Axial symmetry and transverse trace-free tensors in numerical relativity

Transverse trace-free (TT) tensors play an important role in the initial conditions of numerical relativity, containing two of the component freedoms. Expressing a TT tensor entirely, by the choice of two scalar potentials, is not a trivial task however. Assuming the added condition of axial symmetry, expressions are given in both spherical and cylindrical coordinates, for TT tensors in flat space. A coordinate relation is then calculated between the scalar potentials of each coordinate system. This is extended to a non-flat space, though only one potential is found. The remaining equations are reduced to form a second order partial differential equation in two of the tensor components. With the axially symmetric flat space tensors, the choice of potentials giving Bowen-York conformal curvatures, are derived. A restriction is found for the potentials which ensure an axially symmetric TT tensor, which is regular at the origin, and conditions on the potentials, which give an axially symmetric TT tensor with a spherically symmetric scalar product, are also derived. A comparison is made of the extrinsic curvatures of the exact Kerr solution and numerical Bowen-York solution for axially symmetric black hole space-times. The Brill wave, believed to act as the difference between the Kerr and Bowen-York space-times, is also studied, with an approximate numerical solution found for a mass-factor, under different amplitudes of the metric.

Synthesis of iminophosphine and phosphinoiminol cyclometallated Pt (II) and Pt (IV) chloro complexes and studies into their biological and photophysical properties

This thesis focuses on the synthesis and analysis of novel chloride based platinum complexes derived from iminophosphine and phosphinoamide ligands, along with studies on their reactivity towards substitution and oxidation reactions. Also explored here are the potential applications of these complexes for biological and luminescent purposes. Chapter one provides an extensive overview of platinum coordination chemistry with examples of various mixed donor ligands along with the history of platinum anticancer therapy. It also looks at metals in medicine, both for biological functions as well as for therapeutic purposes and gives a background to some other applications for platinum complexes. Chapter two outlines the design and synthetic strategies employed for the development of novel platinum (II) chloride complexes from iminophosphine and phosphinoamide ligands. Also reported is the cyclometallation of these complexes to form stable tridentate mixed donor platinum (II) compounds. In Chapter three the development of a direct method for displacing a chloride from a platinum metal centre with a desired phosphine is reported. Numerous methods for successful oxidation of the platinum (II) complexes will also be explored, leading to novel platinum (IV) complexes being reported here also. The importance of stabilisation of the displaced anion, chloride, by the solvent system will also be discussed in this chapter. Chapter four investigates the reactivity of the platinum (II) complexes towards two different biomolecules to form novel platinum bio-adducts. The potential application of the platinum (II) cyclometallates as chemotherapeutics will also be explored here using in-vitro cancer cell testing. Finally, luminescence studies are also reported here for the ligands and platinum complexes reported in chapter two and three to investigate potential applications in this field also. Chapter five provides a final conclusion and an overall summary of the entire project as well as identifying key areas for future work.