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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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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

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 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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