ar X iv : g r - qc / 0 20 20 81 v 1 2 1 Fe b 20 02 The non - existence of a Lanczos potential for the Weyl curvature tensor in dimensions n ≥ 7

In this paper it is shown that a Lanczos potential for the Weyl curvature tensor does not exist for all spaces of dimension n ≥ 7. Whether there exists a Lanczos potential [1] for Weyl curvature tensors in dimensions n > 4 has still not been determined. Although Lanczos's original proof [1] for existence was flawed [2], there have subsequently been complete proofs for existence in four dimensions [2, 3, 4]. The latter two proofs [3, 4] are in spinors, although in [4] a translation into tensors is given which is explicitly for four dimensions, independent of signature. The detailed and complicated proof given by Bampi and Caviglia [2] is also explicitly for four dimensions, although they also discuss briefly the possibility of existence in higher dimensions. An important aspect of all of these proofs is that they are not only valid for Weyl curvature tensors C abcd , but for Weyl candidate tensors W abcd , i.e., any 4-tensor having the index symmetries of the Weyl curvature tensor, W abcd = W [ab]cd = W ab[cd] = W cdab W a bad = 0. (1a) 0 = W a[bcd] (1b) In a recent paper [5] we have shown that a Lanczos potential for a Weyl candidate tensor does not generally exist for dimensions n > 4. In particular, we have shown that in flat and conformally flat spaces with dimensions n > 4, the assumed existence of a Lanczos potential for an arbitrary Weyl candidate imposes non-trivial conditions on the Weyl candidate. However, this result does not say anything about the existence of a Lanczos potential for a Weyl curvature tensor in spaces with non-zero conformal curvature in dimensions n > 4. In this paper we will address the problem of Weyl curvature tensors directly, and show explicitly, that A Lanczos potential for the Weyl curvature tensor does not exist for all spaces of dimension n ≥ 7.