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This report studies the generation of new solutions to Einstein's field equations in general relativity by the method of sigma models. If, when projected from four to three dimensions, the relativistic action decouples into a gravity term and a non-linear sigma model term, target space isometries of the sigma model can be found that correspond to generating new solutions. We give a self-contained description of the method and relate it to the early articles through which the method was introduced. We discuss the virtues of the method and how it is used today. We find that it is a powerful technique of finding new solutions and can also give insight to the general features of the theory. We also identify some possible further developments of the method.

In a large class of space-times, the specification of the curvature tensor components Rabcd in some coordinate domain of the space-time uniquely determines the metric up to a constant conformal factor. The purpose of this thesis is to investigate the spaces where the metric is not so determined, and to look at the determination of the metric when the components of the derivatives of the Riemann tensor (one index up) are also specified, with special reference to the role of the infinitesimal holonomy group (ihg). In chapter one we set up the mathematical background, describing the Weyl and Ricci tensor classifications, and defining holonomy. In chapter two we look at spaces with Riemann tensors of low rank. This leads us on to decomposable spaces and the connection between decomposable spaces and relativity in three dimensions. We examine the connection between decomposability and the ihg, and relate this to the Weyl and Ricci tensor classifications. In chapter three we discuss the problem of determination of the metric by the Riemann tensor alone, and give a brief review of the history of the problem. In chapter four we go on to look at the determination of the metric by the curvature and its derivatives. It is shown that, with the exception of the generalised pp-waves, we only need look as far as the first derivatives of the Riemann tensor to obtain the best determination of the metric, unless the Riemann tensor is rank 1, when the second derivatives may also be required. The form of the metric ambiguity, the ihg and Petrov types are determined in each case. These results are then reviewed in the final chapter.

Although the study of the initial value problem in General Relativity started in the decade of 1950 with the work of Foures-Bruhat, addressing the problem of global non-linear stability of solutions to the Einstein field equations is in general a hard problem. The first non-linear global stability result in General Relativity was obtained for the de-Sitter spacetime by means of the so-called conformal Einstein field equations introduced by H. Friedrich in the decade of 1980. The latter constitutes the main conceptual and technical tool for the results discussed in this thesis. In Chapter 1 the physical and geometrical motivation for these equations is discussed. In Chapter 2 the conformal Einstein equations are presented and first order hyperbolic reduction strategies are discussed. Chapter 3 contains the first result of this work; a second order hyperbolic reduction of the spinorial formulation of the conformal Einstein field equations. Chapter 4 makes use of the latter equations to give a discussion of the non-linear stability of the Milne universe. Chapter 5 is devoted to the analysis of perturbations of the Schwarzschild-de Sitter spacetime via suitably posed asymptotic initial value problems. Chapter 6 gives a partial generalisation of the results of Chapter 5. Finally a result relating the Newman-Penrose constants at future and past null infinity for spin-1 and spin-2 fields propagating on Minkowski spacetime close to spatial infinity is discussed in Chapter 7 exploiting the framework of the cylinder at spatial in nity. Collectively, these results show how the conformal Einstein field equations and more generally conformal methods can be employed for analysing perturbations of spacetimes of interest and extract information about their conformal structure.

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Thesis (Ph. D.)--University of Missouri-Columbia, 2006. / The entire dissertation/thesis text is included in the research.pdf file; the official abstract appears in the short.pdf file (which also appears in the research.pdf); a non-technical general description, or public abstract, appears in the public.pdf file. Title from title screen of research.pdf file viewed on (Mar. 1, 2007). Vita. Includes bibliographical references.

Singularities have been a long-standing problem in general relativity. In all other fields of physics, singularities can be easily located and avoided; in general relativity, singularities have an impact on the creation of the manifold, but, by definition, are not even part of real spacetime. Moreover, all singularities in general relativity cannot be treated in the same manner; thus, the classification of singularities is essential in order to understand them. One important class of singularities is curvature singularities, which, in some cases, can be subclassified as central, shell focusing or shell crossing singularities. We propose to further classify curvature singularities as either gravitational or non-gravitational. In general relativity, a curvature singularity is ``located'' where the scalar invariants of the spacetime are undefined. The gradient field of a non-zero scalar invariant can then be calculated, and the end points of the associated integral curves can be determined. If integral curves are attracted to (i.e. intersect) the singularity, then it is a gravitational singularity; if the integral curves avoid the singularity, then it is a non-gravitational singularity. We will test our method by analysing several different spacetimes, including Friedman-Lemaitre-Robertson-Walker, Schwarzschild, self-similar Vaidya, self-similar Tolman-Bondi, non-self-similar Vaidya, and Kerr spacetimes. We find that in every case studied, the integral curves have specific end points, therefore they can be used to classify a curvature singularity as gravitational or non-gravitational. In Friedman-Lemaitre-Robertson-Walker and Schwarzschild spacetimes, we determined that the a(t) = 0 and r = 0 singularities, respectively, are gravitational singularities. In Vaidya and Tolman-Bondi spacetime, we determine that the massless shell focusing singularities are non-gravitational singularities and that the central singularities (which have mass) are gravitational singularities. We also find that the non-gravitational singularities are the only singularities that have the possibility of being naked. In summary, we can determine which singularities are gravitational and which are non-gravitational by our method of examining the end points of the integral curves, which are constructed from the gradient field of scalar invariants. / Thesis (Master, Physics, Engineering Physics and Astronomy) -- Queen's University, 2008-07-21 17:13:27.992

The invariant four-volume ($\mathcal V$) of a complete four-dimensional black hole (the volume of the spacetime at and interior to the horizon) diverges. However, if one considers the black hole resulting from the gravitational collapse of an object and integrates only a finite time to the future of the collapse, the resultant volume is well-defined and finite. We show that for non-degenerate black holes, the volume in this case can be written as $\mathcal V \propto \ln|\lambda|$, where lambda is the affine generator of the horizon and we define our volume $\mathcal V^*$ to be the constant of proportionality. In spherical symmetry, this is the Euclidean volume divided by the surface gravity ($\kappa$). More generally, it turns out that $\mathcal V^*$ is the Parikh volume $({}^3 \mathcal V^*)$ divided by $\kappa$. This allows us to define an alternative local and invariant definition of the surface gravity of a stationary black hole. It also encourages us to find a generalization of the Parikh volume (which depends on the existence of an asymptotically timelike Killing vector) to any region of space or spacetime of arbitrary dimension, provided that this space or spacetime contains a Killing vector. We find some properties of this generalized ``Killing volume'' and rewrite our volume as a Killing volume for a particular Killing vector. We revisit the laws of black hole mechanics, considering them in terms of volumes rather than areas, by writing out our volume and the Parikh volume of Kerr-Newman black holes and then considering their variation with respect to the parameters $M$, $J$ and $Q$ to find a modified BH mechanics first law. We also use our new definition of $\kappa$ to develop an alternate demonstration of the BH mechanics third law. We note that the Parikh volume of a Kerr-Newman black hole is equal to $A r_+/3$, where $A$ is the horizon surface area and $r_+$ the value of the radius at the horizon, and we offer some interpretations of this relationship. We review some other relevant work by Parikh as well as some by Cveti\v{c} et al. and by Hayward. We point out some possible next steps to follow up on the work in this thesis. / Thesis (Master, Physics, Engineering Physics and Astronomy) -- Queen's University, 2012-06-04 15:58:03.984

We give a review of the construction and application of spinor fields in general relativity and an account of the spinor-based Geroch-Held-Penrose (GHP) formalism. Specifically, we discuss using the GHP formalism to integrate Einstein's equations as suggested by Held  and developed by Edgar and Ludwig and discuss the similaritites with the Cartan-Karlhede classification of spacetimes. We use this integration method to find a one-parameter subclass and a degenerate case, for which the Cartan-Karlhede algorithm terminates at second order, of the Petrov type III, vacuum Robinson-Trautman metrics. We use the GHP formalism to find the Killing vectors, using theorems by Edgar and Ludwig. The one-parameter family admits exactly two Killing fields, whereas the degenerate case admits three and is Bianchi type VI. Finally we use the Cartan-Karlhede algorithm to show that our class, including the degenerate case, is equivalent to a subclass found by Collinson and French. Our degenerate case corresponds to an example metric given by Robinson and Trautman and is known to be the unique algebraically special vacuum spacetime with diverging rays and a three-dimensional isometry group.

Thesis (M.Sc.)--Victoria University of Wellington, 2005. / Includes bibliographical references.