General Relativity: An Introduction for Physicists · Problem 7.16

The equivalence principle and spacetime curvature — Hobson, Efstathiou & Lasenby

Physics · commutator-covariant-derivatives

Problem

If vav^a are the contravariant components of a vector and TabT^{ab} are the contravariant components of a rank-2 tensor, prove the results ∇c∇bva−∇b∇cva=−Radbcvd,\nabla_c\nabla_b v^a - \nabla_b\nabla_c v^a = -R^a{}_{dbc}v^d, ∇d∇cTab−∇c∇dTab=−RaecdTeb−RbecdTae.\nabla_d\nabla_c T^{ab} - \nabla_c\nabla_d T^{ab} = -R^a{}_{ecd}T^{eb} - R^b{}_{ecd}T^{ae}. Can you guess the corresponding result for the mixed components TabcT^{ab}{}_c of a rank-3 tensor?

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