Characterizing the long-time behavior of solutions to the Einstein field equations remains an active area of research today. In certain types of coordinates the Einstein equations form a coupled system of quasilinear wave equations. The investigation of the nature and properties of solutions to these equations lies in the field of geometric analysis. We make several contributions to the study of solution dynamics near singularities. While singularities are known to occur quite generally in solutions to the Einstein equations, the singular behavior of solutions is not well-understood. A valuable tool in this program has been to prove the existence of families of solutions which are so-called asymptotically velocity term dominated (AVTD). It turns out that a method, known as the Fuchsian method, is well-suited to proving the existence of families of such solutions. We formulate and prove a Fuchsian-type theorem for a class of quasilinear hyperbolic partial differential equations and show that the Einstein equations can be formulated as such a Fuchsian system in certain gauges, notably wave gauges. This formulation of Einstein equations provides a convenient general framework with which to study solutions within particular symmetry classes. The theorem mentioned above is applied to the class of solutions with two spatial symmetries -- both in the polarized and in the Gowdy cases -- in order to prove the existence of families of AVTD solutions. In the polarized case we find families of solutions in the smooth and Sobolev regularity classes in the areal gauge. In the Gowdy case we find a family of wave gauges, which contain the areal gauge, such that there exists a family of smooth AVTD solutions in each gauge.

Durante os últimos 90 anos temos visto o grande esplendor que a teoria da relatividade geral de Einstein alcançou em suas diversas previsões. Esta dissertação é um estudo a respeito desta teoria e suas extrapolações. Falaremos de início acerca da primeira solução das equações de Einstein para buracos negros obtida por Karl Schwarzschild em 1916: o buraco negro esfericamente simétrico e sem carga. Trataremos das possíveis órbitas neste tipo de solução bem como de perturbações gravitacionais e escalares. Ainda utilizando a solução de Schwarzschild, adentraremos os tópicos desenvolvidos recentemente em um tratamento semi-clássico da relatividade: a termodinâmica dos buracos negros. Posteriormente estudaremos as novas teorias com base na relatividade geral, que resolvem o problema da hierarquia buscando por dimensões extras em nosso Universo. Em tal contexto analisamos por exemplo como se comportam os buracos negros nestes Universos com mais do que 4 dimensões. Porém, estudamos perturbações gravitacionais em uma corda negra chegando a averiguar a presença de uma instabilidade para modos com comprimento de onda maiores do que o horizonte da corda (em uma aproximação linear), e demonstramos que em uma das possíveis soluções do problema da Hierarquia (Universos de Randall Sundrum) não existem atalhos gravitacionais mesmo para branas não planas (extrapolação dos Universos de Randall-Sundrum). / Over the last 90 years Einstein`s Theory of General Relativity has had a tremendous success in all its predictions. This dissertation is concerned with the study of this theory and its extrapolations. We begin with the first solution of the Einstein equations for black holes obtained by Karl Schwarzschild at 1916: the spherically symmetric black hole without charge, obtaining the orbits and the scalar and gravitational perturbations around the metric. We also consider the recent developments in black hole thermodynamics via a semiclassical approach to the theory. Subsequently, we study the new theories based on general relativity extrapolations, which solve the hierarchy problem proposing extra dimensions in our Universe. In this context we analyze for example the behavior of black holes in Universes with more than 4 dimensions. Finally, we study the gravitational perturbations in a black string showing the presence of unstable modes with wave length bigger than the black string horizon. We also show that in one of the possible Universes which solve the hierarchy problem there are no gravitational shortcuts even for non-at branes (an extrapolation of Randall-Sundrum Universes).

The theory of General Relativity has been in existence for 90 years and has stood up to all tests it has been subjected to in that time. The PPN parameter is a measure of the accuracy of theories of gravity and assumes different values in different theories. By measuring the Shapiro time delay of light it is possible to constrain and thereby constrain gravitational theories. This Shapiro time delay can be measured in our solar system but it is only in the vicinity of extremely compact objects such as pulsars and black holes that it can be tested under the immense gravitational fields that can only be found there. A pulsar in a binary orbit about another compact object is the ideal system in which to test this effect. In this work we have gone from Kepler’s laws of simple planetary motion to deriving the equations that explain binary orbits to incorporating General Relativity into these equations in order to obtain the equations for relativistic particle orbits. We then evolved this theory even further so as to be able to explain relativistic light ray orbits and then used this knowledge to model the Shapiro delay in a binary system. With a working model it became possible to simulate the Shapiro delay in a wide range of possible systems and then to use these simulations to say something about what type of system should be focussed on in future so as to measure the Shapiro delay and thereby constrain more tightly the parameter / Dr. C.A. Engelbrecht Dr. F.A.M. Frescura

This dissertation examines two fundamentally different types of isolated systems in general relativity. In part 1, an exact solution of the Einstein-Maxwell equations representing the exterior field of two arbitrary charged essentially spherically symmetric (Reissner-Nordström) bodies in equilibrium is studied. Approximate solutions representing the gravitational- electrostatic balance of two arbitrary point sources in general relativity have led to contradictory arguments in the literature with respect to the condition of balance. Up to the present time, the only known exact solutions which can be interpreted as the nonlinear superposition of two Reissner-Nordström bodies without an intervening strut has been for critically charged masses, [special characters omitted]. In this dissertation . the invariant physical charge for each source is found by direct integration of Maxwell's equations. The physical mass for each source is invariantly defined in a manner similar to which the charge was found. It is shown that balance without tension or strut can occur for non-critically charged bodies. It is demonstrated that other authors have not identified the correct physical parameters for the masses and charges of the sources. Examination of the fundamental parameters of the space-time suggests a refinement of the nomenclature used to describe the physical properties is necessary. Such a refinement is introduced. Further properties of the solution, including the multipole structure and comparison with other parameterizations, are examined. Part 2 investigates the viability of constructing gravitational and electromagnetic geons: zero-rest-mass field concentrations, consisting of gravitational or electromagnetic waves, held together for long periods of time by their gravitational attraction. In contrast to an exact solution, the method studied involves solving the Einstein or Einstein-Maxwell equations for perturbations on a static background metric in a self-consistent manner. The Brill-Hartle gravitational geon construct as a spherical shell of small amplitude, high-frequency gravitational waves is reviewed and critically analyzed. The spherical shell in the proposed Brill-Hartle geon cannot be regarded as an adequate geon construct because it does not meet the regularity conditions required for a non-singular source. An attempt is made to build a non- singular solution to meet the requirements of a gravitational geon. Construction of a geon requires gravitational waves of high-frequency and the field equations are decomposed accordingly. A geon must also possess the property of quasi-stability on a time-scale longer than the period of the comprising waves. It is found that only unstable equilibrium solutions to the gravitational and electromagnetic geon problem exist. A perturbation analysis to test the requirement of quasi-stability resulted in a contradiction. Thus it could not be concluded that either electromagnetic or gravitational geons meet all the requirements for existence. The broader implications of the result are discussed with particular reference to the problem of with particular reference to the problem of gravitational energy. / Graduate

The first part of this thesis is concerned with the question of global uniqueness of solutions to the initial value problem in general relativity. In 1969, Choquet-Bruhat and Geroch proved, that in the class of globally hyperbolic Cauchy developments, there is a unique maximal Cauchy development. The original proof, however, has the peculiar feature that it appeals to Zorn’s lemma in order to guarantee the existence of this maximal development; in particular, the proof is not constructive. In the first part of this thesis we give a proof of the above mentioned theorem that avoids the use of Zorn’s lemma. The second part of this thesis investigates the behaviour of so-called Gaussian beam solutions of the wave equation - highly oscillatory and localised solutions which travel, for some time, along null geodesics. The main result of this part of the thesis is a characterisation of the temporal behaviour of the energy of such Gaussian beams in terms of the underlying null geodesic. We conclude by giving applications of this result to black hole spacetimes. Recalling that the wave equation can be considered a “poor man’s” linearisation of the Einstein equations, these applications are of interest for a better understanding of the black hole stability conjecture, which states that the exterior of our explicit black hole solutions is stable to small perturbations, while the interior is expected to be unstable. The last part of the thesis is concerned with the wave equation in the interior of a black hole. In particular, we show that under certain conditions on the black hole parameters, waves that are compactly supported on the event horizon, have finite energy near the Cauchy horizon. This result is again motivated by the investigation of the conjectured instability of the interior of our explicit black hole solutions.

The chief aim of this work is to investigate cosmological consequences of a hypothesis put forward by Morrison and Gold in 1956. These authors postulate the existence of equal amounts of matter and antimatter in our universe. Abandoning the principle of equivalence, they attribute negative gravitational mass to anti-nucleons. The result is a drastic alteration in the field equation for the gravitational potential. In the first three chapters Newtonian Cosmology is developed from basic principles. The equations describing a universe consisting of matter are set up and solved. In chapter IV the hypothesis of Morrison and Gold is introduced, and the resulting model for the universe is compared with models obtained in chapter III. It is concluded that within the framework of the model considered, the hypothesis of Morrison and Gold is incompatible with the observational evidence, because it leads to an age of the universe of between 1.3 and 1.9 billion years, which is less than the age derived from other geological and astrophysical data. / Science, Faculty of / Physics and Astronomy, Department of / Graduate

This thesis examines the relationship between Leibniz’s Principle of Identity of Indiscernibles and symmetry. In his 1717 correspondence with Samuel Clarke, Leibniz argued that “There is no such thing as a pair of individuals that are indiscernible from each other” (Leibniz 16). In other words, any objects sharing all their properties are in fact one and the same object. This is Leibniz’s Principle of Identity of Indiscernibles (the “PII”). The principle and its converse Leibniz’s Law express a conditional relationship between the identity of an object and its properties. Our investigation will use applications of Leibniz’s principles from the history of philosophy to examine this relationship and we’ll find that imperfect applications result in either perfect qualitative identity between multiple objects (“multiple indiscernibles”), imperfect qualitative identity between multiple objects (“incongruent counterparts”), or, finally, a relative identity between two facets of one object (a “singular discernible”). My project will also trace the historical thread leading from Leibniz to the development of symmetry groups in mathematics. Leibniz’s principles are embedded in science’s ability to distinguish the objective from the subjective, owing to their usefulness discerning an object’s intrinsic properties (properties belonging to the object itself) from extrinsic properties (properties based in relations the object is in with other objects). Symmetry is the relativity of identity, and the PII is an exploratory instrument illuminating this relationship: it injects structure into investigations of identity, but also affords the opportunity to capture pre-existing convictions about identity a thinker brings to the application.

This thesis, a Bibliography of Relativity, has begun with a very meager knowledge of the amount of literature which has been written on the subject of relativity either s explanation or commentary. The project was to 11at all the books and periodical articles written on or about the subject Relativity which are on file in the libraries of the College of the Pacific, University of California and stanford up to and including the year 1920.

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