1996/01/19 by S. Brian Edgar, Edgar, S. Brian, A. Höglund +1
Earth and Planetary Sciences · Physics and Astronomy · #Black Holes and Theoretical Physics #Cosmology and Gravitation Theories #FOS: Physical sciences #General Relativity and Quantum Cosmology (gr-qc) #Geophysics and Gravity Measurements #gr-qc
paper · pdf · doi:10.48550/arxiv.gr-qc/9601029
34 pages, plain TeX
arxiv created 1996/01/19 · openalex publication_date 1996/01/19 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In the last few years renewed interest in the 3-tensor potential Labc proposed by Lanczos for the Weyl curvature tensor has not only clarified and corrected Lanczos's original work, but generalised the concept in a number of ways. In this paper we carefully summarise and extend some aspects of these results, and clarify some misunderstandings in the literature. We also clarify some comments in a recent paper by Dolan and Kim; in addition, we correct some internal inconsistencies in their paper and extend their results. The following new results are also presented. The (computer checked) complicated second order partial differential equation for the 3-potential, in arbitrary gauge, for Weyl candidates satisfying Bianchi-type equations is given -- in those n -dimensional spaces (with arbitrary signature) for which the potential exists; this is easily specialised to Lanczos potentials for the Weyl curvature tensor. It is found that it is \it only in 4-dimensional spaces (with arbitrary signature and gauge), that the non-linear terms disappear and that the awkward second order derivative terms cancel; for 4-dimensional spacetimes (with Lorentz signature), this remarkably simple form was originally found by Illge, using spinor methods. It is also shown that, for most 4-dimensional vacuum spacetimes, any 3-potential in the Lanczos gauges which satisfies a simple homogeneous wave equation must be a Lanczos potential for the Weyl curvature tensor of the background vacuum spacetime. This result is used to prove that the form of a \it possible Lanczos potential proposed by Dolan and Kim for a class of vacuum spacetimes is in fact a genuine Lanczos potential for these spacetimes.