Thesis (Ph. D.)--University of Wisconsin--Madison, 1981. / Typescript. Vita. eContent provider-neutral record in process. Description based on print version record. Includes bibliographical references (leaves 294-331).

Thesis (Ph. D.)--University of Maryland, 1973. / Includes vita. Includes bibliographical references (leaves 156-163).

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.

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.

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Brown and York's gravitational microcanonical density of states is extended to general spacetime dimension and shown to be dependent upon features of the 4 dimensional gravitational action for its interpretation. Black hole entropy is calculated from the density of states path integral in general spacetime dimension, and the interpretation is shown to be likewise dependent upon the dimension of spacetime. The entropy of de Sitter and Rindler horizons are calculated using the black hole density of states and the notion of local horizon entropy density is shown to be supported. The applicability of the microcanonical ensemble to black hole mechanics is discussed at a fundamental level focussing on the absence of angular velocity as an external parameter in the gravitational Hamiltonian. The rotational ensemble and a new ensemble - the angular momentum ensemble - are introduced following Jaynes' information theory approach to statistical mechanics and proposed as more compelling candidates to calculate black hole entropy as a function of state. A program to calculate the density of states path integral non-perturbatively using causal dynamical triangulations is initiated. Regge calculus expressions for extrinsic curvature are extended to the case of Lorentzian hypersurfaces and used to derive Regge calculus expressions for quasilocal energy-momentum. The Regge version of the black hole density of states action is derived and specialised to the 3d and 4d spacetime constructions of causal dynamical triangulations. Finally, the recent suggestion that entropy is observer dependent is shown to be incompatible with the Tolman law for the equilibrium temperature in a gravitational field.