The authors of the Rainich-Misner-Wheeler theory no longer believe that everything physical can be accounted for in terms of the topological-geometrical structure of ordinary spacetime. However, many physicists and philosophers entertain the possibility that a geometrodynamics (a theory which accounts for sources as well as fields in terms of topological-geometrical structure) may be feasible in the context of a more general topology. In this dissertation I consider two topological-geometrical models (based upon a single suggestive formalism) in which a geometrodynamics is both feasible and pedagogically advantageous. Specifically I consider the topology which is constituted by the real domains of the two broad classes of rotation groups: those characterized by the commutator and anti-commutator algebras. I then adopt a Riemannian geometric structure and show that the monistically geometric interpretation of this formalism restricts displacements on the proposed manifold to integral multiples of a universal constant. Secondly I demonstrate that in the context under consideration, this constraint affects a very interesting ontological reduction: the unification of quantum mechanics with a discrete, multidimensional extension of general relativity. A particularly interesting feature of this unification is that it includes and (for the world which is characterized by energy levels which range in magnitude from low to intermediately high) requires the choice cf an SL(2,R)xSU(3)-symmetric realization of the proposed, generic formalism which is a lattice of spins π and π/2. (This is in the context of the same universally constant scale factor as that which yields the quantization conditions described above.) If the vertices of this lattice are associated with the fundamental particles, then the resulting theory predicts and precludes the same interactions as the standard supersymmetry theory. In addition to the ontological reduction which is provided, and the restriction to supersymmetry, the proposed theory may also represent a scientifically useful extension of conventional theory in that it suggests a means of understanding the apparently large energy productions of the quasars and relates Planck's constant to the size of the universe.

Some branches of quantum gravity demand the existence of higher dimensions and the addition of higher curvature terms to the gravitational Lagrangian in the form of the Lovelock polynomials. In this thesis we investigate some of the classical properties of Lovelock gravity. We first derive the Hamiltonian for Lovelock gravity and find that it takes the same form as in general relativity when written in terms of the Misner-Sharp mass function. We then minimally couple the action to matter fields to find Hamilton’s equations of motion. These are gauge fixed to be in the Painleve-Gullstrand co–ordinates and are well suited to numerical studies of black hole formation. We then use these equations of motion for the massless scalar field to study the formation of general relativistic black holes in four to eight dimensions and Einstein-Gauss-Bonnet black holes in five and six dimensions. We study Choptuik scaling, a phenomenon which relates the initial conditions of a matter distribution to the final observables of small black holes. In both higher dimensional general relativity and Einstein-Gauss-Bonnet gravity we confirm the existence of cusps in the mass scaling relation which had previously only been observed in four dimensional general relativity. In the general relativistic case we then calculate the critical exponents for four to eight dimensions and find agreement with previous calculations by Bland et al but not Sorkin et al who both worked in null co–ordinates. For the Einstein-Gauss-Bonnet case we find that the self-similar behaviour seen in the general relativistic case is destroyed. We find that it is replaced by some other form of scaling structure. In five dimensions we find that the period of the critical solution at the origin is proportional to roughly the cube root of the Gauss-Bonnet parameter and that there is evidence for a minimum black hole radius. In six dimensions we see evidence for a new type of scaling. We also show, from the equations of motion, that there is reason to expect qualitative differences between five and higher dimensions.

Some branches of quantum gravity demand the existence of higher dimensions and the addition of higher curvature terms to the gravitational Lagrangian in the form of the Lovelock polynomials. In this thesis we investigate some of the classical properties of Lovelock gravity. We first derive the Hamiltonian for Lovelock gravity and find that it takes the same form as in general relativity when written in terms of the Misner-Sharp mass function. We then minimally couple the action to matter fields to find Hamilton’s equations of motion. These are gauge fixed to be in the Painleve-Gullstrand co–ordinates and are well suited to numerical studies of black hole formation. We then use these equations of motion for the massless scalar field to study the formation of general relativistic black holes in four to eight dimensions and Einstein-Gauss-Bonnet black holes in five and six dimensions. We study Choptuik scaling, a phenomenon which relates the initial conditions of a matter distribution to the final observables of small black holes. In both higher dimensional general relativity and Einstein-Gauss-Bonnet gravity we confirm the existence of cusps in the mass scaling relation which had previously only been observed in four dimensional general relativity. In the general relativistic case we then calculate the critical exponents for four to eight dimensions and find agreement with previous calculations by Bland et al but not Sorkin et al who both worked in null co–ordinates. For the Einstein-Gauss-Bonnet case we find that the self-similar behaviour seen in the general relativistic case is destroyed. We find that it is replaced by some other form of scaling structure. In five dimensions we find that the period of the critical solution at the origin is proportional to roughly the cube root of the Gauss-Bonnet parameter and that there is evidence for a minimum black hole radius. In six dimensions we see evidence for a new type of scaling. We also show, from the equations of motion, that there is reason to expect qualitative differences between five and higher dimensions.

Ausgehend von der Forderung, dass die Dynamik klassischer Materiefelder auf einer glatten Mannigfaltigkeit prädiktiv und quantisierbar sein muss, leiten wir einen Satz von „Mastergleichungen“ her, deren Lösungen die Dynamik (in Form einer Lagrangedichte) der den Materiegleichungen zugrundeliegenden Geometrie beschreiben. Es gelingt also das physikalische Problem der Suche nach geeigneten Gravitationsdynamiken für eine beliebige tensorielle Raumzeitgeometrie, die physikalische Materie tragen kann, in die bloß noch mathematische Frage nach der Lösung eines Systems von linearen partiellen Differentialgleichungen zu reformulieren. Dieses Ergebnis fußt auf der Einsicht, dass die Forderung nach der Prädiktivität und Quantisierbarkeit einer Materietheorie zunächst die möglichen Klassen der zugrundeliegenden Raumzeitgeometrien auf solche beschränkt, die bi-hyperbolisch sind und die Unterscheidung von positiven und negativen Energien zulassen. Gleichzeitig stellen solche Materietheorien bereits alle kinematischen Strukturen zur Verfügung, die nötig sind, um die Dynamik der Geometrie als Anfangswertproblem zu formulieren. Die Mastergleichungen stellen dann einen Ausdruck dafür dar, dass die Lagrangefunktion der Gravitationsdynamik, die die zeitliche Entwicklung von geometrischen Anfangsdaten beschreibt, eine Darstellung der Hyperflächendeformationsalgebra sein muss, welche sich ausgehend von der Dynamik der Materietheorie direkt berechnen lässt. Wir geben eine allgemeine Vorgehensweise an, mit der sich die Mastergleichungen für eine beliebige tensorielle Raumzeitgeometrie herleiten lassen und illustrieren dieses Verfahren anhand von vier physikalisch relevanten Beispielen. Die Arbeit wird abgerundet durch ein Studium von Energie-Impuls-Tensoren von Materie auf tensoriellen Raumzeiten. / Starting from classical matter dynamics on a smooth manifold that are required to be predictive and quantizable, we derive a set of `gravitational master equations'' that determine the Lagrangian describing the dynamics of the geometry on which the matter dynamics are defined. We thus convert the physical problem of finding admissible gravitational dynamics for any tensorial geometry that can support physical matter equations into the clear mathematical task of solving a system of linear partial differential equations. This result builds on the insight that predictive and quantizable matter dynamics, on the one hand, restrict the class of admissible spacetime geometries to those that are bi-hyperbolic and energy-distinguishing, and, on the other hand, provide the necessary kinematical structure needed to formulate spacetime geometry dynamics as an initial value problem. The gravitational master equations then express the fact that the Lagrangian of the gravitational dynamics must arise as a representation of the algebra of hypersurface deformations---which can be calculated from the kinematical structure imprinted on the geometry by the matter field dynamics---on a suitable geometric phase space. We provide a general prescription of how to obtain the gravitational master equations for any candidate geometry and illustrate our procedure by way of four instructive examples. We solve the master equations for metric geometry supporting Maxwell theory, finding Einstein-Hilbert dynamics as the unique solution, and for a non-trivial composite geometry supporting modified Dirac dynamics. We also discuss generalized energy-momentum tensors of matter fields and their role as sources of the gravitational dynamics obtained from the gravitational master equations.