Academic literature on the topic 'Raychaudhuri'

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.

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.

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.

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.

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;

Made available in DSpace on 2014-12-17T15:14:52Z (GMT). No. of bitstreams: 1 CrislaneSS_DISSERT.pdf: 1091298 bytes, checksum: 831e6bef52e8fad49a4683ec16886d4d (MD5) Previous issue date: 2011-04-14
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.

Submitted by Automa??o e Estat?stica (sst@bczm.ufrn.br) on 2017-07-17T13:08:15Z No. of bitstreams: 1 CrislaneDeSouzaSantos_TESE.pdf: 859033 bytes, checksum: 6c4933c54ee1e77f0ea6be3839fc13c1 (MD5)
Approved for entry into archive by Arlan Eloi Leite Silva (eloihistoriador@yahoo.com.br) on 2017-07-18T14:28:54Z (GMT) No. of bitstreams: 1 CrislaneDeSouzaSantos_TESE.pdf: 859033 bytes, checksum: 6c4933c54ee1e77f0ea6be3839fc13c1 (MD5)
Made available in DSpace on 2017-07-18T14:28:54Z (GMT). No. of bitstreams: 1 CrislaneDeSouzaSantos_TESE.pdf: 859033 bytes, checksum: 6c4933c54ee1e77f0ea6be3839fc13c1 (MD5) Previous issue date: 2017-03-24
Coordena??o de Aperfei?oamento de Pessoal de N?vel Superior (CAPES)
A evid?ncia observacional da expans?o acelerada do Universo tem sido a principal raz?o para uma revis?o da evolu??o cosmol?gica como previsto pela Relatividade Geral (RG). Atualmente existe duas principais abordagens para resolver este problema: pela introdu??o nas equa??es de Einstein de um termo o qual representa um novo tipo de fluido (a chamada energia escura) possuindo caracter?sticas ex?ticas ou pela modifica??o da teoria de gravita??o. Nesta tese n?s focamos na segunda abordagem, particularmente, as teorias conhecidas como teorias f(R) de gravidade as quais t?m recebido muita aten??o nos ?ltimos anos. Neste contexto, a equa??o de Raychaudhuri permite examinar a estrutura do espa?o-tempo como um todo sem solu??es espec?ficas das equa??es de Einstein, desempenhando assim um papel central para a compreens?o da atra??o gravitacional em Astrof?sica e Cosmologia. Na teoria da Relatividade Geral sem uma constante cosmol?gica, uma contribui??o n?o-positiva da geometria do espa?o-tempo a equa??o de Raychaudhuri ? usualmente interpretada como a manifesta??o do car?ter atrativo da gravidade. Neste caso, condi??es de energia espec?ficas - de fato a condi??o de energia forte - deve ser assumida, a fim de garantir o car?cter atrativo. No contexto das teorias f(R) de gravidade, no entanto, mesmo assumindo as condi??es de energia usuais pode-se ter uma contribui??o positiva para a equa??o de Raychaudhuri. Al?m de nos fornecer uma maneira simples de explicar a observada expans?o acelerada do Universo, este fato abre a possibilidade de um car?ter repulsivo deste tipo de gravidade. Nesta tese n?s abordamos o car?cter atrativo/n?o-atrativo da gravidade f(R) ? luz da equa??o de Raychaudhuri e fazemos uso da condi??o de energia forte, juntamente com estimativas recentes dos par?metros cosmogr?ficos, para colocar limites em uma classe paradigm?tica de teorias f(R) de gravidade.
The observational evidence of the accelerated expansion of the Universe has been the main reason for a revision of the cosmological evolution as predicted by General Relativity (GR). Currently there are two main approaches to solving this problem: by introducing in the Einstein?s equations a term which represent a new kind of fluid (the so-called dark energy possessing exotic features) or by the modification of the gravitation theory. In this thesis we focus on the second approach, particularly the theories know as f(R) theories of gravity, which have received many attention in the last years. In this framework, the Raychaudhuri equation makes possible to examine the whole of spacetime structures without specific solutions of Einstein?s equations, playing so a central role to the understanding of gravitational attraction in Astrophysics and Cosmology. In the general relativity theory of gravity without a cosmological constant, a non-positive contribution from the spacetime to Raychaudhuri?s equation is usually interpreted as manifestation of the attractive character of gravity. In this case, particular energy conditions - indeed the strong energy condition - must be assumed in order to guarantee this attractive character. In the context of f(R) theories of gravity however, even assuming the usual energy conditions we may have a positive contribution to Raychaudhuri?s equation. Besides giving us a simple way to explain the observed accelerated expansion of the Universe, this fact opens the possibility of a repulsive character of this kind of gravity. In this thesis we address the attractive/non-attractive character of f(R) theories of gravity at the light of Raychaudhuri?s equation and make use of the strong energy condition, jointly with recent estimated values for the cosmographic parameters, in order to put bounds on a paradigmatic class of f(R) theories of gravity.

In questa tesi studiamo come lo spin macroscopico della materia possa trovare una giustificazione geometrica nel tensore di torsione e quali siano le implicazioni dovute alla presenza di un fluido con spin in cosmologia, in particolare se possa giustificare almeno in parte l'espansione dell'universo attuale e se possa risolvere il problema della piattezza dell'universo. In primo luogo studiamo la derivata covariante senza ipotizzare simmetria dei coefficienti della connessione affine. Descriviamo la cinematica di particelle test nel formalismo 1+3 e ricaviamo l'equazione di Raychaudhuri. Dopodiché introduciamo l'azione di Einstein-Cartan da cui deriviamo le due equazioni di campo. Da queste ricaviamo la cinematica in un universo di Einstein-Cartan. In seguito introduciamo il fluido di Weyssenhoff e, facendo alcune assunzioni sulla natura del fluido di spin (irrotazionale e a taglio nullo) e dell'universo (omogeneo e isotropo), otteniamo l'equazione di Raychaudhuri per un fluido di spin, analoga a quella del modello di Friedmann-Robertson-Walker classico. Successivamente introduciamo il formalismo Hamiltoniano per valutare le condizioni di espansione al tempo attuale sui parametri di densità e vediamo che il contributo richiesto allo spin per giustificare l'espansione è elevato rispetto a quanto misurato dai dati delle supernovae Ia e dalla radiazione cosmica di fondo. Poi studiamo il problema della piattezza osservando come la presenza del fluido di spin prevenga la formazione della singolarità iniziale, grazie al fatto che l'espansione inizia a t0 quando l'universo ha un raggio finito am = 9×10−6m. Allo stesso tempo la presenza del fluido di spin giustifica la piattezza attuale grazie al parametro di densità di spin ΩS = −8.6 × 10−70, evitando l'utilizzo dell'inflazione cosmica. Tuttavia nuove problematiche si aprono nell'interpretare ciò che accade negli istanti precedenti t0 in cui l'universo si sta contraendo.

This book brings together the two disparate worlds of computational text analysis and biology and presents some of the latest methods and applications to proteomics, sequence analysis and gene expression data. Modern genomics generates large and comprehensive data sets but their interpretation requires an understanding of a vast number of genes, their complex functions, and interactions. Keeping up with the literature on a single gene is a challenge itself-for thousands of genes it is simply impossible. Here, Soumya Raychaudhuri presents the techniques and algorithms needed to access and utilize the vast scientific text, i.e. methods that automatically "read" the literature on all the genes. Including background chapters on the necessary biology, statistics and genomics, in addition to practical examples of interpreting many different types of modern experiments, this book is ideal for students and researchers in computational biology, bioinformatics, genomics, statistics and computer science.

Text about genes can be effectively leveraged to enhance sequence analysis (MacCallum, Kelley et al. 2000; Chang, Raychaudhuri et al. 2001; McCallum and Ganesh 2003; Eskin and Agichtein 2004; Tu, Tang et al. 2004). Most of the emerging methods utilize textual representations similar to the one we introduced in the previous chapter. To analyze sequences, a numeric vector that contains information about the counts of different words in references about that sequence can be used in conjunction with the actual sequence information. Experienced biologists understand the value of using the information in scientific text during sequence searches, and commonly use scientific text and annotations to guide their intuition. For example, after a quick BLAST search, a trained expert might quickly look over the hits and their associated annotations and literature references and assess the validity of the hits. The apparently valid sequence hits can then be used to draw conclusions about the query sequence by transferring information from the hits. In most cases, the text serves as a proxy for structured functional information. High quality functional annotations that succinctly and thoroughly describe the function of a protein are often unavailable. Defining appropriate keywords for a protein requires a considerable amount of effort and expertise, and in most cases, the results are incomplete as there is an evergrowing collection of knowledge about proteins. So, one option is to use text to compare the biological function of different sequences instead. There are different ways in which the functional information in text could be used in the context of sequence analysis. One possibility is to first run a sequence analysis algorithm, and then to use text profiles to summarize or organize results. Functional keywords can be assigned to the whole group of hit sequences. Additionally, given a series of sequences, they can be grouped according to like function. In either case, quick assessment of the content of text associated with sequences offers insight about exactly what we are seeing. These approaches are particularly useful if we are querying a large database of sequences with a novel sequence that we have very little information about.