A simple tensorial proof for the completely symmetric property of the Bel-Robinson tensor

2010/06/16 by Lau Loi So, So, Lau Loi
Computer Science · Engineering · Mathematics · Physics and Astronomy · #Advanced Mathematical Theories and Applications #Computational Physics and Python Applications #Cosmology and Gravitation Theories #Elasticity and Material Modeling #FOS: Physical sciences #General Relativity and Quantum Cosmology (gr-qc) #Relativity and Gravitational Theory #Tensor decomposition and applications #gr-qc

paper · pdf · doi:10.48550/arxiv.1006.3168

6 pages; added a new argument in section 3, results unchanged

openalex publication_date 2010/06/16 · arxiv created 2010/07/22 · arxiv updated 2010/07/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The Bel-Robinson tensor T_αβμν was proposed in 1958. The main application of this tensor is for describing gravitational energy. It is known that T_αβμν has many nice properties such as being completely symmetric. It is easy to prove this property using spinors as shown in Penrose's book. The present paper provides an alternative way, using the tensorial method to prove that T_αβμν is indeed totally symmetric. Moreover, we also found that the well known identity in vacuum, R_αλστR_β^λστ ≡ 1/4g_αβR_ρλστ R^ρλστ, which can be proven by the same tensorial method.

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