Spacetimes with symmetry play a critical role in Einstein's Theory of General Relativity. Missing from the literature is a correct, usable, and computer accessible classification of such spacetimes. This dissertation fills this gap; specifically, we i) give a new and different approach to the classification of spacetimes with symmetry using modern methods and tools such as the Schmidt method and computer algebra systems, resulting in ninety-two spacetimes; ii) create digital databases of the classification for easy access and use for researchers; iii) create software to classify any spacetime metric with symmetry against the new database; iv) compare results of our classification with those of Petrov and find that Petrov missed six cases and incorrectly normalized a significant number of metrics; v) classify spacetimes with symmetry in the book Exact Solutions to Einstein’s Field Equations Second Edition by Stephani, Kramer, Macallum, Hoenselaers, and Herlt and in Komrakov’s paper Einstein-Maxwell equation on four-dimensional homogeneous spaces using the new software.

Einstein's field equations are extremely difficult to solve, and when solved, the solutions are even harder to understand. In this thesis, two analysis tools are developed to explore and visualize the curvature of spacetimes. The first tool is based on a thorough examination of observer independent curvature invariants constructed from different contractions of the Riemann curvature tensor. These invariants are analyzed through their gradient fields, and attention is given to the resulting flow and critical points. Furthermore, we propose a Newtonian analog to some general relativistic invariants based on the underlying physical meaning of these invariants, where they represent the cumulative tidal and frame-dragging effects of the spacetime. This provides us with a novel and intuitive tool to compare Newtonian gravitational fields to exact solutions of Einstein's field equations on equal footing. We analyze the obscure Curzon-Chazy solution using the new approach, and reveal rich structure that resembles the Newtonian gravitational field of a non-rotating ring, as it has been suspected for decades. Next, we examine the important Kerr solution, which describes the gravitational field of rotating black holes. We discover that the observable part of the geometry outside the black hole's event horizon depends significantly on its angular momentum. The fields representing the cumulative tidal and frame-dragging forces change qualitatively at seven specific values of the dimensionless spin parameter of the black hole. The second tool we develop in this thesis is the accurate construction of the Penrose conformal diagrams. These diagrams are a valuable tool to explore the causal structure of spacetimes, where the entire spacetime is compactified to a finite size, and the coordinate choice is fixed such that light rays are straight lines on the diagram. However, for most spacetimes these diagrams can only be constructed as a qualitative guess, since their null geodesics cannot be solved. We developed an algorithm to construct very accurate Penrose diagrams based on numeric solutions to the null geodesics, and applied it to the McVittie metric. These diagrams confirmed the long held suspicion that this spacetime does indeed describe a black hole embedded in an isotropic universe. / Thesis (Ph.D, Physics, Engineering Physics and Astronomy) -- Queen's University, 2013-09-30 14:02:55.865

This thesis contributes with a countryside perspective to queer research by highlighting the countryside as a context where queer lives are lived. In the thesis I problematize the city/countryside divide with a view of the concept of queer as dependent on space and time.  The empirical materials are generated through a workshop on queerness, gathering people living within a countryside context; the materials consist of a discussion and written responses to questions on queerness and the city/countryside binary. Theoretically and methodologically, the thesis is inspired by the notion of agential realism (Barad 2007) and situated knowledge, (Haraway 1988); the use of creative writing, inspired by Richardson (1994 and 2000), has also been central to the development of the thesis. The analysis is carried out within themes focusing on conditions for queerness within city/countryside experienced by people situated in the countryside. The analysis shows how space, time, contexts and intersections are entangled and queering the city/countryside divide.

The scalar wave equation is investigated for a scalar field propagating in a spacetime background ds²=e^{2a}(-dt²+dr²)+R(e^{-2ψ}dφ²+e^{2ψ}dz²). The metric is compactified in the radial direction. The spacetime slices of constant φ and z are foliated into outgoing null hypersurfaces by the null coordinate transformation u=t-r. The scalar field imitates the amplitude behavior of a light ray, or a gravitational wave, traveling along a null hypersurface when the area function R is a constant or is a function of u. These choices for R restrict the gravitational wave factor ψ to being an arbitrary function of u.

In General Relativity, the Einstein field equations allow us to study the evolution of a spacelike 3-manifold, provided that its metric and extrinsic curvature satisfy a system of geometric constraint equations. The so-called Einstein constraint equations, arise as a consequence of the fact that the 3-manifold in question is necessarily a submanifold of the spacetime its evolution defines. This thesis is devoted to a study of the structure of the Einstein constraint system in the special case when the spacelike 3-manifold also satisfies the quasispherical ansatz of Bartnik [B93]. We make no mention of the generality of this gauge; the extent to which the quasispherical ansatz applies remains an open problem. After imposing the quasispherical gauge, we give an argument to show that the resulting Einstein constraint system may be viewed as a coupled system of partial differential equations for the parameters describing the metric and second fundamental form. The hencenamed quasisperical Einstein constraint system, consists of a parabolic equation, a first order elliptic system and (essentially) a system of ordinary differential equations. The question of existence of solutions to this system naturally arises and we provide a partial answer to this question. We give conditions on the initial data and prescribable fields under which we may conclude that the quasispherical Einstein constraint system is uniquley solvable, at least in a region surrounding the unit sphere. The proof of this fact is centred on a linear iterative system of partial differential equations, which also consist of a parabolic equation, a first order elliptic system and a system of ordinary differential equations. We prove that this linear system consistently defines a sequence, and show via a contraction mapping argument, that this sequence must converge to a fixed point of the iteration. The iteration, however, has been specifically designed so that any fixed point of the iteration coincides with a solution of the quasispherical Einstein constraints. The contraction mapping argument mentioned above, relies heavily on a priori estimates for the solutions of linear parabolic equations. We generalise and extend known results 111 concerning parabolic equations to establish special a priori estimates which relate a useful property: the L2-Sobolev regularity of the solution of a parabolic equation is greater than that of the coefficients of the elliptic operator, provided that the initial data is sufficiently regular. This 'smoothing' property of linear parabolic equations along with several estimates from elliptic and ordinary differential equation theory form the crucial ingredients needed in the proof of the existence of a fixed point of the iteration. We begin in chapter one by giving a brief review of the extensive literature concerning the initial value problem in General Relativity. We go on, after mentioning two of the traditional methods for constructing spacetime initial data, to introduce the notion of a quasispherical foliation of a 3-manifold and present the Einstein constraint system written in terms of this gauge. In chapter two we introduce the various inequalities and tracts of analysis we will make use of in subsequent chapters. In particular we define the, perhaps not so familiar, complex differential operator 9 (edth) of Newman and Penrose. In chapter three we develop the appropriate Sobolev-regularity theory for linear parabolic equations required to deal with the quasispherical initial data constraint equations. We include a result due to Polden [P] here, with a corrected proof. This result was essential for deriving the results contained in the later chapters of [P], and it is for this reason we include the result. We don't make use of it explicitly when considering the quasispherical Einstein constraints, but the ideas employed are similar to those we use to tackle the problem of existence for the quasispherical constraints. Chapter four is concerned with the local existence of quasispherical initial data. We firstly consider the question of existence and uniqueness when the mean curvature of the 3-manifold is prescribed, then after introducing the notion of polar curvature, we also present another quasispherical constraint system in which we consider the polar curvature as prescribed. We prove local existence and uniqueness results for both of these alternate formulations of the quasispherical constraints. This thesis was typeset using LATEXwith the package amssymb.

<p>In general relativity, the Riemannian Penrose inequality (RPI) provides a lower bound for the ADM mass of an asymptotically flat manifold of nonnegative scalar curvature in terms of the area of the outermost minimal surface, if one exists. In physical terms, an equivalent statement is that the total mass of an asymptotically flat spacetime admitting a time-symmetric spacelike slice is at least the mass of any black holes that are present, assuming nonnegative energy density. The main goal of this thesis is to deduce geometric lower bounds for the ADM mass of manifolds to which neither the RPI nor the famous positive mass theorem (PMT) apply. This is the case, for instance, for manifolds that contain metric singularities or have boundary components that are not minimal surfaces.</p> <p>The fundamental technique is the use of conformal deformations of a given Riemannian metric to arrive at a new Riemannian manifold to which either the PMT or RPI applies. Along the way we are led to consider the geometry of certain types non-smooth metrics. We prove a result regarding the local structure of area-minimizing hypersurfaces with respect such metrics using geometric measure theory.</p> <p>One application is to the theory of ``zero area singularities,'' a type of singularity that generalizes the degenerate behavior of the Schwarzschild metric of negative mass. Another application deals with constructing and understanding some new invariants of the harmonic conformal class of an asymptotically flat metric.</p> / Dissertation

This thesis is divided into three main parts. In the first of these (comprising chapters 1 and 2) we present the physical context of the research and cover the basic geometric background we will need to use throughout the rest of this thesis. In the second part (comprising chapters 3 to 5) we motivate and develop the strong homogeneity theorem for supergravity backgrounds. We go on to prove it directly for a number of top-dimensional Poincaré supergravities and furthermore demonstrate how it also generically applies to dimensional reductions of those theories. In the third part (comprising chapters 6 and 7) we show how further specialising to the case of symmetric backgrounds allows us to compute complete classifications of such backgrounds. We demonstrate this by classifying all symmetric type IIB supergravity backgrounds. Next we apply an algorithm for computing the supersymmetry of symmetric backgrounds and use this to classify all supersymmetric symmetric M-theory backgrounds.

The thesis starts with a brief introduction to the algebraic classificati- on of tensors and spacetimes in higher dimensions. Attempts to generalize the Goldberg-Sachs theorem are also discussed. There is a summary of main results for optical matrices of algebraically special spacetimes in higher dimensions. The optical matrix for a type III spacetime in six dimensions is found using Bianchi identities. A few properties of type III optical matrices in a general dimension are also found. Various properties of equations obtained from Bianchi identities for type III spacetimes are studied in appendices. 1

The main focus of the work is the construction of a quantum spacetime emerging from theory.

京都大学 / 0048 / 新制・課程博士 / 博士(理学) / 甲第18794号 / 理博第4052号 / 新制||理||1583(附属図書館) / 31745 / 京都大学大学院理学研究科物理学・宇宙物理学専攻 / (主査)教授 川合 光, 准教授 福間 將文, 教授 田中 貴浩 / 学位規則第4条第1項該当 / Doctor of Science / Kyoto University / DFAM