Gravitational Constraint Combinations Generate a Lie Algebra

We find a first--order partial differential equation whose solutions are all ultralocal scalar combinations of gravitational constraints with Abelian Poisson brackets between themselves. This is a generalisation of the Kucha\v{r} idea of finding alternative constraints for canonical gravity. The new scalars may be used in place of the hamiltonian constraint of general relativity and, together with the usual momentum constraints, replace the Dirac algebra for pure gravity with a true Lie algebra: the semidirect product of the Abelian algebra of the new constraint combinations with the algebra of spatial diffeomorphisms.Comment: 10 pages, latex, submitted to Classical and Quantum Gravity. Section 3 is expanded and an additional solution provided, minor errors correcte

Action functionals of single scalar fields and arbitrary--weight gravitational constraints that generate a genuine Lie algebra

We discuss the issue initiated by Kucha\v{r} {\it et al}, of replacing the usual Hamiltonian constraint by alternative combinations of the gravitational constraints (scalar densities of arbitrary weight), whose Poisson brackets strongly vanish and cast the standard constraint-system for vacuum gravity into a form that generates a true Lie algebra. It is shown that any such combination---that satisfies certain reality conditions---may be derived from an action principle involving a single scalar field and a single Lagrange multiplier with a non--derivative coupling to gravity.Comment: 26 pages, plain TE

Hazard Avoidance Alerting With Markov Decision Processes

This thesis describes an approach to designing hazard avoidance alerting systems based on a Markov decision process (MDP) model of the alerting process, and shows its benefits over standard design methods. One benefit of the MDP method is that it accounts for future decision opportunities when choosing whether or not to alert, or in determining resolution guidance. Another benefit is that it provides a means of modeling uncertain state information, such as knowledge about unmeasurable mode variables, so that decisions are more informed. A mode variable is an index for distinct types of behavior that a system exhibits at different times. For example, in many situations normal system behavior is safe, but rare deviations from the normal increase the likelihood of a harmful incident. Accurate modeling of mode information is needed to minimize alerting system errors such as unnecessary or late alerts. The benefits of the method are illustrated with two alerting scenarios where a pair of aircraft must avoid collisions when passing one another. The first scenario has a fully observable state and the second includes an uncertain mode describing whether an intruder aircraft levels off safely above the evader or is in a hazardous blunder mode. In MDP theory, outcome preferences are described in terms of utilities of different state trajectories. In keeping with this, alerting system requirements are stated in the form of a reward function. This is then used with probabilistic dynamic and sensor models to compute an alerting logic (policy) that maximizes expected utility. Performance comparisons are made between the MDP-based logics and alternate logics generated with current methods. It is found that in terms of traditional performance measures (incident rate and unnecessary alert rate), the MDP-based logic can meet or exceed that of alternate logics

Generalized Philosophy of Alerting with Applications for Parallel Approach Collision Prevention

An alerting system is automation designed to reduce the likelihood of undesirable outcomes that are due to rare failures in a human-controlled system. It accomplishes this by monitoring the system, and issuing warning messages to the human operators when thought necessary to head off a problem. On examination of existing and recently proposed logics for alerting it appears that few commonly accepted principles guide the design process. Different logics intended to address the same hazards may take disparate forms and emphasize different aspects of performance, because each reflects the intuitive priorities of a different designer. Because performance must be satisfactory to all users of an alerting system (implying a universal meaning of acceptable performance) and not just one designer, a proposed logic often undergoes significant piecemeal modification before gaining general acceptance. This report is an initial attempt to clarify the common performance goals by which an alerting system is ultimately judged. A better understanding of these goals will hopefully allow designers to reach the final logic in a quicker, more direct and repeatable manner. As a case study, this report compares three alerting logics for collision prevention during independent approaches to parallel runways, and outlines a fourth alternative incorporating elements of the first three, but satisfying stated requirements.NASA grant NAG1-218

Remarks on the Reduced Phase Space of (2+1)-Dimensional Gravity on a Torus in the Ashtekar Formulation

We examine the reduced phase space of the Barbero-Varadarajan solutions of the Ashtekar formulation of (2+1)-dimensional general relativity on a torus. We show that it is a finite-dimensional space due to existence of an infinite dimensional residual gauge invariance which reduces the infinite-dimensional space of solutions to a finite-dimensional space of gauge-inequivalent solutions. This is in agreement with general arguments which imply that the number of physical degrees of freedom for (2+1)-dimensional Ashtekar gravity on a torus is finite.Comment: 13 pages, Latex. More details have been included and the expression for the finite residual gauge transformations has been worked ou

A nonlinear quantum model of the Friedmann universe

A discussion is given of the quantisation of a physical system with finite degrees of freedom subject to a Hamiltonian constraint by treating time as a constrained classical variable interacting with an unconstrained quantum state. This leads to a quantisation scheme that yields a Schrodinger-type equation which is in general nonlinear in evolution. Nevertheless it is compatible with a probabilistic interpretation of quantum mechanics and in particular the construction of a Hilbert space with a Euclidean norm is possible. The new scheme is applied to the quantisation of a Friedmann Universe with a massive scalar field whose dynamical behaviour is investigated numerically.Comment: 11 pages of text + 4 pages for 8 figure

High frequency conductivity in the quantum Hall effect

We present high frequency measurements of the diagonal conductivity sigma_xx of a two dimensional electron system in the integer quantum Hall regime. The width of the sigma_xx peaks between QHE minima is analyzed within the framework of scaling theory using both temperature T=100-700 mK and frequency f <= 6 GHz in a two parameter scaling ansatz. For the plateau transition width we find scaling behaviour for both its temperature dependence as well as its frequency dependence. However, the corresponding scaling exponent for temperature kappa=0.42 significantly differs from the one deduced for frequency scaling (c=0.6). Additionally we use the high frequency experiments to suppress the contact resistances that strongly influences DC measurements. We find an intrinsic critical conductivity sigma_c=0.17e^2/h, virtually independent of temperature and filling factor, and deviating significantly from the proposed universal value 0.5e^2/h.Comment: Proceedings of the '14th international conference on high magnetic fields in semiconductor physics' (Semimag-2000) in Matsue, Japa

Canonical Gravity, Diffeomorphisms and Objective Histories

This paper discusses the implementation of diffeomorphism invariance in purely Hamiltonian formulations of General Relativity. We observe that, if a constrained Hamiltonian formulation derives from a manifestly covariant Lagrangian, the diffeomorphism invariance of the Lagrangian results in the following properties of the constrained Hamiltonian theory: the diffeomorphisms are generated by constraints on the phase space so that a) The algebra of the generators reflects the algebra of the diffeomorphism group. b) The Poisson brackets of the basic fields with the generators reflects the space-time transformation properties of these basic fields. This suggests that in a purely Hamiltonian approach the requirement of diffeomorphism invariance should be interpreted to include b) and not just a) as one might naively suppose. Giving up b) amounts to giving up objective histories, even at the classical level. This observation has implications for Loop Quantum Gravity which are spelled out in a companion paper. We also describe an analogy between canonical gravity and Relativistic particle dynamics to illustrate our main point.Comment: Latex 16 Pages, no figures, revised in the light of referees' comments, accepted for publication in Classical and Quantum Gravit

Action and Hamiltonian for eternal black holes

We present the Hamiltonian, quasilocal energy, and angular momentum for a spacetime region spatially bounded by two timelike surfaces. The results are applied to the particular case of a spacetime representing an eternal black hole. It is shown that in the case when the boundaries are located in two different wedges of the Kruskal diagram, the Hamiltonian is of the form $H = H_+ - H_-$, where $H_+$ and $H_-$ are the Hamiltonian functions for the right and left wedges respectively. The application of the obtained results to the thermofield dynamics description of quantum effects in black holes is briefly discussed.Comment: 24 pages, Revtex, 5 figures (available upon request