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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.

In this thesis we review the status of an approach to Quantum Gravity through lattice toy models, Quantum Graphity. In particular, we describe the two toy models introduced in the literature and describe with a certain level of details the results obtained so far. We emphasize the connection between Quantum Graphity and emergent gravity, and the relation with Variable Speed of Light theories.

Thesis (M.S.)--University of Hawaii at Manoa, 2006. / Includes bibliographical references (leaves 43-46). / vii, 46 leaves, bound ill. (some col.) 29 cm

Bibliography: leaves 131-142 / 142, 6 leaves : ill ; 30 cm. / Title page, contents and abstract only. The complete thesis in print form is available from the University Library. / Thesis (Ph.D.)--University of Adelaide, 1986

Thesis (Ph. D.)--Montana State University--Bozeman, 2007. / Typescript. Chairperson, Graduate Committee: William A. Hiscock. Includes bibliographical references (leaves 55-60).

Thesis (M. Sc.)--University of Adelaide, Mawson Institute for Antarctic Research, 1991. / Includes bibliographical references (leaves 112-123).