Thesis submitted for the degree of Doctor of Philosophy at the University of Newcastle, Department of Mathematics
In this thesis various generalisations of the conformal Killing equation are considered. The generalisation to totally antisymmetric tensors leads to conformal Killing-Yano (CKY) tensors. It is shown how the existence of shear-free congruences is equivalent to the existence of tensors satisfying a generalised CKY equation. Although some of the results are specific to the four-dimensional Lorentzian case, the formalism allows for all dimensions and signatures. The exterior calculus is used throughout.
The above formalism is applied to the Debye potential scheme for finding solutions to Maxwell's equations in curved spacetimes, following on from the work of Cohen and Kegeles. This Debye scheme leads naturally to the existence of symmetry operators on the space of Maxwell fields. It is shown how the Debye scheme is related to symmetry operators found by Kalnins et al. A second order symmetry of the conformally covariant Laplace-Beltrami equation is also found to be related to the Debye potential. In the treatment of these symmetries, tensors which can be regarded as generalisations of both the conformal Killing and CKY equations are introduced.
The problem of finding a force-free magnetic field is restated in terms of finding a vacuum Maxwell field on a particular curved spacetime. The Debye potential scheme is then applied to construct force-free fields. The connection is made between Debye potentials and the Chandrasekhar-Kendall eigenfunctions. Some new force-free fields, with non-constant eigenvalue, are presented.