The difficulties associated with the quantization of the gravitational field suggests a modification of space-time is needed. For example at suffici~ly small length scales the geometry of space-time might better discussed in terms of a noncommutative algebra. In this thesis we discuss a particular example of a noncommutative algebra, namely the symplectic Schonberg algebra, which we treat as a geometric algebra. Thus our investigation has some features in common with recent work that explores how geometry can be formulated in terms of noncommutative structures. The symplectic Schonberg algebra is a geometric algebra associated with the covariant and the contravariant vectors of a general affine space. The "embedding" of this space in a noncommutative algebra leads us to a structure which we regard as a noncommutative affine geometry. The theory in question takes us naturally to stochastic elements without the usual ad-hoc assumptions concerning measurements in physical ensembles that are made in the usual interpretation of quantum mechanics. The probabilistic nature of space is obtained purely from the structure of this algebra. As a consequence, geometric objects like points, lines and etc acquire a kind of fuzzy character. This allowed us to construct the space of physical states within the algebra in terms of its minimum left-ideals as was proposed by Hiley and Frescura [1J. The elements of these ideals replace the ordinary point in the Cartesian geometry. The study of the main inner-automorphisms of the algebra gives rise to the representation of the symplectic group of linear classical canonical transformations. We show that this group acts on the minimum left-ideal of the algebra and in this case manifests itself as the metaplectic group, i.e the double covering of the symplectic group. Thus we are lead to the theory of symplectic spinors as minimum left-ideals in exactly the same way as the orthogonal spinors can be formulated in terms of minimum left-ideals in the Clifford algebra .. The theory of the automorphisms of the symplectic Schonberg algebra allows us to give a geometrical meaning to integral transforms such as: the Fourier transform, the real and complex Gauss Weierstrass transform, the Bargmann (3) transform and the Bilateral Laplace transform. We construct a technique for obtaining a realization of these algebraic transformations in terms of integral kernels. This gives immediately the Feynmann propagators of conventional non-relativistic quantum mechanics for Hamiltonians quadratic in momentum and position. This then links our approach to those used in quantum mechanics and optics. The link between the theory of this noncommutative geometric algebra and the theory of vector bundles is also discussed.

An important aspect for precision guidance, navigation, and control for lunar operations is environmental modeling. In particular, consider gravity field modeling. Available gravity field models for the Moon reach degree and order 165 requiring the use and storage of approximately 26,000 spherical harmonic coefficients. Although the high degree and order provide a means by which to accurately predict trajectories within the influence of the Moon's gravitational field, the size of these models makes using them computationally expensive and restricts their use in design environments with limited computer memory and storage. It is desirable to determine reduced complexity realizations of the gravitational models to lower the computational burden while retaining the structure of the original gravitational field for use in rapid design environments. The extended Kalman filter and the unscented Kalman filter are used to create reduced order models and are compared against a simple truncation based reduction method. Both variations of the Kalman filter out perform the truncation based method as a means by which to reduce the complexity of the gravitational field. The extended Kalman filter and unscented Kalman filter were able to achieve good estimates of position while reducing the number of spherical harmonic coefficients used in gravitational acceleration calculations by approximately 5,400, greatly increasing the speed of the calculations while reducing the required computer allocation. / text

No description available.

One of the applications of the Satellite Laser Ranging (SLR) technique is the derivation of gravity field models; these models have various geophysical and geodynamical applications. Gravity field modelling has reached a new era where the latest satellite missions (CHAMP, GRACE and GOCE) are thought to provide significant improvement of global gravity field information in terms of quality and spatial resolution. In particular, the recent satellite missions carry on-board Global Navigation Satellite System (GNSS) receivers, accelerometers, K/Kaband microwave system (e.g. in GRACE) and gradiometers (e.g. in GOCE) allowing measurements of gravity field with unprecedented accuracy in contrast to the unsteady and fragmented orbit tracking by unevenly distributed SLR ground stations. Numerous gravity field models have been derived based on the newly available data sets by various research groups globally. Due to the availability of high quality SLR and satellite data, some of the older gravity field models are being updated as new models with higher degree and order are developed. Notwithstanding the significant progress in gravity field modelling, research focusing on assessing the accuracy and precision of the existing gravity field models has largely remained insufficient. The difference between the observed and computed satellite orbit (which is often expressed as the O-C range residuals) is used as a parameter for Precise Orbit Determination (POD) of satellites. Furthermore, O-C range residuals computed during SLR analysis are used as proxy parameters for evaluating the accuracy of gravity field models. The work presented in this thesis firstly reviewed and evaluated the accuracy of gravity field models released between 1990 and 2008. The accuracy of the gravity field models was examined by analysing the O-C residuals computed from LAGEOS 1 and 2 data analysis based on a set of twelve gravity field models. The results demonstrated that in general, there has been an improvement in the accuracy of gravity field models released between 1990 and 2008 by a factor of 2 based on improvements in the O-C residuals. Additionally, the influence of SLR tide parameterization (the IERS 2010 solid Earth and pole tide models) on the O-C residuals across five gravity field models has been assessed and results illustrate that the solid Earth and pole tides parameterization influence on the O-C residuals is dependent on the type of gravity field model. In order to ascertain the significance of mean differences in the Standard Deviations (SD) of O-C residuals based on the tide parameterization options, the student’s t-test was used. Results suggest that in general the O-C residuals derived from SLR LAGEOS 1 data have insignificant mean SD differences across the tide parameterizations. On the other hand analysis of SLR observations of LAGEOS 2 resulted in statistically significant mean SD differences in the O-C based on EIGEN-CG03C, EGM2008 and AIUB-GRACE01S gravity field models. The J2 coefficient forms part of the SLR Data Analysis Software (SDAS) package output products and was investigated in this thesis due to its role in understanding mass-redistribution within the Earth system (i.e. the equatorial bulge due to centrifugal force and rotation). In particular, the J 2 coefficient computed from SLR analysis of LAGEOS 1 and 2 data sets and based on the four selected gravity field models were compared with a priori J2 coefficients from the four models and those published in the literature. The results indicated that the J2 coefficients computed from the SDAS package were in agreement with the published coefficients. For geophysical applications, the relationship between the J2 parameter and LOD and AAM was investigated by use of data adaptive analysis methodology (the empirical mode decomposition). The results demonstrated that some degree of synchronization exists between the signal components of J2 and LOD and J2 and AAM. / Thesis (PhD)--University of Pretoria, 2013. / Geography, Geoinformatics and Meteorology / Unrestricted

Study of light rays (light world lines) plays a significant role in many of astro- physical applications. Light rays are mainly studied in terms of so-called grav- itational lensing. However, the majority of studies are mainly focused on light propagation in vacuum. If the refractive and dispersive medium characterised by refractive index n is considered, effects occurring due to the medium presence need to be taken into account, which significantly complicates the problem. In the present thesis, rays propagating through simple refractive and dispersive systems, such as plane differentially sheared medium, are studied. In order to simplify the problem, the Hamiltonian equations of motion are used. The ray trajectories in the vicinity of Kerr black hole as well as accessible regions for the rays are also studied. Radial variation of the medium velocity is considered. Due to the recent increase of publications focused on the gravitational lensing in plasma, a detailed review summarizing the results obtained recently is included. 1

Submitted by Vasti Diniz (vastijpa@hotmail.com) on 2017-09-13T12:46:00Z No. of bitstreams: 1 aquivototal.pdf: 1083129 bytes, checksum: c7786a11a4d811fd716d27cf8c82948d (MD5) / Made available in DSpace on 2017-09-13T12:46:00Z (GMT). No. of bitstreams: 1 aquivototal.pdf: 1083129 bytes, checksum: c7786a11a4d811fd716d27cf8c82948d (MD5) Previous issue date: 2011-11-25 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / In thiswork,weinvestigateaclassofspace-timesbycalculatingtheloopvariables for differentcurves,intheparticularcasewherethepathstakenareclosed(circles),we obtain theso-calledholonomytransformations,fromwhichthetraceprovidesthematrix obtained intheWilsonloopgravitational.Firstwediscussthecasewherethegravita- tional fieldisgeneratedbyacylindricalshell,withoutrotation,intheapproximationof weak field.Thenwestudythespace-timesofGödelandFriedman-Robertson-Walker.We also investigatethespace-timesgeneratedbyacosmicstringwithinternalstructure.Fi- nally wediscusstheKottlerspace-timeandsomemodelsofwormholes.Tocalculatethe loop variablesweusetheperturbationexpansiontechnique,whichwecall perturbative method, andtechniquethroughtheLorentztransformationoflocalcoordinates,which we callthe exact method. Theobtainedresultsshowusthewaythesecontourvariables contain informationsrelatedtothegeometricalandtopologicalpropertiesofeachoneof these space-times. / Neste trabalho,investigamosumaclassedeespaços-tempoatravésdocálculodasva- riáveis decontornoparadiferentescurvas,quenocasoparticular,emqueoscaminhosto- mados sãofechados(círculos),obtêm-seasdenominadastransformaçõesdeholonomias, do qualotraçodamatrizobtidanosforneceoloopdeWilsongravitacional.Primeiro abordamos ocasoemqueocampogravitacionalégeradoporumacascacilíndrica,sem rotação,naaproximaçãodecampofraco.Emseguida,estudamososespaços-tempode Gödel eodeFriedman-Robertson-Walker.Tambéminvestigamosocenáriogeradopor uma cordacósmicacomestruturainterna.Porfim,abordamosoespaço-tempodeKöttle e algunsmodelosdewormholes.Paraocálculodasvariáveisdecontornoutilizamosa técnica daexpansãoperturbativa,oqualdenominamosde método perturbativo, eatéc- nica viatransformaçãodecoordenadasdeLorentzlocal,oqualdenominamosde método exato. Osresultadosnosmostramdequemaneiraessasvariáveisdecontornodetectam as característicasgeométricasetopológicasdecadaumdessesespaços-tempo.

Made available in DSpace on 2015-03-03T15:15:24Z (GMT). No. of bitstreams: 1 ThiagoBRFO_DISSERT.pdf: 776732 bytes, checksum: 79a4002c3c2d724d3d1651680816802b (MD5) Previous issue date: 2010-07-01 / Coordena??o de Aperfei?oamento de Pessoal de N?vel Superior / 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 G?del model being the best known example of such a solution. Here we show that every perfect-fluid G?del-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 G?del geometry, demonstrating, therefore, that these theories present causal anomalies in the form of closed time-like curves. This result extends a theorem on G?del-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 G?del-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 G?del 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 G?del-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 G?deltype solution with no violation of causality. Finally, we show that by combining a perfect fluid plus a scalar field as sources of G?del-type geometries, we obtain both solutions in the form of closed time-like curves, as well as solutions with no violation of causality / Nesta disserta??o, ap?s uma breve revis?o sobre a Teoria da Relatividade Geral de Einstein e suas aplica??es para os modelos cosmol?gicos de Friedmann-Lemaitre- Robertson-Walker (FLRW), apresentamos e discutimos as teorias alternativas de gravidade denominadas de gravidade f(R). Estas teorias surgem quando substitu?mos na a??o de Einstein-Hilbert o escalar de curvatura de Ricci R por qualquer fun??o f(R) n?o-linear bem comportada. Elas fornecem uma maneira alternativa para explicar a acelera??o c?smica atual sem necessitar envolver qualquer componente de energia escura ou a exist?ncia de dimens?es espaciais extras. Quando lidamos com gravidade f(R), dois diferentes princ?pios variacionais podem ser seguidos, a saber, o formalismo m?trico e o de Palatini, os quais levam a equa??es de movimento muito diferentes. Descrevemos brevemente o formalismo m?trico e ent?o nos concentramos no princ?pio variacional de Palatini para a a??o da gravidade. Fazemos uma deriva??o sistem?tica e detalhada das equa??es de campo para a gravidade f(R) de Palatini, as quais generalizam as equa??es de Einstein da Relatividade Geral. Em seguida obtemos as equa??es de Friedmann generalizadas, que podem ser usadas para testes cosmol?gicos. Para exemplificar, usamos compila??es recentes de observa??es de supernovas do tipo Ia e mostramos como a classe de teorias de gravidade f(R) = R ? /Rn explica a recente acelera??o observada do universo quando colocamos v?nculos razo?veis sobre os par?metros livres e n. Examinamos tamb?m a quest?o de como teorias f(R) de gravidade em Palatini permitem espa?os-tempos em que a causalidade, um resultado fundamental em qualquer teoria f?sica [22], ? violada. Como ? bem conhecido, na Relatividade Geral existem solu??es para as equa??es de campo que possuem anomalias causais na forma de curvas tipo-tempo fechadas, sendo o modelo de G?del o exemplo mais bem conhecido de tais solu??es. Aqui mostramos que toda solu??o do tipo-G?del de gravidade f(R) em Palatini com fluido perfeito, caracterizado por densidade e press?o p, satisfazendo a condi??o de energia fraca + p 0, ? necessariamente isom?trica ? geometria de G?del, demonstrando, portanto, que essas teorias apresentam anomalias causais na forma de curvas tipo-tempo fechadas. Esses resultados ampliam um teorema sobre modelos tipo-G?del para a estrutura das teorias de gravidade f(R) de Palatini. Derivamos uma express?o para o raio cr?tico rc (al?m do qual a causalidade ? violada) para uma teoria arbitr?ria de gravidade f(R) de Palatini. A express?o encontrada tornou claro que a viola??o da causalidade depende da forma de f(R) e dos componentes do conte?do de mat?ria. Examinamos objetivamente as solu??es tipo-G?del de um fluido perfeito na classe f(R) = R ? /Rn das teorias de gravidade de Palatini e mostramos que, para uma densidade de mat?ria positiva e para e n em um intervalo permitido pelas observa??es, essas teorias n?o admitem como solu??es de suas equa??es de campo a geometria de G?del juntamente com um fluido perfeito. Nesse sentido, teorias de gravidade f(R) remediam a patologia causal na forma de curvas tipotempo fechadas que ? permitido na Relatividade Geral. Examinamos tamb?m essa viola??o de causalidade ao considerar um campo escalar como conte?do material. Para essa fonte, mostramos que a gravidade f(R) em Palatini d? origem a uma ?nica solu??o do tipo-G?del sem viola??o de causalidade. Finalmente, mostramos que a combina??o de um fluido perfeito mais um campo escalar como fontes de geometrias tipo-G?del, levam a solu??es na forma de curvas tipo-tempo fechadas como a solu??es sem viola??o de causalidade

The aim of this thesis is to create collections of examples for the subject Astronomy taught for students of pedagogical faculties, studying this discipline as a part of physics courses. Due to very different mathematical knowledge of students I have chosen typical and not much difficult examples oriented to several branches of astronomy. Each part of examples begins with a self-contained theoretical introduction. The difficulty rises gradually from trivial to more complicated examples. The examples are mainly focused on motions in radial gravitational fields.