Indefinite quadratic forms and the invariance of the interval in Special Relativity

A simple theorem on proportionality of indefinite real quadratic forms is proved, and is used to clarify the proof of the invariance of the interval in Special Relativity from Einstein's postulate on the universality of the speed of light; students are often rightfully confused by the incomplete or incorrect proofs given in many texts. The result is illuminated and generalized using Hilbert's Nullstellensatz, allowing one form to be a homogeneous polynomial which is not necessarily quadratic. Also a condition for simultaneous diagonalizabilityof semi-definite real quadratic functions is given.Comment: 6 pages, no figure

Problem of Time in Quantum Gravity

The Problem of Time occurs because the `time' of GR and of ordinary Quantum Theory are mutually incompatible notions. This is problematic in trying to replace these two branches of physics with a single framework in situations in which the conditions of both apply, e.g. in black holes or in the very early universe. Emphasis in this Review is on the Problem of Time being multi-faceted and on the nature of each of the eight principal facets. Namely, the Frozen Formalism Problem, Configurational Relationalism Problem (formerly Sandwich Problem), Foliation Dependence Problem, Constraint Closure Problem (formerly Functional Evolution Problem), Multiple Choice Problem, Global Problem of Time, Problem of Beables (alias Problem of Observables) and Spacetime Reconstruction/Replacement Problem. Strategizing in this Review is not just centred about the Frozen Formalism Problem facet, but rather about each of the eight facets. Particular emphasis is placed upon A) relationalism as an underpinning of the facets and as a selector of particular strategies (especially a modification of Barbour relationalism, though also with some consideration of Rovelli relationalism). B) Classifying approaches by the full ordering in which they embrace constrain, quantize, find time/history and find observables, rather than only by partial orderings such as "Dirac-quantize". C) Foliation (in)dependence and Spacetime Reconstruction for a wide range of physical theories, strategizing centred about the Problem of Beables, the Patching Approach to the Global Problem of Time, and the role of the question-types considered in physics. D) The Halliwell- and Gambini-Porto-Pullin-type combined Strategies in the context of semiclassical quantum cosmology.Comment: Invited Review: 26 pages including 2 Figures. This v2 has a number of minor improvements and correction

On metric-connection compatibility and the signature change of space-time

We discuss and investigate the problem of existence of metric-compatible linear connections for a given space-time metric which is, generally, assumed to be semi-pseudo-Riemannian. We prove that under sufficiently general conditions such connections exist iff the rank and signature of the metric are constant. On this base we analyze possible changes of the space-time signature.Comment: 18 standard LaTeX 2e pages. The packages AMS-LaTeX and amsfonts are require

Mixmaster universe in Horava-Lifshitz gravity

We consider spatially homogeneous (but generally non-isotropic) cosmologies in the recently proposed Horava-Lifshitz gravity and compare them to those of general relativity using Hamiltonian methods. In all cases, the problem is described by an effective point particle moving in a potential well with exponentially steep walls. Focusing on the closed-space cosmological model (Bianchi type IX), the mixmaster dynamics is now completely dominated by the quadratic Cotton tensor potential term for very small volume of the universe. Unlike general relativity, where the evolution towards the initial singularity always exhibits chaotic behavior with alternating Kasner epochs, the anisotropic universe in Horava-Lifshitz gravity (with parameter lambda > 1/3) is described by a particle moving in a frozen potential well with fixed (but arbitrary) energy E. Alternating Kasner epochs still provide a good description of the early universe for very large E, but the evolution appears to be non-ergodic. For very small E there are harmonic oscillations around the fully isotropic model. The question of chaos remains open for intermediate energy levels.Comment: 1+35 pages, 4 figure