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I. Metric and Connection
Equation (1.3) defines ${\mathbf{a}} \cdot \nabla $ as the directed
derivative along a vector ${\mathbf{a}}$. When this
directional derivative acts on a tangent vector field there is no
guarantee that the resulting vector also lies entirely in the same
tangent space, even if ${\mathbf{a}}$ does. In order to restrict to
quantities intrinsic to the manifold, one defines a new derivative
$D$ for curved spacetime, the covariant derivative or coderivative
for short, which is the covariant generalization of the gradient
$\nabla $. In the literature, the coderivative is often denoted as
(bold) $\nabla$ and the covariant directional derivative as (bold)
${\nabla _{\mathbf{a}}}$.
The coderivative is usually introduced with reference to the local
coordinate system by defining ${D_\mu }: = {{\mathbf{g}}_\mu } \cdot
D$ as the covariant vector derivative. By definition, its action on
a basis vector returns a new vector in the tangent space which must
be a linear (affine) sum
\[ {D_\mu
}{{\mathbf{g}}_\nu }: = \Gamma _{\mu \nu}^\kappa
{{\mathbf{g}}_\kappa }{\quad}{D_\mu }{{\mathbf{g}}^\nu } = - \Gamma
_{\mu \kappa }^\nu {{\mathbf{g}}^\kappa } \]
for some set of scalars
\[ \Gamma _{\mu \nu
}^\kappa ({\mathbf{x}}): = ({D_\mu }{{\mathbf{g}}_\nu }) \cdot
{{\mathbf{g}}^\kappa }: = {{\mathbf{g}}^\kappa } \cdot {\Gamma
_\mu }({{\mathbf{g}}_\nu }) \]
called the (Levi-Civita) connection coefficients or Christoffel
symbols (of the second kind). These coefficients give the
rate of change of the basis vector ${{\mathbf{g}}_\nu }$ in the
direction of the basis vector ${{\mathbf{g}}_\mu }$ and may be
seen as components of a linear function of the coordinate base
vectors. The placement of the indices follows the convention of
[Wikipedia:
Christoffel Symbols].
The second equation (1.14) follows from the first. Proof:
assume the coefficient in the second equation is some other
coefficient $\bar \Gamma _{\mu \kappa }^\nu $. Using the
orthogonality of reciprocal basis vectors one then deduces they
must be the same:
\[ \begin{gathered} 0 =
{D_\kappa }({{\mathbf{g}}_\mu } \cdot {{\mathbf{g}}^\nu }) =
({D_\kappa }{{\mathbf{g}}_\mu }) \cdot {{\mathbf{g}}^\nu } +
{{\mathbf{g}}_\mu } \cdot ({D_\kappa }{{\mathbf{g}}^\nu }) \\
=\Gamma _{\kappa \mu }^\lambda {{\mathbf{g}}_\lambda } \cdot
{{\mathbf{g}}^\nu } - \bar \Gamma _{\kappa \lambda }^\nu
{{\mathbf{g}}^\lambda } \cdot {{\mathbf{g}}_\mu } = \Gamma
_{\kappa \mu }^\nu - \bar \Gamma _{\kappa \mu }^\nu \hfill \\
\end{gathered} \]