Covariant derivative ambiguity

Solution 1:

I think that the confusion in such a matter is generated by a slight misuse of notation that is usually made by physicists (such as myself).

The correct action of the covariant derivative, say, on a vector $A=A^a e_a$, for a coordinate basis $e_a=\partial_a$, is of course $$ \nabla_bA\equiv \nabla_{b}\left(A^a e_a\right)=\left(\nabla_bA^a\right)e_a+A^a\nabla_be_a =\left(\partial_bA^a+\Gamma^a_{bc}A^c\right)e_a, $$ where $$ \nabla_bA^a=\partial_bA^a $$ and the definition of connection $$ \nabla_ae_b=\Gamma^c_{ab}e_c $$ have been used.

However, since the $a$-th component of the covariant derivative $\nabla_b$ of a vector of components $A^a$ is $\partial_bA^a+\Gamma^a_{bc}A^c$, physics textbooks tend to use the improper notation $$ \nabla_bA^a=\partial_bA^a+\Gamma^a_{bc}A^c $$ in the calculations, with the convention that this is in fact a shorthand for the above computation. In practice, this writing leaves the basis vectors ''implicit''.