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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
operator $D$ for curved spacetime, the covariant vector
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 the directional coderivative
${D_\mu }: = {{\mathbf{g}}_\mu } \cdot D$. By definition, its action
on a basis vector returns a new vector in the tangent space which
must be a linear sum
\[ {D_\mu
}{{\mathbf{g}}_\nu }: = \Gamma _{\mu \nu}^\kappa
{{\mathbf{g}}_\kappa }{\quad}{D_\mu }{{\mathbf{g}}^\nu } = - \Gamma
_{\mu \kappa }^\nu {{\mathbf{g}}^\kappa } \]
with some set of displacement fields
\[ \Gamma _{\mu \nu
}^\kappa ({\mathbf{x}}): = ({D_\mu }{{\mathbf{g}}_\nu }) \cdot
{{\mathbf{g}}^\kappa }: = {{\mathbf{g}}^\kappa } \cdot {\Gamma
_\mu }({{\mathbf{g}}_\nu }) \]
called the (Levi-Civita) connection coefficients or Christoffel
symbols (of the second kind). 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:
\[ \begin{gathered} 0 =
{D_\kappa }({{\mathbf{g}}_\mu } \cdot {{\mathbf{g}}^\nu }) =
({D_\kappa }{{\mathbf{g}}_\mu }) \cdot {{\mathbf{g}}^\nu } +
{{\mathbf{g}}_\mu } \cdot ({D_\kappa }{{\mathbf{g}}^\nu }) \\
=\Gamma _{\kappa \mu }^\lambda {{\mathbf{g}}_\lambda } \cdot
{{\mathbf{g}}^\nu } - \bar \Gamma _{\kappa \lambda }^\nu
{{\mathbf{g}}^\lambda } \cdot {{\mathbf{g}}_\mu } = \Gamma
_{\kappa \mu }^\nu - \bar \Gamma _{\kappa \mu }^\nu \hfill \\
\end{gathered} \]
The connection coefficients (1.15) 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 (affine) map between tangent spaces. The
introduction of this affine structure of the spacetime manifold
is mandatory in order to be able to compare tensor quantities at
different spacetime points.