The geodesic law for test particles is one of the fundamental principles of general relativity and is extensively used. It is thought to be a consequence of the field laws but no rigorous proof exists. This thesis is concerned with a precise formulation of the geodesic law for test particles and with the extent of its validity. It will be shown to be true in certain cases but not in others. A rigorous version of the Infeld/Schild theorem is presented. Several explicit examples of both geodesic and non-geodesic motion of singularities are given. In the case of a test particle derived from a test body with a regular internal stress-energy tensor, a proof of the geodesic law for an ideal fluid test particle under plausible, explicitly stated conditions is given. It is also shown that the geodesic law is not generally true, even for weak fields and slow motion, unless the stress-energy tensor satisfies certain conditions. An explicit example using post-Newtonian theory is given showing how the geodesic law can be violated if these conditions are not satisfied. / Thesis (Ph.D.)-University of Natal, Durban, 1998.

In this thesis we seek new solutions to the anisotropic Einstein field equations which are important in the study of highly dense stellar structures. We first adopt the approach used by Maharaj & Maartens (1989) to obtain an exact anisotropic solution in terms of elementary functions for a particular choice of the energy density. This class of solution contains the Maharaj & Maartens (1989) and Gokhroo & Mehra (1994) models as special cases. In addition, we obtain six other new solutions following the same approach for different choices of the energy density. All the solutions in this section reduce to one with the energy density profile f-L ex r-2 . Two new algorithms are generated, Algorithm A and Algorithm B, which produce a new anisotropic solution to the Einstein field equations from a given isotropic solution. For any new anisotropic solution generated with the help of these algorithms, the original isotropic seed solution is regained as a special case. Two examples of known isotropic solutions are used to demonstrate how Algorithm A and Algorithm B work, and to obtain new anisotropic solutions for the Einstein and de Sitter models. Anisotropic isot~ermal sphere models are generated given the corresponding isotropic (f-L ex r-2 ) solution of the Einstein field equations. Also, anisotropic interior Schwarzschild sphere models are found given the corresponding isotropic (f-L ex constant) solution of the field equations. The exact solutions and line elements are given in each case for both Algorithm A and Algorithm B. Note that the solutions have a simple form and are all expressible in terms of elementary functions. Plots for the anisotropic factor S = J3(Pr - pJJ/2 (where Pr and Pl. are radial and tangential pressure respectively) are generated and these point to physically viable models. / Thesis (Ph.D.)-University of Natal, Durban, 2004.

In this thesis we utilise a form for the Hubble constant first proposed by Berman (1983) to study a variety of cosmological models. In particular we investigate the Robertson-Walker spacetimes, the Bianchi I spacetime, and the scalar-tensor theory of gravitation of Lau and Prokhovnik (1986). The Einstein field equations with variable cosmological constant and gravitational constant are discussed and the Friedmann models are reviewed. The relationship between observation and the Friedmann models is reviewed. We present a number of new solutions to the Einstein field equations with variable cosmological constant and gravitational constant in the Robertson-Walker spacetimes for the assumed form of the Hubble parameter. We explicitly find forms for the scale factor, cosmological constant, gravitational constant, energy density and pressure in each case. Some of the models have an equation of state for an ideal gas. The gravitational constant may be increasing in certain regions of spacetime. The Bianchi I spacetime, which is homogeneous and anisotropic, is shown to be consistent with the Berman (1983) law by defining a function which reduces to the scale factor of Robertson-Walker. We illustrate that the scalar-tensor theory of Lau and Prokhovnik (1986) also admits solutions consistent with the Hubble variation proposed by Berman. This demonstrates that this approach is useful in seeking solutions to the Einstein field equations in general relativity and alternate theories of gravity. / Thesis (M.Sc.)-University of Natal, 1992.

This thesis begins by introducing the concept of a rigid quasilocal frame (RQF) as a geometrically natural way to define an extended system in the context of the dynamical spacetime of general relativity. An RQF is defined as a two-parameter family of timelike worldlines comprising the worldtube boundary of the history of a finite spatial volume with the rigidity conditions that the congruence of worldlines is expansion-free (the ``size'' of the system is not changing) and shear-free (the ``shape'' of the system is not changing). We demonstrate that this frame exists in flat and arbitrary curved spacetimes and, moreover, exhibits the full six motional time-dependent degrees of freedom we are familiar with from Newtonian mechanics. The latter result is intimately connected with the fact that a spatial slice through the RQF - having a two-sphere topology - always admits precisely six conformal Killing vector (CKV) fields (three boosts and three rotations) associated with the action of the Lorentz group on a two-sphere. These CKVs, along with the four-velocity of observers on the RQF, are then used to quasilocally define the energy, momentum, and angular momentum inside an RQF without relying on the pre-general relativistic practice of appealing to spacetime symmetries. These quasilocal definitions for energy, momentum, and angular momentum also involve replacing the local matter-only stress-energy-momentum (SEM) tensor with the Brown-York matter plus gravity boundary SEM tensor. This allows for the construction of completely general conservation laws which describe the changes in a system in terms of fluxes across the boundary. Furthermore, since an RQF is a congruence with zero expansion and shear only relevant fluxes appear in these conservation laws - that is, fluxes due merely to changes in the size or shape of the boundary are eliminated. These resulting fluxes are simple, exact, and quantified in terms of operationally-defined geometrical quantities on the boundary and we show that they explain at a deeper level the mechanisms behind gravitational energy and momentum transfer by way of the equivalence principle. In particular, when we accelerate relative to a mass, the energy changes at a rate proportional to our acceleration times the momentum (and we propose an exact gravitational analogue of the electromagnetic Poynting vector to capture this idea). Similarly, the momentum of that object changes at a rate proportional to our acceleration times the energy. This new insight has fascinating consequences for how we should understand everyday occurrences like a falling apple - that is, the change in energy of the apple involves frame dragging while the change in momentum involves extrinsic curvature effects near the apple. Our naive general relativistic intuition tells us that these quantities should be so tiny that they should be negligible and, indeed, they are tiny but they are multiplied by huge numbers to give rise to macroscopic effects. This is how general relativity universally explains the transfer of energy and momentum but we needed rigid quasilocal frames to uncover this beautiful property of nature. Using the RQF formalism we also investigate a variety of specific problems. In particular, while looking at time-dependent rotations we discover that the reason Ehrenfest's rigid rotating disk paradox has gone unsolved for so long is that rotation introduces a subtle non-locality in time. By this we mean that, in order to maintain rigidity while undergoing time-dependent rotation, one needs to know, not only the instantaneous rotation rate, but the entire history of the motion. This makes it impossible to keep a volume of observers rigid but is doable with an RQF. We also consider RQFs in the small-sphere limit to derive many of our results and one example with particularly interesting consequences involves Bell's spaceship accelerating through an electromagnetic field. Here, we show that the change in electromagnetic energy inside the spaceship is made up of two pieces: the usual electromagnetic Poynting flux accounts for half the change while the gravitational Poynting vector equally contributes to make up the other half. This means that electromagnetism in flat spacetime generically does not tell you what is actually going on. Rather, the curvature due to the electromagnetic field necessitates a fully general relativistic treatment to get the whole story. We also use the RQF linear momentum conservation law in the context of stationary observers and fields to derive, for the first time, an exact fully general relativistic analogue of Archimedes' law. In essence, this law demonstrates that the weight of the matter and gravitational fields contained in a finite region of space is supported by the stresses (buoyant forces) acting on the boundary of that region. Furthermore, in a post-Newtonian approximation, we derive a simple set of quasilocal conservation laws which describe non-relativistic systems bound by mutual gravitational attraction. In turn, we use these laws to obtain expressions for the rates of gravitational energy and angular momentum transfer between two tidally interacting bodies - that is, the tidal heating and tidal torque - without the need to define unphysical pseudotensors. Moreover, the RQF approach explains these transfers of energy and momentum again, not as the difference of forces acting on a tidal bulge, but instead more fundamentally in the language of the equivalence principle in terms of ``accelerations relative to mass''. Throughout this work we demonstrate that the RQF approach always gives very simple, geometrical descriptions of the physical mechanisms at work in general relativity. Given that this approach also includes both matter and gravitational energy, momentum, and angular momentum and does not rely on spacetime symmetries to define these quantities, we argue that we are seeing here strong evidence that the universe is actually quasilocal in nature. We are really deeply ingrained with a local way of thinking, so shifting to a quasilocal mindset will require great effort, but we contend that it ultimately leads to a deeper understanding of the universe.

General relativity has a small number of known, exact solutions which model astronomically relevant systems. These models are highly idealized situations. Either perturbation theory or numerical simulations are typically needed to produce more realistic models. Numerical simulations are time-consuming and suffer from a difficulty in interpreting the results. In addition, global properties of numerical solutions are nearly impossible to uncover. On the other hand, standard perturbation methods are very difficult to implement beyond the second order, which means they barely scratch the surface of non-linear phenomena which distinguishes general relativity from Newtonian gravity. This work develops a method of finding exact solutions, inspired by perturbation theory, which have energy-momentum tensor components that approximately satisfy desired relationships. We find a spherical lump of matter which has a density profile $\mu \propto r^{-2}$ in a Robertson-Walker background; it looks like a galaxy in an expanding universe. We also find a plane-symmetric perturbation of a Bianchi type I metric with a density profile $\mu \propto z^{-2}$; it models a jet impacting a sheet-like structure. The former solution involves a wormhole while the latter involves a two dimensional singularity. These are both non-linear structures which perturbation theory can never produce.

Includes bibliographical references (leaves 168-179). / xiii, 179 leaves ; 30 cm. / Title page, contents and abstract only. The complete thesis in print form is available from the University Library. / Finds that the post-Newtonian approximation seems to be a better approximation of the general relativistic theory than the standard Newtonian theory. / Thesis (Ph.D.)--Adelaide University, Dept. of Physics and Mathematical Physics, 2001

Thesis (Ph. D.)--University of Notre Dame, 2008. / Thesis directed by Don Howard for the Department of History and Philosophy of Science. "July 2008." Includes bibliographical references (leaves 196-233).

Thesis (PhD)--Montana State University--Bozeman, 2009. / Typescript. Chairperson, Graduate Committee: Neil J. Cornish. Includes bibliographical references (leaves 137-140).

Thesis (Ph. D.)--University of Texas at Austin, 2004. / Supervisor: Richard Matzner. Vita. Includes bibliographical references. Available also from UMI company.

Made available in DSpace on 2016-05-17T16:50:55Z (GMT). No. of bitstreams: 0 Previous issue date: 2007-05-04. Added 1 bitstream(s) on 2016-05-17T16:54:22Z : No. of bitstreams: 1 000855801.pdf: 618196 bytes, checksum: 88ac5f6edd9a6e08f839d677db4ca1f7 (MD5) / Nesta dissertação estudamos aspectos clássicos dos modelos análogos à Relatividade Geral em matéria condensada visando sobretudo criar uma nova percepção dos efeitos gravitacionais semi-clássicos, tais como a radiação Hawking, afim de melhor compreendê-los. Neste sentido, demonstramos que as ondas sonoras se propagando em um fluido ideal, barotrópico e irrotacional sobre uma esfera 'S POT. 2' de raio r se comportam como um campo escalar de Klein-Gordon não massivo em um espaço tempo curvo. Analisamos ao longo desta dissertação diversas propriedades deste espaço-tempo efetivo sentido pelo som, cuja geometria é descrita por uma métrica lorentziana dependente das variáveis hidrodinâmicas do fluxo, como a velocidade do fluido, sua densidade e a velocidade local do som, sempre buscando estabelecer correlações entre os conceitos clássicos da dinâmica dos fluidos e conceitos puramente relativísticos. Feita uma análise mais geral destes espaços-tempos, que denominamos de espaços-tempos acústicos, nos propomos a encontrar soluções das variáveis dinâmicas do fluido, uma vez que são elas que determinam a geometria acústica, capazes de modelar espaços-tempos efetivos dotados de horizontes de eventos e singularidades, criando portanto um buraco mudo/surdo, ou seja, um análogo de um buraco negro e de buraco branco da Relatividade Geral. Discutimos ainda alguns pontos da estrutura causal dos espaços-tempos acústicos construindo assim um diagrama de Carter-Penrose do buraco mudo/surdo com o intuito de evidenciar as possíveis trajetórias nulas deste espaço-tempo. Ademais, mostramos que na aproximação da acústica geométrica, ou também aproximação eikonal, os raios de som seguem geodésicas tipo luz do espaço-tempo acústico. Por fim, calculamos a curvatura escalar deste espaço-tempo verificando a presença... / In this dissertation we study the classical aspects of analogue models of General Relativity in condensed matter seeking mainly to create a new perception about semi-classical gravitational effects, such as Hawking radiation, in order to better comprehend them. We demonstrate that sound waves propagating in an ideal barotropic fluid with a non-homogeneous irrotacional flow, over a sphere 'S POT. 2' with radius r behave as a Klein-Gordon massless scalar field in a curved spacetime. Through this dissertation, we analyze several properties of this effective spacetime governing the propagation of sound, whose geometry is described by a Lorentzian metric that depends on the hydrodynamic variables of the flow such as the flow velocity, the density and the local speed of sound, always trying to establish correlations between classical concepts of fluid dynamics and purely relativistic concepts. Once a general analysis of these spacetimes is made, which we denominate acoustic spacetimes, we find solutions of the dynamic variables of the fluid, since they determine the acoustic geometry, capable of modeling effective spacetimes endowed with event horizons and singularities, creating therefore a dumb/deaf hole, i.e., an analogue of a black hole and white hole of the General Relativity. We further discuss some points of the causal structure of the acoustic spacetimes, so constructing a Carter-Penrose diagram of the dumb/deaf hole with the aim of bringing to evidence the possible null trajectories of this spacetime. Furthermore, we show that in the approximation of the acoustic geometry, also called eikonal approximation, the sound rays follow lightlike geodesics of the acoustic spacetime. Finally we calculate the scalar curvature of this spacetime verifying the presence of the non flat structure of the 'S POT. 2' sphere, over which the fluid moves