Thesis (M.S.)--West Virginia University, 2003. / Title from document title page. Document formatted into pages; contains v, 30 p. Includes abstract. Includes bibliographical references (p. 29-30).

Thesis (B.A.)--Haverford College, Dept. of Mathematics, 2002. / Includes bibliographical references.

Title from PDF of title page (University of Missouri--Columbia, viewed on Feb 11, 2010). The entire thesis text is included in the research.pdf file; the official abstract appears in the short.pdf file; a non-technical public abstract appears in the public.pdf file. Dissertation advisor: Dr. Naz E. Islam Vita. Includes bibliographical references.

Although introductory courses in special relativity give an introduction to the causal structure of Minkowski space, it is common for causal structure in general space- times to be regarded as an advanced topic, and omitted from introductory courses in general relativity, although the related topic of gravitational lensing is often included. Here a numerical approach to visualizing the light cones in exterior Schwarzschild space taking advantage of the symmetries of Schwarzschild space and the conformal invariance of null geodesics is formulated, and used to make some of these ideas more accessible. By means of the Matlab software developed, a user is able to produce figures showing how light cones develop in Schwarzschild space, starting from an arbitrary point and developing for any length of time. The user can then interact with the figure, changing their point of view, or zooming in or out, to investigate them. This approach is then generalised, using the symbolic manipulation facility of Matlab, to allow the user to specify a metric as well as an initial point and time of development. Finally, the software is demonstrated with a selection of metrics.

The objective of this dissertation is the application of Minkowskian cross-section measures (i.e., section and projection measures in finite-dimensional linear normed spaces over the real field) to various topics of geometric convexity in Minkowski spaces, such as bodies of constant Minkowskian width, Minkowskian geometry of simplices, geometric inequalities and the corresponding optimization problems for convex bodies. First we examine one-dimensional Minkowskian cross-section measures deriving (in a unified manner) various properties of these measures. Some of these properties are extensions of the corresponding Euclidean properties, while others are purely Minkowskian. Further on, we discover some new results on the geometry of a simplex in Minkowski spaces, involving descriptions of the so-called tangent Minkowskian balls and of simplices with equal Minkowskian heights. We also give some (characteristic) properties of bodies of constant width in Minkowski planes and in higher dimensional Minkowski spaces. This part of investigation has relations to the well known \emph{Borsuk problem} from the combinatorial geometry and to the widely used monotonicity lemma from the theory of Minkowski spaces. Finally, we study bodies of given Minkowskian thickness ($=$ minimal width) having least possible volume. In the planar case a complete description of this class of bodies is given, while in case of arbitrary dimension sharp estimates for the coefficient in the corresponding geometric inequality are found. / Die Dissertation befasst sich mit Problemen fuer spezielle konvexe Koerper in Minkowski-Raeumen (d.h. in endlich-dimensionalen Banach-Raeumen). Es wurden Klassen der Koerper mit verschiedenen metrischen Eigenschaften betrachtet (z.B., Koerper konstante Breite, reduzierte Koerper, Simplexe mit Inhaltsgleichen Facetten usw.) und einige kennzeichnende und andere Eigenschaften fuer diese Klassen herleitet.

The purpose of this study was to see how the fundamental properties of hyperbolic geometry applies in physics.

L'étude des espaces-temps plats singuliers munis d'une surface de Cauchy polyédrale est motivée par leur rôle de model jouet de gravité quantique proposé par Deser, Jackiw et 'T Hooft. Cette thèse porte sur les paramétrisations de certaines classes d'espaces-temps plat singuliers : les espaces-temps plats avec particules massives et BTZ Cauchy-compacts maximaux. Deux paramétrisations sont proposées, l'une reposant sur une extension du théorème de Mess aux espaces-temps plats avec BTZ et la surface de Penner-Epstein, l'autre reposant sur une généralisation du théorème d'Alexandrov aux espaces-temps plats avec particules massives et BTZ. Ce travail propose également une amorce de cadre théorique permettant de considérer des espaces-temps singuliers plus généraux. / The study of singular flat spacetimes with polyhedral Cauchy-surfaces is motivated by the quantum gravity toy model role they play in the seminal work of Deser, Jackiw and 'T Hooft. This thesis study parametrisations of classes of singular flat spacetimes : Cauchy-compact maximal flat spacetimes with massive and BTZ-like singularities. Two parametrisations are constructed. The first is based on an extension of Mess theorem to flat spacetimes with BTZ and Penner-Epstein convex hull construction. The second is based on a generalisation of Alexandrov polyhedron theorem to radiant Cauchy-compact flat spacetimes with massive and BTZ-like singularities. This work also initiate a wider theoretical background that encompass singular spacetimes.

This thesis investigates the interplay between explanatory issues in special relativity and the theory's metaphysical foundations. Special attention is given to the 'dynamical approach' to relativity, promoted primarily by Harvey Brown and collaborators, according to which the symmetries of dynamical laws are explanatory of relativistic effects, inertial motion, and even the Minkowskian geometrical structure of a specially relativistic world. The thesis begins with a review of Einstein's 1905 introduction to special relativity, after which brief historical introductions are given for the standard 'geometrical' approach to relativity and the unorthodox 'dynamical' approach. After a critical review of recent literature on the topic, the dynamical approach is shown to be in need of a metaphysical package that would undergird the explanatory claims mentioned above. It is argued that the dynamical approach is best understood as a form of relationalism - in particular, as a relativistic form of 'regularity relationalism', promoted recently by Nick Huggett. According to this view, some portion of a world's geometrical structure actually supervenes upon the symmetries of the best-system dynamical laws for a material ontology endowed with a primitive sub-metrical structure. To explore the plausibility of this construal of the dynamical approach, a case study is carried out on solutions to the Klein-Gordon equation. Examples are found for which the field values, when purged of all spatiotemporal structure but their induced topology, are still arguably best-systematized by the Klein-Gordon equation itself. This bolsters the plausibility of the claim that some system of field values, endowed with mere sub-metrical structure, might have as its best-systems dynamical laws a (set of) Lorentz-covariant equation(s), on which Minkowski geometrical structure would supervene. The upshot is that the dynamical approach to special relativity can be defended as what might be called an ontologically and ideologically relationalist approach to Minkowski spacetime structure. The chapters refer regularly to three appendices, which include a brief introduction to topological and differentiable spaces.

The objective of this dissertation is the application of Minkowskian cross-section measures (i.e., section and projection measures in finite-dimensional linear normed spaces over the real field) to various topics of geometric convexity in Minkowski spaces, such as bodies of constant Minkowskian width, Minkowskian geometry of simplices, geometric inequalities and the corresponding optimization problems for convex bodies. First we examine one-dimensional Minkowskian cross-section measures deriving (in a unified manner) various properties of these measures. Some of these properties are extensions of the corresponding Euclidean properties, while others are purely Minkowskian. Further on, we discover some new results on the geometry of a simplex in Minkowski spaces, involving descriptions of the so-called tangent Minkowskian balls and of simplices with equal Minkowskian heights. We also give some (characteristic) properties of bodies of constant width in Minkowski planes and in higher dimensional Minkowski spaces. This part of investigation has relations to the well known \emph{Borsuk problem} from the combinatorial geometry and to the widely used monotonicity lemma from the theory of Minkowski spaces. Finally, we study bodies of given Minkowskian thickness ($=$ minimal width) having least possible volume. In the planar case a complete description of this class of bodies is given, while in case of arbitrary dimension sharp estimates for the coefficient in the corresponding geometric inequality are found. / Die Dissertation befasst sich mit Problemen fuer spezielle konvexe Koerper in Minkowski-Raeumen (d.h. in endlich-dimensionalen Banach-Raeumen). Es wurden Klassen der Koerper mit verschiedenen metrischen Eigenschaften betrachtet (z.B., Koerper konstante Breite, reduzierte Koerper, Simplexe mit Inhaltsgleichen Facetten usw.) und einige kennzeichnende und andere Eigenschaften fuer diese Klassen herleitet.

The results presented in this dissertation refer to the geometry of Minkowski spaces, i.e., of real finite-dimensional Banach spaces. First we study geometric properties of radial projections of bisectors in Minkowski spaces, especially the relation between the geometric structure of radial projections and Birkhoff orthogonality. As an application of our results it is shown that for any Minkowski space there exists a number, which plays somehow the role that $\sqrt2$ plays in Euclidean space. This number is referred to as the critical number of any Minkowski space. Lower and upper bounds on the critical number are given, and the cases when these bounds are attained are characterized. Moreover, with the help of the properties of bisectors we show that a linear map from a normed linear space $X$ to another normed linear space $Y$ preserves isosceles orthogonality if and only if it is a scalar multiple of a linear isometry. Further on, we examine the two tangent segments from any exterior point to the unit circle, the relation between the length of a chord of the unit circle and the length of the arc corresponding to it, the distances from the normalization of the sum of two unit vectors to those two vectors, and the extension of the notions of orthocentric systems and orthocenters in Euclidean plane into Minkowski spaces. Also we prove theorems referring to chords of Minkowski circles and balls which are either concurrent or parallel. All these discussions yield many interesting characterizations of the Euclidean spaces among all (strictly convex) Minkowski spaces. In the final chapter we investigate the relation between the length of a closed curve and the length of its midpoint curve as well as the length of its image under the so-called halving pair transformation. We show that the image curve under the halving pair transformation is convex provided the original curve is convex. Moreover, we obtain several inequalities to show the relation between the halving distance and other quantities well known in convex geometry. It is known that the lower bound for the geometric dilation of rectifiable simple closed curves in the Euclidean plane is $\pi/2$, which can be attained only by circles. We extend this result to Minkowski planes by proving that the lower bound for the geometric dilation of rectifiable simple closed curves in a Minkowski plane $X$ is analogously a quarter of the circumference of the unit circle $S_X$ of $X$, but can also be attained by curves that are not Minkowskian circles. In addition we show that the lower bound is attained only by Minkowskian circles if the respective norm is strictly convex. Also we give a sufficient condition for the geometric dilation of a closed convex curve to be larger than a quarter of the perimeter of the unit circle.