Wave Propagation Through Caustics: Applications in Gravitational Wave Physics

Gravitational Waves -- propagating ripples in the fabric of space and time -- are a fundamental prediction of Einstein's theory of General Relativity. Nearly a century has passed since Einstein's masterwork, and yet gravitational waves remain frustratingly elusive! Astronomers have gathered strong evidence for their existence by measuring the rate of rotation of rapidly-spinning neutron stars ( pulsars ). Here on Earth, physicists are optimistic of direct detection within a decade.Gravitational waves (GWs) are important because they are generated by the most violent processes in the known Universe, such as supernovae, black hole mergers, and galaxy collisions. To the frustration of astronomers, such powerful processes are hidden behind shrouds of dust and strong fields. Light cannot penetrate this shroud, but gravitational waves can. By lifting the shroud, GWs will reveal the dynamic heart at the centre of such processes.Astronomy will enter a new era with the launch of the Laser Interferometer Space Antenna (LISA) in 2018 (est). A key aim of LISA is to detect gravitational waves emitted by binary black hole systems. In theory, the motion of a compact body orbiting a black hole can be reconstructed from the gravitational wave signal alone. A recent NASA-ESA report concludes that LISA will provide unambiguous and clean tests of the theory of general relativity in the strong field dynamical regime and be able to make detailed maps of spacetime near black holes. This is an exciting prospect, yet much theoretical work is needed if we are to separate out a small GW signal from a noisy background.In this project I will model so-called Extreme Mass Ratio Inspiral (EMRI) events, in which a small compact body spirals into a large black hole (e.g. Sag A*, residing at the centre of our galaxy). The gravitational wave signal emitted by EMRIs is a key target for LISA; hence this project is both timely and potentially significant.The most promising method for modelling EMRIs requires the calculation of a gravitational self-force which acts upon the small body. The self-force leads to a loss of orbital energy, causing the small body to spiral inwards, slowly at first, but with increasing rapidity. In curved spacetime -- for example, in the immediate vicinity of a black hole -- it turns out to be surprisingly difficult to compute the self-force, because it depends on the entire history of the small body's motion! In this project I will develop new mathematical methods to calculate the self-force, to complement the existing approach pioneered by members of the Southampton Relativity Group.The self-force arises from the interaction between a body and perturbations in its own gravitational field. Recent work suggests that, for EMRIs, the gravitational self-force may be primarily due to a non-local component which arises from perturbations that are lensed multiple times around the central black hole. To investigate this idea, I will make a mathematical study of the propagation of gravitational waves through focal points (also known as caustics). My primary motivation is to clarify the physical origin of the self-force, but other applications may also arise from this work.Since wave phenomena (diffraction, refraction, rainbows, glories, etc.) are ubiquitous across the physical sciences, I propose to make a multi-disciplinary study of the effects of caustics upon wave propagation. I will draw upon theoretical developments in seismology; catastrophe theory in optics; distribution theory in mathematics; and a range of other fields. I will also draw upon the breadth and depth of experience in the School of Mathematics at Southampton. Together with co-workers, I hope to make a significant contribution to the future development of precision Gravitational Wave Astronomy.