I do research in General Relativity (which for me, nearly always involves research on various types of differential equation), and have recently become interested in research in Mathematics Education.
Isolated objects in an expanding universe
A series of theorems established in the latter half of the 1960's by Penrose, Hawking, Geroch, Ellis, Schmidt and others showed that space-times which are subject to a certain set of conditions must inevitably be singular at some point in their evolution. In the different theorems, these conditions break down into two sets. The first set roughly says that (i) the matter/energy content of the space-time is positive at each point; (ii) the space-time is NOT highly specialised or symmetric (notice that this means the theorems INCLUDE rather than EXCLUDE physically realistic situations); (iii) a causality condition holds, for example that no observer can travel into his or her own past. The second set gives just one condition, and is essentially that the gravitational field is very strong in a particular region of the space-time. Such conditions occur when a star exhausts its nuclear fuel and all other sources of outward pressure, and so collapses under its own mass. These conditions also hold in the early universe. Thus our universe is subject to the conditions of the singularity theorems, and so we must expect that singularities will occur.
However the nature of these singularities is not described by the theorems, and it is this issue which I study. Currently, my research focuses on the question of
Cosmic censorship
The Cosmic Censorship Hypothesis (CCH) was formulated by Roger Penrose in 1969 as a 'boundary condition' for general relativity which disqualifies the singularities which inevitably form in gravitational collapse from taking any further part in the evolution of the space-time model of the universe. Violation of the CCH leads to such unlikely scenarios as the availability of infinite amounts of energy from finite sources and the fundamental inability of physical theory to predict the evolution of the universe. A proof of the hypothesis has thus far eluded relativists, and this problem remains the outstanding open problem of classical gravitation theory. Evidence in favour of the CCH currently outweighs evidence against it. The starting point of a refutation of the hypothesis is to show that singularities arising in the gravitational collapse of compact objects may be visible to observers (i.e. may be naked or uncensored). Such singularities can then be the source of arbitrary data which leads to a breakdown in predictability.
Many examples of naked singularities exist; however they generally assume that the collapsing object is highly symmetric and hence unrealistic. Numerous authors have given examples of spherically symmetric, self-similar space-times which admit naked singularities. However little work has been done on the question of whether or not these examples provide serious evidence against the CCH. In order to address this question, I am currently studying various types of perturbations of the spherically symmetric self-similar space-times; configurations which are nearly spherically symmetric and nearly self-similar (in an appropriately defined sense) are physically realistic. The fundamental question is if the 'nakedness' of the singularity is stable against these perturbations. If so, then one has a class of space-times which provide a serious counter-example to the CCH. If not, one has added significantly to the mounting evidence in favour of the CCH. For recent papers on this topic, follow these links: gr-qc/0505036; gr-qc/0406048; gr-qc/0210035; gr-qc/0010032.
Strengths of singularities
Another topic of importance in the study of space-time singularities is that of the strength of the singularity. Ellis, Schmidt and Tipler initiated this work in the mid-1970's. Tipler defined a singularity to be 'gravitationally strong' if it crushes an object which impinges upon it. It is important to realise that not all singularities are strong, and the mathematical (and consequently physical) ramifications of encountering a singularity which is gravitationally weak are not yet completely understood. I am currently working on the connection between the gravitational strength of singularities and the mathematical structure of the metric. In particular, I am studying the connection between weak singularities and weak solutions of Einstein's equation; see gr-qc/0301028 for a recent paper and this link for a conference presentation on the topic.
Gravitational Collapse
According to the theorems mentioned above, once gravitational collapse has progressed to the stage where the field may be considered strong (technically, this corresponds to the formation of a trapped surface), the later appearence of a singularity is inevitable. An important question is then to characterize initial data of regular configurations in terms of (i) whether or not trapped surfaces form and (ii) the nature of any singularities that do form. I have studied this question for the collapse of different spherically and cylindrically symmetric configurations. See gr-qc/0407041; gr-qc/0405041; gr-qc/0204036; gr-qc/0203078; gr-qc/0108008.
Isolated objects in an expanding universe
The studies of spatially compact bodies and local systems, i.e. astrophysical objects such as stars, planetary systems, galaxies, clusters, etc, in General Relativity (GR) have usually focused on the case when they are truly isolated. This means that the models constructed describe the object embedded in an exterior vacuum region where the gravitational field decays to zero when moving far away from the boundary of the object. The aim of this project is to study the case where the compact objects are somehow ``isolated'' but embedded in a cosmological background, which is taken to be a standard spatially homogeneous and isotropic Friedman-Lemaitre-Robertson-Walker (FLRW) model. This is a more realistic situation for any astrophysical object, since moving away from the boundary one should eventually reach a large scale region which ought to be not flat but described by a dynamical cosmological model. The ultimate motivation is to determine the influence of cosmological dynamics on local systems, and to assess how this influence may affect cosmological observations. This topic has been called the study of ``Finite Infinity", and has been identified by G.F.R. Ellis as a key open question of modern cosmology. See gr-qc/0505093; gr-qc/9907018; gr-qc/9805041.
I am particularly interested in any area of maths education research that feeds directly into teaching practice, e.g. topics such as assessment for learning, guided inquiry and other activity-based learning methods. I am currently involved in a project using Sol Friedberg's ideas about the use of Case Studies in maths tutor training. A potential future research project (for which funding is being sought) involves a study of the maths used and needed in the Irish high-tech sector, and how current third level maths education practice meets these requirements. I am also interested in assessing the effect on student learning in my own teaching practice of the bits and pieces of teaching and learning methods I have picked up at conferences and workshops.
Current Students
Orlaith Mannion (1st year MSc student) is working on the influence of cosmological expansion on local physics. In particular, she is studying null and time-like orbits in McVittie's space-time - which represents the Schwarzschild field embedding in an isotropic universe.
Eoin Condron (1st year PhD track student) is working on cylindrical gravitational collapse. He is studying self-similar collapse of a cylindrical non-linear scalar field, and looking at alternative definitions of Thorne's C-energy.
Emily Duffy (2nd year PhD student) is studying the stability of Cauchy horizons in self-similar collapse. She is applying energy method for hyperbolic systems to study the linear stability of Cauchy horizons in self-similar dust and perfect fluid collapse.
Richard Hoban (3rd year PhD student) is studying the transfer of mathematical knowledge and skills by third level Chemistry students. He is studying the underlying pedagogical sources of successful transfer.
Previous Students
Louise Nolan (no relation) worked on cylindrical gravitational collapse. She studied the cylindrical version of the Oppenheimer-Snyder model (collapse of an isotropic cylindrical star) - see gr-qc/0407041, and the state space of self-similar cylindrical dust collapse. Louise completed her PhD in September 2007 and is currently lecturing in the School of Mathematical Sciences in UCD.
Thomas Waters worked on the stability of Cauchy horizons (and other similarity horizons) in self-similar collapse. He studied scalar waves in a general self-similar spherically symmetric space-time admitting a naked singularity, as well as perturbations of the self-similar Vaidya and Lemaitre-Tolman-Bondi space-times. See gr-qc/0505036, gr-qc/0210035 and arXiv:0903.3243 Thomas completed his PhD in December 2005. Since then he has worked as a post-doc in the University of Strathclyde and is currently lecturing in the Department of Mathematical Physics, NUI Galway.
Prospective students. If you are interested in undertaking postgraduate research on any of the topics above, please contact me by e-mail.
Last modified: January 18, 2010 11:26