Symmetries of Differrential equations and Applications in Relativistic Physics

Σε αυτή την εργασία μελετάμε τους μονοπαραμετρικούς μετασχηματισμούς κάτω από τους οποίους οι διαφορικές εξισώσεις είναι αναλλοίωτες. Ειδικότερα μελετάμε τις σημειακές συμμετρίες Lie και Noether διαφορικών εξισώσεων τάξεως. Αναπτύσσουμε μια γεωμετρική μέθοδο για τον υπολογισμό των συμμετριών η οποία συνδέει τις σημειακές συμμετρίες των διαφορικών εξισώσεων με τις συμμετρίες του χώρου που πραγματοποιείται η κίνηση. Η γεωμετρική μέθοδος εφαρμόζεται σε διάφορα προβλήματα όπως: η κατηγοριοποίηση των συμμετριών Νευτώνειων συστημάτων δύο και τριών διαστάσεων, η γενίκευση του συστήματος Kepler-Ermakov σε καμπύλους χώρους, η σύνδεση των συμμετριών ανάμεσα σε κλασσικά και κβαντικά συστήματα και η αναζήτηση Τύπου ΙΙ κρυφών συμμετριών στην κυματική εξίσωση και στην εξίσωση διάδοσης θερμότητας σε καμπύλους χώρους. Τέλος, η γεωμετρική μέθοδος εφαρμόστηκε σαν γεωμετρικό κριτήριο για την επιλογή διάφορων μοντέλων στις εναλλακτικές θεωρίες βαρύτητας.In this thesis, we study the one parameter point transformations which leave invariant the differential equations. In particular we study the Lie and the Noether point symmetries of second order differential equations. We establish a new geometric method which relates the point symmetries of the differential equations with the collineations of the underlying manifold where the motion occurs. This geometric method is applied in order the two and three dimensional Newtonian dynamical systems to be classified in relation to the point symmetries; to generalize the Newtonian Kepler-Ermakov system in Riemannian spaces; to study the symmetries between classical and quantum systems and to investigate the geometric origin of the Type II hidden symmetries for the homogeneous heat equation and for the Laplace equation in Riemannian spaces. At last but not least, we apply this geometric approach in order to determine the dark energy models by use the Noether symmetries as a geometric criterion in modified theories of gravity

Evolution and Regularisation of Vacuum Brill Gravitational Waves in Spherical Polar Coordinates

In this thesis the universal collapse of vacuum Brill waves is demonstrated numerically and analytically. This thesis presents the mathematical and numerical methods necessary to regularise and evolve Brill Gravitational Waves in spherical polar coordinates. A Cauchy ADM formulation is used for the time evolution. We find strong evidence that all IVP formulations of pure vacuum Brill gravitational waves collapse to form singularities/black holes, and we do not observe critical black hole mass scaling phenomena in the IVP parameter phase space that has been characterised in non-vacuum systems. A theoretical framework to prove this result analytically is presented. We discuss the meaning of Brill metric variables, the topology of trapped surfaces for various scenarios, and verify other results in the field related to critical values of initial value parameters and black hole formation approaching spatial infinity. The instability of Minkowski (flat) space under Brill wave and more general perturbations is demonstrated. The main numerical tools employed to achieve a stable evolution code are (1) derivation of appropriate regularity conditions on the lapse function and metric function q, (2) the move to a 4th order correct discretisation scheme with appropriate boundary conditions, (3) the use of exponential metric terms, (4) an understanding of the right mix of free versus constrained evolution and (5) the development of appropriate numerical techniques for discretisation and differencing to reduce numerical error, along with a characterisation of condition numbers.Comment: PhD Thesis, 2014, University of Calgary, 368 page

The Fifteenth Marcel Grossmann Meeting

The three volumes of the proceedings of MG15 give a broad view of all aspects of gravitational physics and astrophysics, from mathematical issues to recent observations and experiments. The scientific program of the meeting included 40 morning plenary talks over 6 days, 5 evening popular talks and nearly 100 parallel sessions on 71 topics spread over 4 afternoons. These proceedings are a representative sample of the very many oral and poster presentations made at the meeting.Part A contains plenary and review articles and the contributions from some parallel sessions, while Parts B and C consist of those from the remaining parallel sessions. The contents range from the mathematical foundations of classical and quantum gravitational theories including recent developments in string theory, to precision tests of general relativity including progress towards the detection of gravitational waves, and from supernova cosmology to relativistic astrophysics, including topics such as gamma ray bursts, black hole physics both in our galaxy and in active galactic nuclei in other galaxies, and neutron star, pulsar and white dwarf astrophysics. Parallel sessions touch on dark matter, neutrinos, X-ray sources, astrophysical black holes, neutron stars, white dwarfs, binary systems, radiative transfer, accretion disks, quasars, gamma ray bursts, supernovas, alternative gravitational theories, perturbations of collapsed objects, analog models, black hole thermodynamics, numerical relativity, gravitational lensing, large scale structure, observational cosmology, early universe models and cosmic microwave background anisotropies, inhomogeneous cosmology, inflation, global structure, singularities, chaos, Einstein-Maxwell systems, wormholes, exact solutions of Einstein's equations, gravitational waves, gravitational wave detectors and data analysis, precision gravitational measurements, quantum gravity and loop quantum gravity, quantum cosmology, strings and branes, self-gravitating systems, gamma ray astronomy, cosmic rays and the history of general relativity

Analogue gravity in nonlocal fluids of light

Analogue gravity designates the study of curved spacetime in a laboratory environment and allows to test concepts of General Relativity. This analogy is established via a conformal identity between the flow of curved spacetime and inhomogeneous flows in hydrodynamics, which predicts that small waves on a fluid behave exactly as scalar fields in a curved spacetime metric. Atomic quantum fluids such as Bose-Einstein Condensates (BEC) are a widespread workbench for studying artificial black holes and many-body physics but face considerably large experimental challenges. In recent years, quantum fluids of light became a promising alternative at less technical expense, where the many-body dynamics in a laser beam are established via photon-photon interactions mediated through an optical nonlinearity. Whereas recent works considered strongly confined laser fields in microcavities, this work presents a photon fluid in a propagating geometry, i.e. a paraxially propagating laser beam in a bulk nonlinear medium. In this scenario, the propagating direction maps onto a time coordinate and the photon fluid is established in the transverse beam profile. The thermal nonlinearity is excited through heating of the absorbed laser power that introduces a nonlocal response of the medium and adds another level of complexity. It is experimentally shown that the dynamics of small amplitude excitations are governed by the Bogoliubov dispersion relation and allows to observe superfluidity at sufficiently large wavelengths. This is confirmed by the onset of persistent currents and the nucleation of quantized vortices in sub- and supercritical flows around an extended obstacle, which is a direct observation of superfluidity in a room-temperature system. The superfluid regime is a requirement for building analogue spacetime metrics and is thus of paramount importance. The spacetime of a rotating black and white whole was then created by shaping the topology of the spatial phase using diffractive phase masks. The experimental measurements of the inhomogeneous flows revealed, for the first time conclusive evidence of a (2+1) dimensional acoustic horizon and ergosphere. Such a system promises to study Penrose superradiance, where first experimental and numerical results for its observation are presented. Finally, nonlinear wave dynamics such as self-steepening and shock formation are studied where the dynamics can be interpreted in terms of a self-induced spacetime. Furthermore, the dynamics of a sea of incoherent waves is studied with respect to the long-range interactions provided by the nonlocality, where a novel transition from individual dispersive shock waves towards a collective giant shock wave is observed

The Fifteenth Marcel Grossmann Meeting

The three volumes of the proceedings of MG15 give a broad view of all aspects of gravitational physics and astrophysics, from mathematical issues to recent observations and experiments. The scientific program of the meeting included 40 morning plenary talks over 6 days, 5 evening popular talks and nearly 100 parallel sessions on 71 topics spread over 4 afternoons. These proceedings are a representative sample of the very many oral and poster presentations made at the meeting.Part A contains plenary and review articles and the contributions from some parallel sessions, while Parts B and C consist of those from the remaining parallel sessions. The contents range from the mathematical foundations of classical and quantum gravitational theories including recent developments in string theory, to precision tests of general relativity including progress towards the detection of gravitational waves, and from supernova cosmology to relativistic astrophysics, including topics such as gamma ray bursts, black hole physics both in our galaxy and in active galactic nuclei in other galaxies, and neutron star, pulsar and white dwarf astrophysics. Parallel sessions touch on dark matter, neutrinos, X-ray sources, astrophysical black holes, neutron stars, white dwarfs, binary systems, radiative transfer, accretion disks, quasars, gamma ray bursts, supernovas, alternative gravitational theories, perturbations of collapsed objects, analog models, black hole thermodynamics, numerical relativity, gravitational lensing, large scale structure, observational cosmology, early universe models and cosmic microwave background anisotropies, inhomogeneous cosmology, inflation, global structure, singularities, chaos, Einstein-Maxwell systems, wormholes, exact solutions of Einstein's equations, gravitational waves, gravitational wave detectors and data analysis, precision gravitational measurements, quantum gravity and loop quantum gravity, quantum cosmology, strings and branes, self-gravitating systems, gamma ray astronomy, cosmic rays and the history of general relativity

Routines and Applications of Symbolic Algebra Software

Computing has become an essential resource in modern research and has found application across a wide range of scientific disciplines. Developments in symbolic algebra tools have been particularly valuable in physics where calculations in fields such as general relativity, quantum field theory and physics beyond the standard model are becoming increasing complex and unpractical to work with by hand. The computer algebra system Cadabra is a tensor-first approach to symbolic algebra based on the programming language Python which has been used extensively in research in these fields while also having a shallow learning curve making it an excellent way to introduce students to methods in computer algebra. The work in this thesis has been concentrated on developing Cadabra, which has involved looking at two different elements which make up a computer algebra program. Firstly, the implementation of algebraic routines is discussed. This has primarily been focused on the introduction of an algorithm for detecting the equivalence of tensorial expressions related by index permutation symmetries. The method employed differs considerably from traditional canonicalisation routines which are commonly used for this purpose by using Young projection operators to make such symmetries manifest. The other element of writing a computer algebra program which is covered is the infrastruc- ture and environment. The importance of this aspect of software design is often overlooked by funding committees and academic software users resulting in an anti-pattern of code not being shared and contributed to in the way in which research itself is published and promulgated. The focus in this area has been on implementing a packaging system for Cadabra which allows the writing of generic libraries which can be shared by the community, and interfacing with other scientific computing packages to increase the capabilities of Cadabra

Research on Teaching and Learning In Biology, Chemistry and Physics In ESERA 2013 Conference

This paper provides an overview of the topics in educational research that were published in the ESERA 2013 conference proceedings. The aim of the research was to identify what aspects of the teacher-student-content interaction were investigated frequently and what have been studied rarely. We used the categorization system developed by Kinnunen, Lampiselkä, Malmi and Meisalo (2016) and altogether 184 articles were analyzed. The analysis focused on secondary and tertiary level biology, chemistry, physics, and science education. The results showed that most of the studies focus on either the teacher’s pedagogical actions or on the student - content relationship. All other aspects were studied considerably less. For example, the teachers’ thoughts about the students’ perceptions and attitudes towards the goals and the content, and the teachers’ conceptions of the students’ actions towards achieving the goals were studied only rarely. Discussion about the scope and the coverage of the research in science education in Europe is needed.Peer reviewe

MS FT-2-2 7 Orthogonal polynomials and quadrature: Theory, computation, and applications

Quadrature rules find many applications in science and engineering. Their analysis is a classical area of applied mathematics and continues to attract considerable attention. This seminar brings together speakers with expertise in a large variety of quadrature rules. It is the aim of the seminar to provide an overview of recent developments in the analysis of quadrature rules. The computation of error estimates and novel applications also are described