General Relativity/Christoffel symbols

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Definition of Christoffel Symbols

Consider an arbitrary contravariant vector field defined all over a Lorentzian manifold, and take Aⁱ at xⁱ, and at a neighbouring point, the vector is Aⁱ + dAⁱ at xⁱ + dxⁱ.

Next parallel transport Aⁱ from xⁱ to xⁱ + dxⁱ, and suppose the change in the vector is δAⁱ. Define:

DAⁱ = dAⁱ − δAⁱ

The components of δAⁱ must have a linear dependence on the components of Aⁱ. Define Christoffel symbols $\Gamma^{i}_{kl}$:

$\delta A^{i} = -\Gamma^{i}_{kl} A^{k} dx^{l}$

Note that these Christoffel symbols are:

• dependent on the coordinate system (hence they are NOT tensors)
• functions of the coordinates

Now consider arbitrary contravariant and covariant vectors Aⁱ and B_i respectively. Since AⁱB_i is a scalar, δ(AⁱB_i) = 0, one arrives at:

B_iδAⁱ + AⁱδB_i = 0

$\Rightarrow A^{i}\delta B_{i} = \Gamma^{i}_{kl} A^{k} B_{i} dx^{l}$

$\Rightarrow A^{i}\delta B_{i} = \Gamma^{k}_{il} A^{i} B_{k} dx^{l}$

$\Rightarrow \delta B_{i} = \Gamma^{k}_{il} B_{k} dx^{l}$

Connection Between Covariant And Regular Derivatives

From above, one can obtain the relations between covariant derivatives and regular derivatives:

${A^{i}}_{;l} = \frac{\partial A^{i}}{\partial x^{l}} + \Gamma^{i}_{kl} A^{k}$

$A_{i;l} = \frac{\partial A_{i}}{\partial x^{l}} - \Gamma^{k}_{il} A_{k}$

Analogously, for tensors:

${A^{ik}}_{;l} = \frac{\partial A^{ik}}{\partial x^{l}} + \Gamma^{i}_{ml}A^{mk} + \Gamma^{k}_{ml}A^{im}$

Calculation of Christoffel Symbols

From $g_{ik}DA^{k} + A^{k}Dg_{ik} = D\left(g_{ik}A^{k}\right) = DA_{i} = g_{ik} DA^{k}$, one can conclude that g_ik;l = 0.

However, since g_ik is a tensor, its covariant derivative can be expressed in terms of regular partial derivatives and Christoffel symbols:

$g_{ik;l} = \frac{\partial g_{ik}}{\partial x^{l}} - g_{mk}\Gamma^{m}_{il} - g_{im}\Gamma^{m}_{kl} = 0$

Rewriting the expression above, and then performing permutation on i, k and l:

$\frac{\partial g_{ik}}{\partial x^{l}} = g_{mk}\Gamma^{m}_{il} + g_{im}\Gamma^{m}_{kl}$

$\frac{\partial g_{kl}}{\partial x^{i}} = g_{ml}\Gamma^{m}_{ki} + g_{km}\Gamma^{m}_{li}$

$-\frac{\partial g_{li}}{\partial x^{k}} = - g_{mi}\Gamma^{m}_{lk} - g_{lm}\Gamma^{m}_{ik}$

Adding up the three expressions above, one arrives at (using the notation $\frac{\partial A^{i}}{\partial x^{j}} = {A^{i}}_{,j}$):

$2 g_{mk}\Gamma^{m}_{il} = g_{ik,l} + g_{kl,i} - g_{li,k}$

Multiplying both sides by $\frac{1}{2} g^{kn}$:

$\Rightarrow \Gamma^{n}_{il} = \delta^{n}_{m}\Gamma^{m}_{il} = \frac{1}{2} g^{kn}\left(g_{ik,l} + g_{kl,i} - g_{li,k}\right)$

Hence if the metric is known, the Christoffel symbols can be calculated.