DIMENSIONAL REDUCTION AND THE LANCZOS TENSOR

The Lanczos tensor H αβγ is a potential for the Weyl tensor. Given the symmetries of these tensors it would be expected that the identification H αβ5 = Fαβ would give a reduction of the five dimensional vacuum field equations into equations related to the Einstein Maxwell equation, it is shown that this does not happen; furthermore it is shown that there is no dimensional reduction scheme involving the Lanczos tensor which agrees with the one devised by Kaluza and Klein in the weak field limit. The covariant derivative of the Weyl tensor can be expressed as a type of non-linear wave equation in the Lanczos tensor, the literature contains two incorrect expressions for this equation, here the correct expression is given for the first time. The expression for the Lanczos tensor in the case of weak fields is generalized. Some remarks are made on other approaches to include electro-magnetic theory into the theory of the Lanczos tensor.