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:

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

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

If we now swap and , we get, using the commutativity of ordinary derivatives and the symmetry of :

Subtracting these two equations gives

Using the definition of the Riemann tensor:

we have

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

### Like this:

Like Loading...

*Related*