueiras em Gravita??o e Cosmologia, as dedu??es foram feitas inicialmente para um tensor energia-momento de um fluido perfeito generalizado e depois estendidas aos campos escalares com acoplamento m?nimo e n?o-m?nimo ao campo gravitacional. Apresentamos tamb?m um estudo sobre as poss?veis viola??es de algumas dessas condi??es de energia, visando o estudo da natureza singular de algumas solu??es exatas da Relatividade Geral de Einstein, em 1955, o f?sico indiano Raychaudhuri derivou uma equa??o que hoje ? considerada fundamental para o estudo da atra??o gravitacional da mat?ria, a qual ficou conhecida como equa??o de Raychaudhuri. Essa c?lebre equa??o ? considerada o alicerce da compreens?o da atra??o gravitacional em Astrof?sica e Cosmologia e dos teoremas de Singularidades, como por exemplo, o teorema de Hawking e Penrose sobre a singularidade do colapso gravitacional. Nesta disserta??o derivamos a equa??o de Raychaudhuri, o teorema de Frobenius e o teorema da Focaliza??o para congru?ncias tipo-tempo e tipo-nulas de uma variedade pseudo-riemanniana. Discutimos o significado geom?trico e f?sico dessa equa??o, sua conex?o com as condi??es de energia, e algumas de suas in?meras aplica??es.