Covariant derivative: commutativity

Reference: Moore, Thomas A., A General Relativity Workbook, University Science Books (2013) – Chapter 18; 8.

The second absolute gradient (or covariant derivative) of a four-vector is not commutative, as we can show by a direct derivation. Starting with the formula for the absolute gradient of a four-vector:

\displaystyle \nabla_{j}A^{k}\equiv\frac{\partial A^{k}}{\partial x^{j}}+A^{i}\Gamma_{\; ij}^{k} \ \ \ \ \ (1)

and the formula for the absolute gradient of a mixed tensor:

\displaystyle \nabla_{l}C_{j}^{i}=\partial_{l}C_{j}^{i}+\Gamma_{lm}^{i}C_{j}^{m}-\Gamma_{lj}^{m}C_{m}^{i} \ \ \ \ \ (2)

we can write out the second absolute gradient of a four-vector:

\displaystyle \nabla_{i}\left(\nabla_{j}A^{k}\right)=\partial_{i}\partial_{j}A^{k}+\Gamma_{j\ell}^{k}\partial_{i}A^{\ell}+A^{\ell}\partial_{i}\Gamma_{j\ell}^{k}-\Gamma_{ji}^{m}\left(\partial_{m}A^{k}+A^{\ell}\Gamma_{m\ell}^{k}\right)+\Gamma_{im}^{k}\left(\partial_{j}A^{m}+A^{\ell}\Gamma_{j\ell}^{m}\right) \ \ \ \ \ (3)

If we now swap {i} and {j}, we get, using the commutativity of ordinary derivatives and the symmetry of {\Gamma_{ji}^{m}}:

\displaystyle \nabla_{j}\left(\nabla_{i}A^{k}\right)=\partial_{i}\partial_{j}A^{k}+\Gamma_{i\ell}^{k}\partial_{j}A^{\ell}+A^{\ell}\partial_{j}\Gamma_{i\ell}^{k}-\Gamma_{ji}^{m}\left(\partial_{m}A^{k}+A^{\ell}\Gamma_{m\ell}^{k}\right)+\Gamma_{jm}^{k}\left(\partial_{i}A^{m}+A^{\ell}\Gamma_{i\ell}^{m}\right) \ \ \ \ \ (4)

Subtracting these two equations gives

\displaystyle \left(\nabla_{i}\nabla_{j}-\nabla_{j}\nabla_{i}\right)A^{k}=\left(\partial_{i}\Gamma_{j\ell}^{k}-\partial_{j}\Gamma_{i\ell}^{k}+\Gamma_{im}^{k}\Gamma_{j\ell}^{m}-\Gamma_{jm}^{k}\Gamma_{i\ell}^{m}\right)A^{\ell} \ \ \ \ \ (5)

Using the definition of the Riemann tensor:

\displaystyle R_{\; j\ell m}^{i}\equiv\partial_{\ell}\Gamma_{\; mj}^{i}-\partial_{m}\Gamma_{\;\ell j}^{i}+\Gamma_{\; mj}^{k}\Gamma_{\;\ell k}^{i}-\Gamma_{\;\ell j}^{k}\Gamma_{\; km}^{i} \ \ \ \ \ (6)

 


we have

\displaystyle \left(\nabla_{i}\nabla_{j}-\nabla_{j}\nabla_{i}\right)A^{k}=R_{\;\ell ij}^{k}A^{\ell} \ \ \ \ \ (7)

Thus the covariant derivative commutes only if the Riemann tensor is zero, which occurs only in flat spacetime.

Leave a Reply

Your email address will not be published. Required fields are marked *