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I. Metric and Connection
On scalar
functions the coderivative ${D_\mu } = {{\mathbf{g}}_\mu } \cdot D$,
$D = {{\mathbf{g}}^\mu }{D_\mu }$, acts as an ordinary spacetime
derivative ${D_\mu }f({\mathbf{x}}): = {\partial _\mu }f({\mathbf{x}})$ . With
this rule and the Levi-Civita connection as defined in (1.14), covariant
derivatives of a vector ${\mathbf{A}}$ are calculated as:
\[ {D_\mu }{\mathbf{A}} = {D_\mu
}({A^\nu } {{\mathbf{g}}_\nu }) = ({\partial _\mu }{A^\nu } + \Gamma _{\mu
\kappa }^\nu {A^\kappa }) {{\mathbf{g}}_\nu }: = {A^\nu }_{;\mu
}{{\mathbf{g}}_\nu } \]
\[ D \cdot {\mathbf{A}} =
{{\mathbf{g}}^\mu } {D_\mu } \cdot ({A^\nu }{{\mathbf{g}}_\nu }) = {\partial
_\mu }{A^\mu } + \Gamma _{\mu \kappa }^\mu {A^\kappa } \]
\[ {\mathbf{a}} \cdot
D{\mathbf{A}} = {a^\mu }{D_\mu }({A^\nu }{{\mathbf{g}}_\nu }) = {a^\mu
}({\partial _\mu }{A^\nu } + \Gamma _{\mu \kappa }^\nu {A^\kappa
}){{\mathbf{g}}_\nu } \]
\[ D \wedge {\mathbf{A}} =
{{\mathbf{g}}^\mu }{D_\mu } \wedge ({A_\nu }{{\mathbf{g}}^\nu })= {\partial
_\mu }{A_\nu } {{\mathbf{g}}^\mu } \wedge {{\mathbf{g}}^\nu } + {A_\nu }D
\wedge {{\mathbf{g}}^\nu } \]
The last
term of (1.20) is zero in GRT; see equation (1.26). If and only if
that is the case, the covariant curl $D \wedge $ is equivalent to the exterior
derivative ${\text{d}}$ in (1.7). Note the standard notation ‘;’ for the
covariant derivative in the component formalism.
If
${\mathbf{F}}$ is the 2-form (1.8), then:
\[ \begin{gathered} D \cdot
{\mathbf{F}} = {{\mathbf{g}}^\mu } \cdot \left[ {{D_\mu }\left( {\frac{1}{2}
{F^{\kappa \lambda }}{{\mathbf{g}}_\kappa } \wedge {{\mathbf{g}}_\lambda }}
\right)} \right] \\ \qquad \qquad \qquad = \left( {{\partial _\kappa
}{F^{\kappa \lambda }} + \Gamma _{\nu \kappa }^\nu {F^{\kappa \lambda }}}
\right){{\mathbf{g}}_\lambda } {\text{:}} = {F^{\kappa \lambda }}_{;\kappa
}{{\mathbf{g}}_\lambda } \end{gathered} \]
\[ D \cdot D \cdot {\mathbf{F}} =
{g^\mu } \cdot {D_\mu }({F^{\kappa \lambda }}_{;\kappa }{{\mathbf{g}}_\lambda
}) = {F^{\mu \nu }}{\partial _\nu }\Gamma _{\kappa \mu }^\kappa \]
The last term of (1.22) actually vanishes because ${F^{\mu \nu }}$ is
antisymmetric in its indices, whereas the second factor turns out to be
symmetric; see (1.29).