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

Authors
Journal
Journal of Geometry and Physics
0393-0440
Publisher
Elsevier
Publication Date
Volume
54
Issue
3
Identifiers
DOI: 10.1016/j.geomphys.2004.09.008
Keywords
  • Lanczos Potential
  • Weyl Tensor
  • Trace-Free Forms
  • Dimensionally Dependent Tensor Identities
Disciplines
  • Mathematics

Abstract

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 2 m dimensions.

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