## Generalised Conformal Killing-Yano Tensors:

Applications to Electrodynamics

####
Jonathan Kress, B. Sc. (Hons. ) (Ma. )

Thesis submitted for the degree of Doctor of Philosophy at the University
of Newcastle, Department of Mathematics

November 1997

### Abstract

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.

The thesis is is typeset for printing on A4 paper:

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