The state space and physical interpretation of self-similar spherically symmetric perfect-fluid models

The purpose of this paper is to further investigate the solution space of self-similar spherically symmetric perfect-fluid models and gain deeper understanding of the physical aspects of these solutions. We achieve this by combining the state space description of the homothetic approach with the use of the physically interesting quantities arising in the comoving approach. We focus on three types of models. First, we consider models that are natural inhomogeneous generalizations of the Friedmann Universe; such models are asymptotically Friedmann in their past and evolve fluctuations in the energy density at later times. Second, we consider so-called quasi-static models. This class includes models that undergo self-similar gravitational collapse and is important for studying the formation of naked singularities. If naked singularities do form, they have profound implications for the predictability of general relativity as a theory. Third, we consider a new class of asymptotically Minkowski self-similar spacetimes, emphasizing that some of them are associated with the self-similar solutions associated with the critical behaviour observed in recent gravitational collapse calculations.Comment: 24 pages, 12 figure

Self-similar spherically symmetric cosmological models with a perfect fluid and a scalar field

Self-similar, spherically symmetric cosmological models with a perfect fluid and a scalar field with an exponential potential are investigated. New variables are defined which lead to a compact state space, and dynamical systems methods are utilised to analyse the models. Due to the existence of monotone functions global dynamical results can be deduced. In particular, all of the future and past attractors for these models are obtained and the global results are discussed. The essential physical results are that initially expanding models always evolve away from a massless scalar field model with an initial singularity and, depending on the parameters of the models, either recollapse to a second singularity or expand forever towards a flat power-law inflationary model. The special cases in which there is no barotropic fluid and in which the scalar field is massless are considered in more detail in order to illustrate the asymptotic results. Some phase portraits are presented and the intermediate dynamics and hence the physical properties of the models are discussed.Comment: 31 pages, 4 figure

A unified treatment of cubic invariants at fixed and arbitrary energy

Cubic invariants for two-dimensional Hamiltonian systems are investigated using the Jacobi geometrization procedure. This approach allows for a unified treatment of invariants at both fixed and arbitrary energy. In the geometric picture the invariant generally corresponds to a third rank Killing tensor, whose existence at a fixed energy value forces the metric to satisfy a nonlinear integrability condition expressed in terms of a Kahler potential. Further conditions, leading to a system of equations which is overdetermined except for singular cases, are added when the energy is arbitrary. As solutions to these equations we obtain several new superintegrable cases in addition to the previously known cases. We also discover a superintegrable case where the cubic invariant is of a new type which can be represented by an energy dependent linear invariant. A complete list of all known systems which admit a cubic invariant at arbitrary energy is given.Comment: 16 pages, LaTeX2e, slightly revised version. To appear in J. Math. Phys. vol 41, pp 370-384 (2000

Timelike self-similar spherically symmetric perfect-fluid models

Einstein's field equations for timelike self-similar spherically symmetric perfect-fluid models are investigated. The field equations are rewritten as a first-order system of autonomous differential equations. Dimensionless variables are chosen in such a way that the number of equations in the coupled system is reduced as far as possible and so that the reduced phase space becomes compact and regular. The system is subsequently analysed qualitatively using the theory of dynamical systems.Comment: 23 pages, 6 eps-figure

Lax pair tensors in arbitrary dimensions

A recipe is presented for obtaining Lax tensors for any n-dimensional Hamiltonian system admitting a Lax representation of dimension n. Our approach is to use the Jacobi geometry and coupling-constant metamorphosis to obtain a geometric Lax formulation. We also exploit the results to construct integrable spacetimes, satisfying the weak energy condition.Comment: 8 pages, uses IOP style files. Minor correction. Submitted to J. Phys

Equipping medical graduates to address health systems challenges in South Africa: An expressed need for curriculum change

Background: Stellenbosch University Rural Medical Education Partnership Initiative (SURMEPI) aims to enhance health systems knowledge and skills to empower medical graduates to address health systems challenges especially in rural and underserved areas.Objectives: To assess the content of health systems research (HSR) and strengthening, and understand perceptions of medical graduates and faculty about HSR in the undergraduate medical curriculum at Stellenbosch University.Methods: We defined HSR and strengthening competencies for medical graduates through a literature review and expert consultations. Learning outcomes in terms of knowledge, skill or attitude in the 64 module guides of the curriculum were compared with the competencies required. A survey of recent medical graduates assessed whether their training equipped them to address health systems challenges. Interviews with faculty assessed their views on teaching health systems competencies.Results: HSR foundational competencies were covered at a basic knowledge level, with little progression of learning levels, and several key competencies were not taught at all. Teaching was not integrated throughout the curriculum. Of 189 graduates, 63 (33.3%) agreed while 67 (35.4%) disagreed that their training prepared them to address health system challenges; 128 (67.7%) agreed on the importance of learning health systems competencies as undergraduates, and proposed learning areas of health system knowledge, leadership and management, problem solving, community service, evaluation methods and health economics. They wanted more practical, problem-oriented HSR training. Faculty supported the relevance and inclusion of HSR and strengthening in the curriculum.Conclusion: The curriculum needs adaptation to better equip students with HSR and strengthening competencies