Brien C. Nolan

A point mass in an isotropic universe: II. Global properties

The global structure of McVitties solution representing a point mass embedded in a spatially flat Robertson-Walker universe is investigated. The scalar curvature singularity at proper radius R=2m, where m (constant) is the Schwarzschild mass, and the apparent horizon which surrounds it are studied. The conformal diagram for the spacetime is obtained via a qualitative analysis of the radial null geodesics. Particular attention is paid to the physical interpretation of this spacetime; previous work on this issue is reviewed, and to how recent quasi-local definitions of black and white holes relate to this spacetime.

A point mass in an isotropic universe: III. The region Rleq2m

McVitties solution of Einsteins field equations, representing a point mass embedded in an isotropic universe, possesses a scalar curvature singularity at proper radius R = 2m. The singularity is spacelike and precedes, in the expanding case, all other events in the spacetime. It is shown here that this singularity is gravitationally weak, and the possible structure of the region R2m is investigated. A characterization of this solution which does not involve asymptotics is given.

Naked singularities in cylindrical collapse of counterrotating dust shells

≤ 0, (2)which we assume henceforth. The matter content ofthe space-time may be described as infalling null dust.There is a curvature singularity along r = 0. Since thesolution is Petrov type N with a pure radiation energy-momentum tensor, all curvature invariants vanish. It hasbeen shown in [1] that this is a parallel propagated curva-ture singularity. Additionally, scalars such as R

Strengths of singularities in spherical symmetry

Covariant equations characterizing the strength of a singularity in spherical symmetry are derived and several models are investigated. The difference between central and non-central singularities is emphasised. A slight modification to the definition of singularity strength is suggested. The gravitational weakness of shell crossing singularities in collapsing spherical dust is proven for timelike geodesics, closing a gap in the proof.

Transfer in chemistry: a study of students’ abilities in transferring mathematical knowledge to chemistry

It is recognized that there is a mathematics problem in chemistry, whereby, for example, undergraduate students appear to be unable to utilize basic calculus knowledge in a chemistry context – calculus knowledge – which would have been taught to these students in a mathematics context. However, there appears to be a scarcity of literature addressing the possible reasons for this problem. This dearth of literature has spurred the following two questions: (1) Can students transfer mathematical knowledge to chemistry?; and (2) What are the possible factors associated with students being able to successfully transfer mathematical knowledge to a chemistry context? These questions were investigated in relation to the basic mathematical knowledge which chemistry students need for chemical kinetics and thermodynamics, using the traditional view of the transfer of learning. Two studies were undertaken amongst two samples of undergraduate students attending Dublin City University. Findings suggest that the mathematical difficulties which students encounter in a chemistry context may not be because of an inability to transfer the knowledge, but may instead be due to insufficient mathematical understanding and/or knowledge of mathematical concepts relevant to chemical kinetics and thermodynamics.

Non-radial null geodesics in spherical dust collapse

The issue of the local visibility of the shell-focusing singularity in marginally bound spherical dust collapse is considered from the point of view of the existence of future-directed null geodesics with angular momentum which emanate from the singularity. The initial data (i.e. the initial density profile) at the onset of collapse is taken to be of class C 3 . Simple necessary and sufficient conditions for the existence of a naked singularity are derived in terms of the data. It is shown that there exist future-directed non-radial null geodesics emanating from the singularity if and only if there exist future-directed radial null geodesics emanating from the singularity. This result can be interpreted as indicating the robustness of previous results on radial geodesics, with respect to the presence of angular momentum.

Yang's Gravitational Theory

Yangs pure space equations generalize Einsteins gravitational equations, while coming from gauge theory. We study these equations from a number of vantage points: summarizing the work done previously, comparing them with the Einstein equations and investigating their properties. In particular, the initial value problem is discussed and a number of results are presented for these equations with common energy-momentum tensors.

On the existence of dyons and dyonic black holes in Einstein?Yang?Mills theory

We study dyonic soliton and black hole solutions of the Einstein–Yang–Mills equations in asymptotically anti-de Sitter space. We prove the existence of non-trivial dyonic soliton and black hole solutions in a neighbourhood of the trivial solution. For these solutions the magnetic gauge field function has no zeros and we conjecture that at least some of these non-trivial solutions will be stable. The global existence proof uses local existence results and a nonlinear perturbation argument based on the (Banach space) implicit function theorem.

Geometry and topology of singularities in spherical dust collapse

We derive some more results on the nature of the singularities arising in the collapse of inhomogeneous dust spheres. (i) It is shown that there are future-pointing radial and non-radial time-like geodesics emerging from the singularity if and only if there are future-pointing radial null geodesics emerging from the singularity. (ii) Limits of various spacetime invariants and other useful quantities (relating to Thornes point–cigar–barrel–pancake classification and to isotropy/entropy measures) are studied in the approach to the singularity. (iii) The topology of the singularity is studied from the point of view of ideal boundary structure. In each case, the different nature of the visible and censored region of the singularity is emphasized.

Particle and photon orbits in McVittie spacetimes

McVittie spacetimes represent an embedding of the Schwarzschild field in isotropic cosmological backgrounds. Depending on the scale factor of the background, the resulting spacetime may contain black and white hole horizons, as well as other interesting boundary features. In order to further clarify the nature of these spacetimes, we address this question: do there exist bound particle and photon orbits in McVittie spacetimes? Considering first circular photon orbits (CPOs), we obtain an explicit characterization of all McVittie spacetimes for which such orbits exist: there is a two-parameter class of such spacetimes, and so the existence of a CPO is a highly specialized feature of a McVittie spacetime. However, we prove that in two large classes of McVittie spacetimes, there are bound particle and photon orbits: future-complete non-radial timelike and null geodesics along which the areal radius r has a finite upper bound. These geodesics are asymptotic at large times to circular orbits of a corresponding Schwarzschild or Schwarzschild?de Sitter spacetime. The existence of these geodesics lays the foundations for and shows the theoretical possibility of the formation of accretion disks in McVittie spacetimes. We also summarize and extend some previous results on the global structure of McVittie spacetimes. The results on bound orbits are established using centre manifold and other techniques from the theory of dynamical systems.