Numerical relativity is concerned with the generation of solutions to Einstein's equations by numerical means. In general, the construction of such a spacetime is accomplished in two stages: 1) the determination of initial data which is specified on a single spacelike hypersurface and satisfies four initial value equations, and 2) the evolution of the initial data to generate the spacetime or some portion of it. One of the key problems is the development of efficient algorithms for the solutions of these equations, as they are sufficiently complex to tax the fastest present computers. This thesis presents a comparison of various algorithms for the solution of the initial value equations, concentrating on the recently developed multi-grid method. The specific problem examined has been previously studied by Bowen, Piran and York. Their initial data has been interpreted as representing "snapshots" of three new families of black holes. Three of the four initial value equations possess analytic solutions. The remaining 2-dimensional nonlinear partial differential equation is solved numerically in this thesis using finite difference techniques. The performance of the multi-grid method, with respect to three more well-known methods, is evaluated through numerical experiments. The speed and reliability of the multi-grid algorithm are found to be very good. In addition, the results which had been previously calculated numerically by Piran are essentially reproduced, with the correction of some errors in that work. Possible extensions of the work to more complex initial value problems are also discussed. / Science, Faculty of / Physics and Astronomy, Department of / Graduate

Thesis (MSc)--University of Stellenbosch, 2006. / ENGLISH ABSTRACT: Entropy bounds arise from Black hole thermodynamics and are a significant departure from the conventional understanding of the information in a given region. This shift in paradigm is a consequence of the the fact that there is an unexpected relationship between the area and the entropy of a given region of spacetime. Entropy bounds are simplified formulations which are ultimately attempting to be developed into the complete and broad conjecture of the Holographic Principle. This hasn’t been achieved successfully as yet. In this thesis the aim is to introduce how the notion of an entropy bound was first suggested and it’s subsequent development into more robust formulations. The shortcomings of these conjectures are highlighted along with their strengths. A foundational introduction of the mathematical requirements for General Relativity is addressed along with an overview of Einstein’s theory of gravity. This is illustrated by showing the curvature of relative geodesics as being a consequence of gravity. This is contrasted with Newtonian theory where gravity is also shown to manifests as the curvature of relative geodesics. The working background is concluded with a discussion of Einstein’s field equations along with simple and common solutions often used and required. / AFRIKAANSE OPSOMMING: Swartgat Termodinamika impliseer grense op die entropie, en dus inligting, in ’n gegewe ruimtetyd volume, wat ’n drastiese afwyking van die tradisionele denkwyse oor inligting impliseer. Hierdie paradigma skuif het sy oorsprong in ’n onverwagte verband tussen die oppervlakte van, en entropie bevat, in ’n gegewe ruimte tyd volume. Entropie grense is eenvoudige formulerings van hierdie verwantskap wat uiteindelik beslag moet kry in die vollediger en wyer holografiese beginsel. Hierdie doelwit is nog nie bereik nie. Die doel van hierdie tesis is om die oorsprong en verdere formalisering van entropie grense te verduidelik. Beide die sterk en swak punte van die formulerings word bespreek. Algemene relatiwiteits teorie as ’n teorie van gravitasie, sowel as die wiskundige onderbou daarvan, word oorsigtelik bespreek. Die geometries onderbou van gravitasie word geillustreer aan die hand van die buiging van relatiewe geodete. Dit word met Newton se gravitasie teorie vergelyk wat ook in die buiging van relatiewe geodete gemanifesteer word. Hierdie oorsigtelike agtergrond word afgesluit met ’n oorsig van Einstein se vergelykings, asook eenvoudige en algemene oplossings wat dikwels nodig is en gebruik word.

The starting point of any general relativistic numerical simulation is a solution of the Hamiltonian and momentum constraints that (ideally) represents an astrophysically realistic scenario. This dissertation presents a new method to produce initial data sets for binary neutron stars with arbitrary spins and orbital eccentricities. The method only provides approximate solutions to the constraints. However, it was shown that the corresponding constraint violations subside after a few orbits, becoming comparable to those found in evolutions of standard conformally flat, helically symmetric binary initial data. This dissertation presents the first spinning neutron star binary simulations in circular orbits with a orbital eccentricity less then 0.01. The initial data sets corresponding to binaries with spins aligned, zero and anti-aligned with the orbital angular momentum were evolved in time. These simulations show the orbital “hang-up” effect previously seen in binary black holes. Additionally, they show orbital eccentricities that can be up to one order of magnitude smaller than those found in helically symmetric initial sets evolutions. / Includes bibliography. / Dissertation (Ph.D.)--Florida Atlantic University, 2013.

We study the evolution of binary black hole initial data schemes as alternatives to the standard puncture construction. These alternatives are based on post-Newtonian expansions that contain realistic gravitational waves. The first scheme is based on a second order post-Newtonian expansion in Arnowitt, Deser, and Misner transverse-traceless (ADMTT) gauge that has been re-summed to approach standard puncture data at the black holes. The other schemes are based on asymptotic matching of the 4-metrics of two tidally perturbed Schwarzschild solutions to post-Newtonian expansions at different orders. The alternatives are encouraging and lead to quasi-circular orbits and show gravitational radiation from the onset, as well as a reduction of spurious radiation. Deficiencies compared to punctures include more eccentricity during inspiral and larger constraint violations, since the alternative data sets only approximate solutions of Einstein's equations. / by George C. Reifenberger. / Thesis (Ph.D.)--Florida Atlantic University, 2013. / Includes bibliography. / Mode of access: World Wide Web. / System requirements: Adobe Reader.

In this dissertation the dynamics of general relativity is studied via the spin-foam approach to quantum gravity. Spin-foams are a proposal to compute a transition amplitude from a triangulated space-time manifold for the evolution of quantum 3d geometry via path integral. Any path integral formulation of a quantum theory has two important parts, the measure factor and a phase part. The correct measure factor is obtained by careful canonical analysis at the continuum level. The basic variables in the Plebanski-Holst formulation of gravity from which spin-foam is derived are a Lorentz connection and a Lorentz-algebra valued two-form, called the Plebanski two-form. However, in the final spin-foam sum, one usually sums over only spins and intertwiners, which label eigenstates of the Plebanski two-form alone. The spin-foam sum is therefore a discretized version of a Plebanski-Holst path integral in which only the Plebanski two-form appears, and in which the conne ction degrees of freedom have been integrated out. Calculating the measure factor for Plebanksi Holst formulation without the connection degrees of freedom is one of the aims of this dissertation. This analysis is at the continuum level and in order to be implemented in spin-foams one needs to properly discretize and quantize this measure factor. The correct phase is determined by semi-classical behavior. In asymptotic analysis of the Engle-Pereira-Rovelli-Livine spin-foam model, due to the inclusion of more than the usual gravitational sector, more than the usual Regge term appears in the asymptotics of the vertex amplitude. As a consequence, solutions to the classical equations of motion of GR fail to dominate in the semi-classical limit. One solution to this problem has been proposed in which one quantum mechanically imposes restriction to a single gravitational sector, yielding what has been called the “proper” spin-foam model. However, this revised model of quantum gravity, like any proposal for a theory of quantum gravity, must pass certain tests. In the regime of small curvature, one expects a given model of quantum gravity to reproduce the predictions of the linearized theory. As a consistency check we calculate the graviton two-point function predicted by the Lorentzian proper vertex and examine its semiclassical limit. / Includes bibliography. / Dissertation (Ph.D.)--Florida Atlantic University, 2015. / FAU Electronic Theses and Dissertations Collection

Orientadores: Ricardo Antonio Mosna, Márcio José Menon / Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Física Gleb Wataghin / Made available in DSpace on 2018-09-02T03:12:41Z (GMT). No. of bitstreams: 1 QuispeAruquipa_David_M.pdf: 2372918 bytes, checksum: bb030f9f8aecc210d608f2f9005a7a49 (MD5) Previous issue date: 2017 / Resumo: Os modelos análogos são uma ferramenta muito útil quando se quer entender ou testar fenômenos em um sistema físico em termos e conceitos de outro, esses sistemas podem ser mais familiares ou mais facilmente reproduzíveis por experimentos. Este trabalho explora esta questão no contexto de modelos análogos à gravitação baseados na mecânica dos fluidos. Particularmente, estamos interessados em fornecer um modelo análogo para um fluxo radial com uma fonte/sorvedouro na origem. Começamos por considerar o caso em que a velocidade do fluxo (radial) é constante. O modelo análogo resultante é então o espaço-tempo Anti-de Sitter (ou AdS) que é conhecido por ser não-globalmente hiperbólico. Como resultado, a dinâmica dos campos neste contexto não está bem definida até que sejam estabelecidas condições adicionais na fronteira no infinito espacial do espaço-tempo AdS. A contrapartida destas condições de fronteira extra na mecânica dos fluidos proporciona uma descrição efetiva da fonte/sorvedouro que está na origem. Depois disso, nós consideramos regularizações para o modelo análogo perto da fonte/sorvedouro na origem. Logo, impomos condições sobre eles, a fim de que a dinâmica seja bem definida de modo que não são mais necessárias as condições na fronteira. Calculamos como as quantidades físicas, como a diferença de fase entre as ondas que entram e saem, são afetadas pelas regularizações. Estes resultados são então comparados com o caso AdS para compreender as principais implicações do processo de regularização, que tem o efeito de deformar a região perto do infinito espacial do AdS. Mostramos também que, sob certas condições, a diferença de fase obtida para esses espaços deformados do AdS coincide com a obtida no caso do espaço-tempo AdS / Abstract: Analogue models are a useful tool when one wants to understand or probe phenomena in one physical system in terms of concepts from another, which may be more familiar or more easily accessed by experiments. This work explores this framework in the context of analogue models of gravity based on fluid dynamics. Particularly, we are interested in providing an analogue model for a radial fluid flow with a point source/sink at the origin. We start by considering the case where the (radial) flow velocity is constant. The resulting analogue model is then the Anti-de Sitter spacetime (AdS) which is known to be non-globally hyperbolic. As a result, the dynamics of fields in this background is not well defined until extra boundary conditions at its spatial boundary are prescribed. The fluid dynamics counterpart of these extra boundary conditions provide an effective description of the point source/sink at the origin. After that, we consider regularizations of this model near the source/sink at the origin. We then impose conditions on them in order that the dynamics is well defined so that no extra boundary conditions are required. We calculate how physical quantities, like the phase difference between ingoing and outgoing scattered waves are affected by the regularizations. These results are then compared with the AdS case to understand the main implications of the regularization, which has the effect of deforming the AdS space near its spatial infinity. We also show that, under certain conditions, the phase difference obtained for these deformed AdS spaces agrees with that obtained in the AdS case / Mestrado / Física / Mestre em Física / 1490213/2015 / CAPES

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The influence of matter described by the Vlasov equation, on the evolution of anisotropy in the spatially-homogeneous universes, called the Bianchi cosmologies, is studied. Due to the spatial-homogeneity, the Einstein equations for each Bianchi Type are reduced to a set of coupled ordinary differential equations, which has Hamiltonian form with the metric components being the canonical coordinates. In the vacuum Bianchi cosmologies, it is known that a curvature potential, which comes from the symmetries of the three-dimensional Lie groups, determines the basic properties of the evolution of anisotropy. In this work, matter potentials are constructed for Vlasov matter. They are obtained by first introducing a new matter action principle for the Vlasov equation, in terms of a conjugate pair of functions, and then enforcing the symmetry to obtain a reduction. This yields an expression for the matter potential in terms of the phase space density, which is further reduced by assuming cold streaming matter. Some vacuum Bianchi cosmologies and Type I with Vlasov matter are compared. It is shown that the Vlasov-matter potential for cold streaming matter results in qualitatively distinct dynamics from the well-known vacuum Bianchi cosmologies. / text

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