Space and time

Though general relativity (GR) is proven to be a successful theory in describing the macroscopical nature of our universe, it still has several problems to be resolved. One of them is known as the time problem of GR. GR is a pure constraint theory, and the time evolution of the system is a gauge transformation, without carrying any physical information. One potential resolution to this issue is the relational formalism, which considers the dynamics of a material frame by coupling it to gravity. This approach allows for constructing gauge invariant observables and subsequent quantization.
One realization of the relational formalism is the Brown-Kuchaˇr formalism. In this formalism, the gravity couples Brown-Kuchaˇr dust fields, and the Brown-Kuchaˇr dust fields play the roles as a family of observers. Then, one can introduce a gauge fixing scheme to the system and construct gauge invariant observables (Dirac observables) in the reduced phase Space. The probe time of the dust plays the role as the physical time of each point of the spacetime. In this thesis, we consider the Brown-Kuchaˇr formalism in an asymptotically flat background. A set of boundary conditions for the asymptotic flatness are formulated for Dirac observables on the reduced phase space. We compute the boundary term of the physical Hamiltonian, which is identical to the ADM mass. We construct a set of the symmetry charges on the reduced phase space, which encompass both the bulk terms and the boundary terms are conserved by the physical Hamiltonian evolution. The symmetry charges generate transformations preserving the asymptotically flat boundary conditions. Under the reduced-phase space Poisson bracket, the symmetry charges form an infinite dimensional Lie algebra AG after adding a central charge. A suitable quotient of AG is analogous to the BMS algebra at spatial infinity by Henneaux and Troessaert.

The unmanned aerial vehicle (UAV) technology has evolved considerably in recent years and the global demand for package delivery is expected to grow even more during COVID-19 and the social distance era. The low cost of acquisition, payload capacity, maneuverability, and the ability to y at low-altitude with a very low cost of operation, make UAVs a perfect fit to revolutionize the payload transportation of small items. The large-scale adoption of drone package delivery in high-density urban areas can be challenging and the Unmanned Aircraft Systems (UAS) operators must ensure safety, security, efficiency and equity of the airspace system. In order to address some of these challenges, FAA and NASA have developed a new architecture that will support a set of services to enable cooperative management of low-altitude operations between UAS operators. The architecture is still in its conceptual stage and designing a mechanism that ensures the fair distribution of the available airspace to commercial applications has become increasingly important. Considering that, the path planning is one of the most important problems to be explored. The objective is not only to find an optimal and shortest path but also to provide a collision-free environment to the UAVs. Taking into consideration all these important aspects and others such as serving on-demand requests, flight duration limitation due to energy constraints, maintaining the safety distance to avoid collisions, and using warehouses as starting and ending points in parcel delivery, this dissertation proposes: (i) an energy-constrained scheduling mechanism using a multi-source A* algorithm variant, and (ii) a generalized path planning mechanism using a space-time graph with multi-source multi-destination BFS generalization to ensure pre-flight UAV collision-free trajectories. This dissertation also uses the generalized path planning mechanism to solve the energy-constrained drone delivery problem. The experimental results show that the proposed algorithms are computationally efficient and scalable with the number of requests and graph size.

Unmanned Aircraft Systems (UAS) have grown in popularity due to their widespread potential applications, including efficient package delivery, monitoring, surveillance, search and rescue operations, agricultural uses, along with many others. As UAS become more integrated into our society and airspace, it is anticipated that the development and maintenance of a path planning collision-free system will become imperative, as the safety and efficiency of the airspace represents a priority. The dissertation defines this problem as the UAS Collision-free Path Planning Problem.
The overall objective of the dissertation is to design an on-demand, efficient and scalable aerial highway path planning system for UAS. The dissertation explores two solutions to this problem. The first solution proposes a space-time algorithm that searches for shortest paths in a space-time graph. The solution maps the aerial traffic map to a space-time graph that is discretized on the inter-vehicle safety distance. This helps compute safe trajectories by design. The mechanism uses space-time edge pruning to maintain the dynamic availability of edges as vehicles move on a trajectory. Pruning edges is critical to protect active UAS from collisions and safety hazards. The dissertation compares the solution with another related work to evaluate improvements in delay, run time scalability, and admission success while observing up to 9000 flight requests in the network. The second solution to the path planning problem uses a batch planning algorithm. This is a new mechanism that processes a batch of flight requests with prioritization on the current slack time. This approach aims to improve the planning success ratio. The batch planning algorithm is compared with the space-time algorithm to ascertain improvements in admission ratio, delay ratio, and running time, in scenarios with up to 10000 flight requests.

Visual motion can be conveyed by a variety of information sources in the environment, and those types of information may be detected at various levels by different motion-perceiving mechanisms in the visual system. High-level visual information has been demonstrated to have 3rd order, or salience-based properties (Lu & Sperling, 1995). The perceptual system they describe that computes motion from these types of information shares several characteristics with Hock and colleagues' counterchange detection system, notably flexibility with respect to types of input from which motion can be computed, which comes at the cost of diminished processing speed. The mechanism of counterchange detection is well suited to processing visual features often present in environmental scenes, e.g., objects and surfaces, and may be a mechanism of 3rd order motion. Consistent with reported properties of 3rd order motion, the current experiments tested count erchange-, luminance-, and color-based motion stimuli with 3 objectives: to identify whether the 3 systems framework generalizes beyond the stimulus type with which it was defined, to test whether counterchange shares similarities with the 3rd order system with respect to dichoptic integration, and perception of isoluminant color-based motion, and to test subjectively objectless sources of motion-defining information (spreading luminance and hue) to see if they display properties of the 1st order system derived from sine wave gratings. Results indicate that counterchange-based stimuli displayed predicted properties of dichoptic integration, and perception at isoluminance, but putative 1st order (spreading) stimuli also displayed these properties. This may suggest that object-like surfaces, even when not directly the source of motion information, can contribute to computation of motion. Further, these results highlight the difficulty of generalizing from one theoretical framework to another, and specifically, of psychophysically testing high-level information while isolating contributions from low level information upon which high level visual stimuli are built.

The development of geographic information systems (GIS) has changed the way in which geographers are able to visualize and investigate spatial topics. Current research has now shown a need to incorporate the element of time into a GIS for the purpose of better understanding the processes that are related to change. This study investigates two methods of creating spatiotemporal databases, using the evolution of an airline route system as an example. Also discussed are the ways in which a user-friendly interface may be incorporated for easier data exploration.

The concept of time in Stanley Kubrick's 1968 film 2001: A Space Odyssey is examined from social, biological, psychological, and spiritual perspectives. In Arthur C. Clarke's novel, his version of the film, he treats the nature of time as a cyclical process. He eventually explains that the notion of physical time is non-existent or an impermanent illusion. While Clarke's novel interprets time, the film projects and manipulates the nature of space and time, which spectators may experience as reality. Time's direction can be viewed or experienced as a cycle from an Eastern philosophical perspective. However, a Western interpretation requires a compromise between two separate directions of time, one as a cycle, the other as linear. The film and novel ultimately negates the direction of linear time through the appearance of the mysterious monolith, which transcends and reincarnates human beings.

It has been argued that the perception of apparent motion is based on the detection of counterchange (oppositely signed changes in luminance contrast at pairs of spatial locations) rather than motion energy (spatiotemporal changes in luminance). A constraint in furthering this distinction is that both counterchange and motion energy are present for most motion stimuli. Three experiments used illusory-contour and luminance-based stimuli to segregate (experiments 1 and 2) and combine (experiment 3) counterchange and motion energy information. Motion specified by counterchange was perceived for translating illusory squares over a wide range of frame durations, and preferentially for short motion paths. Motion specified by motion energy was diminished by relatively long frame durations, but was not affected by the length of the motion path. Results for the combined stimulus were consistent with counterchange as the basis for apparent motion perception, despite the presence of motion energy.

A discrete formalism for General Relativity was introduced in 1961 by Tulio Regge in the form of a piecewise-linear manifold as an approximation to (pseudo-)Riemannian manifolds. This formalism, known as Regge Calculus, has primarily been used to study vacuum spacetimes as both an approximation for classical General Relativity and as a framework for quantum gravity. However, there has been no consistent effort to include arbitrary non-gravitational sources into Regge Calculus or examine the structural details of how this is done. This manuscript explores the underlying framework of Regge Calculus in an effort elucidate the structural properties of the lattice geometry most useful for incorporating particles and fields. Correspondingly, we first derive the contracted Bianchi identity as a guide towards understanding how particles and fields can be coupled to the lattice so as to automatically ensure conservation of source. In doing so, we derive a Kirchhoff-like conservation principle that identifies the flow of energy and momentum as a flux through the circumcentric dual boundaries. This circuit construction arises naturally from the topological structure suggested by the contracted Bianchi identity. Using the results of the contracted Bianchi identity we explore the generic properties of the local topology in Regge Calculus for arbitrary triangulations and suggest a first-principles definition that is consistent with the inclusion of source. This prescription for extending vacuum Regge Calculus is sufficiently general to be applicable to other approaches to discrete quantum gravity. We discuss how these findings bear on a quantized theory of gravity in which the coupling to source provides a physical interpretation for the approximate invariance principles of the discrete theory.