Connection Coefficient

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

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

(long form) or

$$\Gamma^{\alpha}_{\beta\gamma} \equiv {\vec e}\,^\alpha\cdot(\nabla_\gamma \vec e_\beta),$$ (2)

(abbreviated form), and satisfy

$$\nabla_{\vec e_\gamma} {\vec e_\beta} = \Gamma^{\vec e_\alpha}_{\vec e_\beta \vec e_\gamma} {\vec e}_\alpha$$ (3)

(long form) and

$$\nabla_\gamma \vec e_\beta = \Gamma^{\alpha}_{\beta\gamma} {\vec e}_\alpha$$ (4)

(abbreviated form).

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

$$\Gamma_{\alpha\beta\gamma} \equiv {\textstyle{1\over 2}}(g_{... ...lpha\beta\gamma}+c_{\alpha\gamma\beta}-c_{\beta\gamma\alpha}),$$ (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

$$\Gamma_{\alpha\beta\gamma}=\Gamma_{\alpha\beta}^{\mu}g_{\mu\... ...pha\beta\gamma}+ c_{\alpha\gamma\beta}-c_{\beta\gamma\alpha})$$ (6)

and

$$\Gamma_{ijk} = 0 \qquad {\rm for\ } i\not= j\not= k$$ (7)

$$\Gamma_{iik} = -{1\over 2} {\partial g_{ii}\over\partial x^k} \qquad {\rm for\ } i\not=k$$ (8)

$$\Gamma_{iji} = \Gamma_{jii}={1\over 2}{\partial g_{ii}\over\partial x^j}$$ (9)

$$\Gamma_{ij}^k = 0 \qquad {\rm for\ } i\not= j\not= k$$ (10)

$$\Gamma_{ii}^k = -{1\over 2g_{kk}} {\partial g_{ii}\over\partial x^k} \qquad {\rm for\ } i\not=k$$ (11)

$$\Gamma_{ij}^i = \Gamma_{ji}^i = {1\over 2g_{ii}} {\partial g_{ii}\over\partial x^j} = {1\over 2} {\partial \ln g_{ii}\over \partial x^j}.$$ (12)

For Tensors of Rank 3, the connection Coefficients may be concisely summarized in Matrix form:

$$\Gamma^\theta \equiv \left[{\matrix{ \Gamma^\theta_{rr} & \G... ...^\theta_{\phi \phi}\cr}}\right]. \hrule width 0pt height 5.9pt$$ (13)

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

$$d\tau^2=-\eta_{\alpha\beta}\,d\xi^\alpha\,d\xi^\beta,$$ (14)

or

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

Expanding,

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

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

But

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

so

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

where

$$\Gamma^\lambda_{\mu\nu} \equiv {\partial^2\xi^\alpha\over\pa... ...\partial x^\nu} {\partial x^\lambda \over \partial\xi^\alpha}.$$ (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