General Relativity: An Introduction for Physicists · Problem 4.8

Tensor calculus on manifolds — Hobson, Efstathiou & Lasenby

Physics · covariant-derivative-antisymmetric-tensor

Problem

v_a

are the covariant components of a vector and

A_{ab}

are the components of an antisymmetric rank-2 tensor, show that

\nabla_a v_b - \nabla_b v_a = \partial_a v_b - \partial_b v_a,

\nabla_a A_{bc} + \nabla_c A_{ab} + \nabla_b A_{ca} = \partial_a A_{bc} + \partial_c A_{ab} + \partial_b A_{ca}.

Determine the symmetry properties of the rank-3 tensor

B_{abc} = \partial_a A_{bc} + \partial_c A_{ab} + \partial_b A_{ca}.