The Lanczos potential, dimensionally dependent identities and algebraic Rainich theory
Abstract: In this thesis three major topics are dealt with: the Lanczos potential, dimensionally dependent identities and algebraic Rainich theory.The Lanczos potential is a potential for the \Vey! curvature tensor or for a tensor v-·ith the same algebraic symmetries. In this thesis we investigate wave equations for the Lanczos potential, not only for four dimensions as has been done before but in arbitrary dimension and also in arbitrary gauges. It is found that it is only in four dimensions that the wave equation has a simple form.We also investigate the existence of the Lanczos potential in dimensions higher than four. It is found that the Lanczos potential for the Wey! curvature tensor does not exist in all spaces of seven dimensions and higher, and, when not restricted to the Weyl curvature tensor, in five dimensions and higher.Dimensionally dependent identities are identities which are valid only in some dimensions. In this thesis we generalise a class of dimensionally dependent identities found by Lovelock. \Ve find necessary and sufficient conditions on tensors to satisfy these identities. This kind of identities are used extensively in the other parts of the thesis.Algebraic Rainich theory is about finding necessary and sufficient conditions for a tensor to be the superenergy tensor of a particular type. We find links between algebraic Rainich theory and identities by antisymmetrisation. Among the results is the necessary and sufficient condition on a two index tensor to be the superenergy tensor of a 2-form of rank four. This, together with already known results, completes the five dimensional case of algebraic Rainich theory for forms.
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