Christoffel Symbols

Relevant parts to questions...

• Second kind::${\Gamma }_{jk}^i={\mathbf{e}}^i\cdot \frac{\partial {\mathbf{e}}_j}{\partial x^k}$; the expansion coefficients of $\frac{\partial {\mathbf{e}}_j}{\partial x^k}={\Gamma }_{jk}^i{\mathbf{e}}_i$.
• First kind::${\Gamma }_{ijk}={\mathbf{e}}_i\cdot \frac{\partial {\mathbf{e}}_j}{\partial x^k}$.
• From the metric::${\Gamma }_{ijk}=\frac{1}{2}({\partial }_k{g}_{ij}+{\partial }_j{g}_{ik}-{\partial }_i{g}_{jk})$, and ${\Gamma }_{jk}^i=g^{i\ell }{\Gamma }_{\ell kj}$.
• Symmetric in $j,k$::${\Gamma }_{jk}^i={\Gamma }_{kj}^i$ and ${\Gamma }_{ijk}={\Gamma }_{ikj}$.
• Not a tensor - transforms with an extra inhomogeneous term.

The Christoffel Symbols are the correction coefficients that make Covariant Differentiation produce tensors in a Local Basis. They package the information about how basis vectors change with position.

Definitions

Second kind ${\Gamma }_{jk}^i$::the contravariant expansion coefficients of $\frac{\partial {\mathbf{e}}_j}{\partial x^k}$:

$$\boxed{{\Gamma }_{jk}^i={\mathbf{e}}^i\cdot \frac{\partial {\mathbf{e}}_j}{\partial x^k},\frac{\partial {\mathbf{e}}_j}{\partial x^k}={\Gamma }_{jk}^i{\mathbf{e}}_i}$$

First kind ${\Gamma }_{ijk}$::the covariant expansion coefficients (dot with ${\mathbf{e}}_i$ instead of ${\mathbf{e}}^i$):

$$\boxed{{\Gamma }_{ijk}={\mathbf{e}}_i\cdot \frac{\partial {\mathbf{e}}_j}{\partial x^k}}$$

In 3D each symbol has $3^3=27$ components, but the symmetry ${\Gamma }_{jk}^i={\Gamma }_{kj}^i$ reduces this to $3\cdot 6=18$ independent ones.

Formulae from the Metric

Using $g_{ij}={\mathbf{e}}_i\cdot {\mathbf{e}}_j$ and expanding $\frac{\partial g_{ij}}{\partial x^k}$ via the product rule, plus the symmetry ${\Gamma }_{ijk}={\Gamma }_{ikj}$, we can derive each symbol entirely from the Metric Tensor:

$$\boxed{{\Gamma }_{ijk}=\frac{1}{2}\left(\frac{\partial g_{ij}}{\partial x^k}+\frac{\partial g_{ik}}{\partial x^j}-\frac{\partial g_{jk}}{\partial x^i}\right)}$$ $$\boxed{{\Gamma }_{jk}^i=\frac{1}{2}{g}^{i\ell }\left(\frac{\partial g_{\ell j}}{\partial x^k}+\frac{\partial g_{\ell k}}{\partial x^j}-\frac{\partial g_{jk}}{\partial x^{\ell }}\right)}$$

No further geometric data is needed: the metric determines everything.

Properties

• Symmetry in the lower two indices::${\Gamma }_{jk}^i={\Gamma }_{kj}^i$ and ${\Gamma }_{ijk}={\Gamma }_{ikj}$ - follows from the equality of mixed partials $\frac{{\partial }^2\mathbf{r}}{\partial x^j\partial x^k}=\frac{{\partial }^2\mathbf{r}}{\partial x^k\partial x^j}$.
• Metric raises/lowers the first index::${\Gamma }_{ijk}=g_{i\ell }{\Gamma }_{jk}^{\ell }$ and ${\Gamma }_{jk}^i=g^{i\ell }{\Gamma }_{\ell kj}$.
• Vanish for a fixed basis::if $\frac{\partial {\mathbf{e}}_j}{\partial x^k}=0$, all $\Gamma =0$. This is the Cartesian special case.
• Not a tensor - see warning below.
• Orthogonal-coordinate antisymmetry (first kind only)::if $g_{ik}$ is diagonal, ${\Gamma }_{ik\ell }=-{\Gamma }_{ki\ell }$ for $i\ne k$ (consequence of Ricci’s Theorem).

Christoffel symbols are not tensors

Despite their index notation, they transform with an extra term:

${\Gamma }_{jk}^{\prime i}=L_m^{i^{\prime }}{L}_{j^{\prime }}^n{L}_{k^{\prime }}^p{\Gamma }_{np}^m+L_m^{i^{\prime }}\frac{\partial L_{j^{\prime }}^m}{\partial x^p}{L}_{k^{\prime }}^p$

The second term spoils the tensor transformation law. This is expected: Christoffel symbols encode how the basis changes, which is coordinate-dependent information, not intrinsic geometry.

Applications

1. Computing covariant derivatives of vectors and tensors in curvilinear coordinates.
2. Measuring curvature via the Riemann-Christoffel Tensor $R_{ijk}^r$, which is built from $\Gamma$ and its derivatives.
3. Geodesic equations ${\ddot{x}}^i+{\Gamma }_{jk}^i{\dot{x}}^j{\dot{x}}^k=0$ describing straight lines in curved spaces (not covered here but motivated by this machinery).

Christoffel symbols in a diagonal metric

For $(\rho ,\varphi ,\theta )$ with $g_{11}=1$, $g_{22}=r^2$, $g_{33}=\sin \varphi \sin \theta$, only three metric derivatives are nonzero. Applying the first-kind formula directly:

${\Gamma }_{122}=-\frac{1}{2}\frac{\partial g_{22}}{\partial \rho }=-r$, ${\Gamma }_{212}={\Gamma }_{221}=r$.

Raising the first index with $g^{11}=1,g^{22}=1/r^2$ gives ${\Gamma }_{22}^1=-r$ and ${\Gamma }_{12}^2={\Gamma }_{21}^2=\boxed{1/r}$.

Second-kind = upper, expands in ${\mathbf{e}}_i$. First-kind = all lower, expands in ${\mathbf{e}}^i$. Metric bridges them.