Kwong, Kwok Kun. / Thesis (Ph.D.)--Chinese University of Hong Kong, 2011. / Includes bibliographical references (leaves 66-70) and index. / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Abstract also in Chinese.

Master of Science / General Relativity (GR) is founded on the revolutionary idea that space and time are merely parts of a greater, unified whole: spacetime. Furthermore, the force we know as gravity results from the bending and stretching of the geometry of spacetime by its energetic contents. GR is notorious for its mathematical complexity and subtlety, meaning that an intuitive understanding of a spacetime is difficult. One of the best approaches to studying the properties of a given spacetime is to consider its geodesic structure—that is, to consider the motion of unaccelerated, “free-falling” particles. This report presents the results of such a study into two important spacetimes — the Kerr solution for a rotating black hole, and the Robertson-Walker solution for a homogeneous universe.

In this thesis we first apply the 1+3 covariant description of general relativity to analyze n-fluid Friedmann-Lemaitre (FL) cosmologies; that is, homogeneous and isotropic cosmologies whose matter-energy content consists of n non-interacting fluids. We are motivated to study FL models of this type as observations suggest the physical universe is closely described by a FL model with a matter content consisting of radiation, dust and a cosmological constant. Secondly, we use the 1+3 covariant description to analyse scalar, vector and tensor perturbations of FL cosmologies containing a perfect fluid and a cosmological constant. In particular, we provide a thorough discussion of the behaviour of perturbations in the physically interesting cases of a dust or radiation background.

Two different works in mathematical physics are presented: A construction of conformal infinity in null and spatial directions is constructed for the Rainbow-flat space-time corresponding to doubly special relativity. From this construction a definition of asymptotic DSRness is put forward which is com- patible with the correspondence principle of Rainbow gravity. Furthermore a result equating asymptotically flat space-times with asymptotically DSR spacetimes is presented. An overview of microlocality in braided ribbon networks is presented. Follow- ing this, a series of definitions are presented to explore the concept of microlocality and the topology of ribbon networks. Isolated substructure of ribbon networks are introduced, and a theorem is proven that allows them to be relocated. This is fol- lowed by a demonstration of microlocal translations. Additionally, an investigation into macrolocality and the implications of invariants in braided ribbon networks are presented.

In this thesis we first apply the 1+3 covariant description of general relativity to analyze n-fluid Friedmann-Lemaitre (FL) cosmologies; that is, homogeneous and isotropic cosmologies whose matter-energy content consists of n non-interacting fluids. We are motivated to study FL models of this type as observations suggest the physical universe is closely described by a FL model with a matter content consisting of radiation, dust and a cosmological constant. Secondly, we use the 1+3 covariant description to analyse scalar, vector and tensor perturbations of FL cosmologies containing a perfect fluid and a cosmological constant. In particular, we provide a thorough discussion of the behaviour of perturbations in the physically interesting cases of a dust or radiation background.

Two different works in mathematical physics are presented: A construction of conformal infinity in null and spatial directions is constructed for the Rainbow-flat space-time corresponding to doubly special relativity. From this construction a definition of asymptotic DSRness is put forward which is com- patible with the correspondence principle of Rainbow gravity. Furthermore a result equating asymptotically flat space-times with asymptotically DSR spacetimes is presented. An overview of microlocality in braided ribbon networks is presented. Follow- ing this, a series of definitions are presented to explore the concept of microlocality and the topology of ribbon networks. Isolated substructure of ribbon networks are introduced, and a theorem is proven that allows them to be relocated. This is fol- lowed by a demonstration of microlocal translations. Additionally, an investigation into macrolocality and the implications of invariants in braided ribbon networks are presented.

The continuous, kinematical Hilbert space of loop quantum gravity is built upon a family of spaces $\mathcal{H}_\Gamma$, each associated to a different \textit{graph} $\Gamma$, i.e. a network of interconnected one-dimensional links $\l$, embedded within a spatial geometry. The kinematics of loop quantum gravity are well-established, but difficult problems remain for the dynamics. There are two steps in getting to the quantum theory from the classical one: first, the embedded graphs are used to define a smearing of the continuous gravitational fields to obtain a holonomy $h_\l$ and flux $\X_\l$ for each link of the graph, giving a phase space $P_\Gamma$; second, this phase space is quantized to yield a finite dimensional Hilbert space $\mathcal{H}_\Gamma$. The intermediate classical theory in terms of $P_\Gamma$ phase spaces remains largely unexplored, and here we endeavour to develop it. If we can find such a theory that is consistent with general relativity, then we will have a theory of gravity based upon finite-dimensional phase spaces that is nicely set up for quantization \`a la loop quantum gravity. To begin, we first review the basic elements of the quantum theory before introducing the classical phase space structure. Within this framework we show that there is a one-to-one correspondence between the data on a graph and an equivalence class of continuous geometries. We find that a particular member of each class, the spinning geometry, makes a promising candidate as a gauge choice to represent the $(h_\l, \X_\l)$ data in the continuous theory, helping us to formulate a dynamics for the discrete theory. Considering all of the possible graphs, it is important to know how we can evolve from one phase space into another, and how the dynamics in $P_\Gamma$ relates to the continuous evolution. There is a geometrical description of phase spaces where dynamics appears as a class of subspaces within a symplectic manifold. We use this picture to formulate a dynamics between $P_\Gamma$ phase spaces, and demonstrate this process on a simple model that mimics the case of full gravity. Following this, we study a system of point particles in three-dimensional gravity which provides an illuminating demonstration of what we hope to accomplish for full gravity. We develop the classical theory of point particles and show that it can be described by an evolving triangulation where discrete bistellar flips can occur. From here we define the loop gravity theory and show that it agrees with the continuous theory, having two-to-two moves on the graph which mirror the bistellar flips in the triangulation. The results are promising for finding a dynamics for four-dimensional loop gravity, and if the full theory is developed further, we expect it will lead to a breakthrough in the quantum dynamics.

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.

Master of Science / General Relativity (GR) is founded on the revolutionary idea that space and time are merely parts of a greater, unified whole: spacetime. Furthermore, the force we know as gravity results from the bending and stretching of the geometry of spacetime by its energetic contents. GR is notorious for its mathematical complexity and subtlety, meaning that an intuitive understanding of a spacetime is difficult. One of the best approaches to studying the properties of a given spacetime is to consider its geodesic structure—that is, to consider the motion of unaccelerated, “free-falling” particles. This report presents the results of such a study into two important spacetimes — the Kerr solution for a rotating black hole, and the Robertson-Walker solution for a homogeneous universe.

Thesis (Ph.D.)--Aberdeen University, 2009. / Title from web page (viewed on Oct. 8, 2009). Includes bibliographical references.