A quantity also known as a Christoffel Symbol of the Second Kind.
Connection Coefficients are defined by
![\begin{displaymath}
\Gamma^{\vec e_\alpha}_{\vec e_\beta \vec e_\gamma} \equiv \vec e\,^\alpha\cdot(\nabla_{\vec e_\gamma} {\vec e_\beta})
\end{displaymath}](c2_1372.gif) |
(1) |
(long form) or
![\begin{displaymath}
\Gamma^{\alpha}_{\beta\gamma} \equiv {\vec e}\,^\alpha\cdot(\nabla_\gamma \vec e_\beta),
\end{displaymath}](c2_1373.gif) |
(2) |
(abbreviated form), and satisfy
![\begin{displaymath}
\nabla_{\vec e_\gamma} {\vec e_\beta} = \Gamma^{\vec e_\alpha}_{\vec e_\beta \vec e_\gamma} {\vec e}_\alpha
\end{displaymath}](c2_1374.gif) |
(3) |
(long form) and
![\begin{displaymath}
\nabla_\gamma \vec e_\beta = \Gamma^{\alpha}_{\beta\gamma} {\vec e}_\alpha
\end{displaymath}](c2_1375.gif) |
(4) |
(abbreviated form).
Connection Coefficients are not Tensors, but have Tensor-like Contravariant and Covariant indices. A fully Covariant
connection Coefficient is given by
![\begin{displaymath}
\Gamma_{\alpha\beta\gamma} \equiv {\textstyle{1\over 2}}(g_{...
...lpha\beta\gamma}+c_{\alpha\gamma\beta}-c_{\beta\gamma\alpha}),
\end{displaymath}](c2_1376.gif) |
(5) |
where the
s are the Metric Tensors, the
s are Commutation Coefficients, and the commas indicate the Comma Derivative. In an Orthonormal Basis,
and
, so
![\begin{displaymath}
\Gamma_{\alpha\beta\gamma}=\Gamma_{\alpha\beta}^{\mu}g_{\mu\...
...pha\beta\gamma}+
c_{\alpha\gamma\beta}-c_{\beta\gamma\alpha})
\end{displaymath}](c2_1379.gif) |
(6) |
and
![$\displaystyle \Gamma_{ijk}$](c2_1380.gif) |
![$\textstyle =$](c2_7.gif) |
![$\displaystyle 0 \qquad {\rm for\ } i\not= j\not= k$](c2_1381.gif) |
(7) |
![$\displaystyle \Gamma_{iik}$](c2_1382.gif) |
![$\textstyle =$](c2_7.gif) |
![$\displaystyle -{1\over 2} {\partial g_{ii}\over\partial x^k} \qquad {\rm for\ } i\not=k$](c2_1383.gif) |
(8) |
![$\displaystyle \Gamma_{iji}$](c2_1384.gif) |
![$\textstyle =$](c2_7.gif) |
![$\displaystyle \Gamma_{jii}={1\over 2}{\partial g_{ii}\over\partial x^j}$](c2_1385.gif) |
(9) |
![$\displaystyle \Gamma_{ij}^k$](c2_1386.gif) |
![$\textstyle =$](c2_7.gif) |
![$\displaystyle 0 \qquad {\rm for\ } i\not= j\not= k$](c2_1381.gif) |
(10) |
![$\displaystyle \Gamma_{ii}^k$](c2_1387.gif) |
![$\textstyle =$](c2_7.gif) |
![$\displaystyle -{1\over 2g_{kk}} {\partial g_{ii}\over\partial x^k} \qquad {\rm for\ } i\not=k$](c2_1388.gif) |
(11) |
![$\displaystyle \Gamma_{ij}^i$](c2_1389.gif) |
![$\textstyle =$](c2_7.gif) |
![$\displaystyle \Gamma_{ji}^i = {1\over 2g_{ii}} {\partial g_{ii}\over\partial x^j} = {1\over 2}
{\partial \ln g_{ii}\over \partial x^j}.$](c2_1390.gif) |
(12) |
For Tensors of Rank 3, the connection Coefficients may be
concisely summarized in Matrix form:
![\begin{displaymath}
\Gamma^\theta \equiv \left[{\matrix{ \Gamma^\theta_{rr} & \G...
...^\theta_{\phi \phi}\cr}}\right].
\hrule width 0pt height 5.9pt
\end{displaymath}](c2_1391.gif) |
(13) |
Connection Coefficients arise in the computation of Geodesics. The Geodesic
Equation of free motion is
![\begin{displaymath}
d\tau^2=-\eta_{\alpha\beta}\,d\xi^\alpha\,d\xi^\beta,
\end{displaymath}](c2_1392.gif) |
(14) |
or
![\begin{displaymath}
{d^2\xi^\alpha\over d\tau^2} = 0.
\end{displaymath}](c2_1393.gif) |
(15) |
Expanding,
![\begin{displaymath}
{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}](c2_1394.gif) |
(16) |
![\begin{displaymath}
{\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}](c2_1395.gif) |
(17) |
But
![\begin{displaymath}
{\partial\xi^\alpha\over\partial x^\nu}{\partial x^\lambda\over\partial\xi^\alpha}= \delta^\lambda_\mu,
\end{displaymath}](c2_1396.gif) |
(18) |
so
![\begin{displaymath}
\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}](c2_1397.gif) |
(19) |
where
![\begin{displaymath}
\Gamma^\lambda_{\mu\nu} \equiv {\partial^2\xi^\alpha\over\pa...
...\partial x^\nu} {\partial x^\lambda \over \partial\xi^\alpha}.
\end{displaymath}](c2_1398.gif) |
(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
© 1996-9 Eric W. Weisstein
1999-05-26