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Connection Coefficient

A quantity also known as a Christoffel Symbol of the Second Kind. Connection Coefficients are defined by

\Gamma^{\vec e_\alpha}_{\vec e_\beta \vec e_\gamma} \equiv \vec e\,^\alpha\cdot(\nabla_{\vec e_\gamma} {\vec e_\beta})
\end{displaymath} (1)

(long form) or
\Gamma^{\alpha}_{\beta\gamma} \equiv {\vec e}\,^\alpha\cdot(\nabla_\gamma \vec e_\beta),
\end{displaymath} (2)

(abbreviated form), and satisfy
\nabla_{\vec e_\gamma} {\vec e_\beta} = \Gamma^{\vec e_\alpha}_{\vec e_\beta \vec e_\gamma} {\vec e}_\alpha
\end{displaymath} (3)

(long form) and
\nabla_\gamma \vec e_\beta = \Gamma^{\alpha}_{\beta\gamma} {\vec e}_\alpha
\end{displaymath} (4)

(abbreviated form).

Connection Coefficients are not Tensors, but have Tensor-like Contravariant and Covariant indices. A fully Covariant connection Coefficient is given by

\Gamma_{\alpha\beta\gamma} \equiv {\textstyle{1\over 2}}(g_{...
\end{displaymath} (5)

where the $g$s are the Metric Tensors, the $c$s are Commutation Coefficients, and the commas indicate the Comma Derivative. In an Orthonormal Basis, $g_{\alpha\beta,\gamma}=0$ and $g_{\mu\gamma} =\delta_{\mu\gamma}$, so
\end{displaymath} (6)

$\displaystyle \Gamma_{ijk}$ $\textstyle =$ $\displaystyle 0 \qquad {\rm for\ } i\not= j\not= k$ (7)
$\displaystyle \Gamma_{iik}$ $\textstyle =$ $\displaystyle -{1\over 2} {\partial g_{ii}\over\partial x^k} \qquad {\rm for\ } i\not=k$ (8)
$\displaystyle \Gamma_{iji}$ $\textstyle =$ $\displaystyle \Gamma_{jii}={1\over 2}{\partial g_{ii}\over\partial x^j}$ (9)
$\displaystyle \Gamma_{ij}^k$ $\textstyle =$ $\displaystyle 0 \qquad {\rm for\ } i\not= j\not= k$ (10)
$\displaystyle \Gamma_{ii}^k$ $\textstyle =$ $\displaystyle -{1\over 2g_{kk}} {\partial g_{ii}\over\partial x^k} \qquad {\rm for\ } i\not=k$ (11)
$\displaystyle \Gamma_{ij}^i$ $\textstyle =$ $\displaystyle \Gamma_{ji}^i = {1\over 2g_{ii}} {\partial g_{ii}\over\partial x^j} = {1\over 2}
{\partial \ln g_{ii}\over \partial x^j}.$ (12)

For Tensors of Rank 3, the connection Coefficients may be concisely summarized in Matrix form:
\Gamma^\theta \equiv \left[{\matrix{ \Gamma^\theta_{rr} & \G...
...^\theta_{\phi \phi}\cr}}\right].
\hrule width 0pt height 5.9pt
\end{displaymath} (13)

Connection Coefficients arise in the computation of Geodesics. The Geodesic Equation of free motion is

\end{displaymath} (14)

{d^2\xi^\alpha\over d\tau^2} = 0.
\end{displaymath} (15)

{d\over d\tau}\left({{\partial\xi^\alpha\over\partial x^\mu}...
...u\partial x^\nu}
{dx^\mu\over d\tau} {dx^\nu\over d\tau} = 0
\end{displaymath} (16)

{\partial\xi^\alpha\over\partial x^\mu}{d^2x^\mu\over d\tau^...
...\over d\tau} {\partial x^\lambda\over\partial \xi^\alpha} = 0.
\end{displaymath} (17)

{\partial\xi^\alpha\over\partial x^\nu}{\partial x^\lambda\over\partial\xi^\alpha}= \delta^\lambda_\mu,
\end{displaymath} (18)


\delta_\mu^\lambda{d^2 x^\mu\over d\tau^2}+\left({{\partial^...
...Gamma^\lambda_{\mu\nu} {dx^\mu\over d\tau}{dx^\nu\over d\tau},
\end{displaymath} (19)

\Gamma^\lambda_{\mu\nu} \equiv {\partial^2\xi^\alpha\over\pa...
...\partial x^\nu} {\partial x^\lambda \over \partial\xi^\alpha}.
\end{displaymath} (20)

See also Cartan Torsion Coefficient, Christoffel Symbol of the First Kind, Christoffel Symbol of the Second Kind, Comma Derivative, Commutation Coefficient, Curvilinear Coordinates, Semicolon Derivative, Tensor

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© 1996-9 Eric W. Weisstein