2003/10/29 by Ola Wingbrant, Wingbrant, Ola
Mathematics · Physics and Astronomy · #Black Holes and Theoretical Physics #FOS: Physical sciences #General Relativity and Quantum Cosmology (gr-qc) #Particle physics theoretical and experimental studies #Tensor decomposition and applications #gr-qc
paper · pdf · doi:10.48550/arxiv.gr-qc/0310120
LaTeX2e, 18 pages
arxiv created 2003/10/29 · openalex publication_date 2003/10/29 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In an n dimensional vector space, any tensor which is antisymmetric in k>n arguments must vanish; this is a trivial consequence of the limited number of dimensions. However, when other possible properties of tensors, for example trace-freeness, are taken into account, such identities may be heavily disguised. Tensor identities of this kind were first considered by Lovelock, and later by Edgar and Hoeglund. In this paper we continue their work. We obtain dimensionally dependent identities for highly structured expressions of products of (2,2)-forms. For tensors possessing more symmetries, such as block symmetry Wabcd = Wcdab, or the first Bianchi identity Wa[bcd] = 0, we derive identities for less structured expressions. These identities are important tools when studying super-energy tensors, and, in turn, deriving identities for them. As an application we are able to show that the Bel-Robinson tensor, the super-energy tensor for the Weyl tensor, satisfies the equation TabcyTabcx = 1/4gxyTabcdTabcd in four dimensions, irrespective of the signature of the space.