Proofs of existence of local potentials for trace-free symmetric 2-forms using dimensionally dependent identities

2004/07/01 by S. Brian Edgar, Edgar, S. Brian
Mathematics · Physics and Astronomy · #Black Holes and Theoretical Physics #Differential Geometry (math.DG) #FOS: Mathematics #FOS: Physical sciences #General Relativity and Quantum Cosmology (gr-qc) #Quantum Chromodynamics and Particle Interactions #Quantum chaos and dynamical systems #gr-qc #math.DG

paper · pdf · doi:10.48550/arxiv.gr-qc/0407002

8 pages; Tex file

arxiv created 2004/07/01 · openalex publication_date 2004/07/01 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We exploit four-dimensional tensor identities to give a very simple proof of the existence of a Lanczos potential for a Weyl tensor in four dimensions with any signature, and to show that the potential satisfies a simple linear second order differential equation, e.g., a wave equation in Lorentz signature. Furthermore, we exploit higher dimensional tensor identities to obtain the analogous results for (m,m)-forms in 2m dimensions.

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