I am a member of the Mathematical Physics Group and have a number of interests all of which involve the application of modern differential geometry to spacetime physics and Newtonian field theories.

### Relativistic continuum mechanics

Many of the laws of classical (i.e. non-relativistic) continuum mechanics are formulated in terms of Newtonian concepts and require the Euclidean structure of Galilean space-time. Such ideas are incompatible with the fundamental postulates of General Relativity and can only appear in a non-relativistic reduction of a relativistic theory. Cosserat rod theories afford an economical description of non-relativistic slender media but are formulated in terms of balance laws using Euclidean structures. My PhD research concentrated on the interaction of slender elastic media and the gravitational field, in both Newtonian and General Relativistic contexts, and I developed a General Relativistic theory of Cosserat rods. The aim was to investigate novel methods of gravitational wave detection and I showed that space-based schemes with the mathematical concept of strings and rods at their core are both useful and highly desirable. In recent years, I revisited relativistic continuum mechanics and worked on a theory of relativistic anelasticity describing the material evolution of defects and growth compatible with General Relativity.

### Vortex-Induced Vibration

After completing my PhD in 2000 I worked on the effective simulation of vortex-induced vibrations of marine risers in collaboration with Orcina Ltd. A realistic, predictive and computationally efficient mathematical model of a fluid interacting with one or more solid bodies is both desirable, from an engineering perspective, and challenging for the physicist to construct. The sort of physical situations I examined consisted of a slender solid body immersed in a fluid with finite inflow speed and possessed a Reynolds number of the order of 100,000. The fluid sector of the calculations employed discrete-vortex methods and has also been used to model light wind-rain induced vibrations of cable-stayed bridges.

### Twisted modes and gravitomagnetic fields

This work involves the geometrical modelling of electromagnetic modes in closed tubular cavities, e.g. a toroidal cavity, on background spacetimes. We call such cavities "wavetubes" because they possess propagating modes (waves). The aim was to understand how the wavetube and spacetime geometry alters the behaviour of the contained electromagnetic field. Most analyses in this field tend to rely on geometrical ray-optics arguments and completely neglect the vectorial nature of the electromagnetic field. Using the covariant Maxwell equations for accelerating media we showed how the spacetime and wavetube geometries influence the mode spectrum. The shape of the wavetube is specified by a closed space-curve on 3-dimensional Euclidean space. Any such space-curve is specified on Euclidean space, up to rigid motions, by its Frenet curvature and torsion. The Frenet torsion is a measure of the non-planarity of the space-curve and we showed how its mean value contributes to the mode spectrum. This work is of relevance to those who intend to measure gravitomagnetic phenomena, such as the Lense-Thirring effect, using ring-lasers.

### High-field plasmas and radiation reaction

Recent developments in high-energy laser science have seen a surge of interest in the collective behaviour of high-density charged relativistic matter. However, the dynamical behaviour of relativistic charged matter reacting to its own electromagnetic field has been a source of intense debate ever since the problem was first tackled by Dirac in the 1930s. Until recently, this area was of purely theoretical interest but contemporary developments in high-energy laser science have ensured that it is no longer the case. Our recent work has focussed on the development of many-body theories of radiating charged matter, which we are using to explore the non-linear collective behaviour of intense radiating matter and probe parameter regimes in which the radiation reaction of plasmas and beams is significant.

This work is supported by the Cockcroft Institute.