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- TL;DR
- Reason why the traceless part of the torsion tensor is usually set to zero in theories that extend general relativity to include electromagnetism?
When extending general relativity to include electromagnetism, several authors (e.g. Novello, Sabbata ecc.) assume that the traceless part of the torsion tensor vanishes or is deliberately set to zero. Then, either the trace or axial part of the torsion is used in association with the electromagnetic potential (coupling). Is there any reason why, besides mathematical convenience, the leftover part of the torsion is set to zero?
Is it related to gauge invariance?
Furthermore, is it correct to consider the decomposition of the torsion tensor into three components - i.e., trace part, axial part, and traceless part - as the most general one?
The decomposition I'm referring to is the following: $$T^λ_{μν} = \bar{T}^λ_{μν}+\frac{1}{6}ϵ_{λμνρ}V^ρ+\frac{1}{3}(g_{λν}T_μ − g_{λμ}T_ν)$$ where ##\bar{T}^λ_{μν}## is the traceless part of torsion, ##V^ρ## is the axial torsion vector or "pseudo-trace" and ##T_μ## is the torsion trace vector. This is found for example in Sur and Bhatia (Appendix, A-7 to A-10).
Is it related to gauge invariance?
Furthermore, is it correct to consider the decomposition of the torsion tensor into three components - i.e., trace part, axial part, and traceless part - as the most general one?
The decomposition I'm referring to is the following: $$T^λ_{μν} = \bar{T}^λ_{μν}+\frac{1}{6}ϵ_{λμνρ}V^ρ+\frac{1}{3}(g_{λν}T_μ − g_{λμ}T_ν)$$ where ##\bar{T}^λ_{μν}## is the traceless part of torsion, ##V^ρ## is the axial torsion vector or "pseudo-trace" and ##T_μ## is the torsion trace vector. This is found for example in Sur and Bhatia (Appendix, A-7 to A-10).