Christoffel symbols

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In mathematics and physics, the Christoffel symbols, named for Elwin Bruno Christoffel (1829–1900), are coordinate-space expressions for the Levi-Civita connection derived from the metric tensor. The Christoffel symbols may be used for performing practical calculations in differential geometry. Unfortunately, the calculations are usually quite lengthy and complex, and require careful attention to detail. By contrast, the index-less, formal notation for the Levi-Civita connection is terse, and allows theorems to be stated in an elegant way, but requires more advanced techniques for practical calculations.

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The definitions given below are valid for both Riemannian manifolds and pseudo-Riemannian manifolds, such as those of general relativity, with careful distinction being made between upper and lower indices (contra-variant and co-variant indices). The formulas hold for either sign convention, unless otherwise noted. The Christoffel symbols are defined by:

\nabla_ie_j=\Gamma_{ij}^ke_k.

The Christoffel symbols can be derived from the vanishing of the covariant derivative of the metric tensor g_{ik}\ :

\nabla_\ell g_{ik}=\frac{\partial g_{ik}}{\partial x^\ell}- g_{mk}\Gamma^m {}_{i\ell} - g_{im}\Gamma^m {}_{k\ell}=0.\

As a shorthand notation, the nabla symbol and the partial derivative symbols are frequently dropped, and instead a semi-colon and a comma are used to set off the index that is being used for the derivative. Thus, the above is sometimes written as

\,g_{ik;\ell} = g_{ik,\ell} - g_{mk} \Gamma^m {}_{i\ell} - g_{im} \Gamma^m {}_{k\ell}. \

By permuting the indices, and resumming, one can solve explicitly for the Christoffel symbols:

\Gamma^i {}_{k\ell}=\frac{1}{2}g^{im} \left(\frac{\partial g_{mk}}{\partial x^\ell} + \frac{\partial g_{m\ell}}{\partial x^k} - \frac{\partial g_{k\ell}}{\partial x^m} \right) = {1 \over 2} g^{im} (g_{mk,\ell} + g_{m\ell,k} - g_{k\ell,m}), \

where the matrix (g^{jk}\ ) is an inverse of the matrix (g_{jk}\ ), defined as (using the Kronecker delta, and Einstein notation for summation) g^{j i} g_{i k}= \delta^j {}_k\ . Although the Christoffel symbols are written in the same notation as tensors with index notation, they are not tensors. Indeed, they do not transform like tensors under a change of coordinates; see below.

NB. Note that most authors choose to define the Christoffel symbols in a holonomic basis, which is the convention followed here. In an anholonomic basis, the Christoffel symbols take the more complex form

\Gamma^i {}_{k\ell}=\frac{1}{2}g^{im} \left( \frac{\partial g_{mk}}{\partial x^\ell} + \frac{\partial g_{m\ell}}{\partial x^k} - \frac{\partial g_{k\ell}}{\partial x^m} + c_{mk\ell}+c_{m\ell k} - c_{k\ell m} \right) \

where c_{k\ell m}=g_{mp} {c_{k\ell}}^p\ are the commutation coefficients of the basis; that is,

[e_k,e_\ell] = c_{k\ell}{}^m e_m\,\

where ek are the basis vectors and [,]\ is the Lie bracket. An example of an anholonomic basis with non-vanishing commutation coefficients are the unit vectors in spherical and cylindrical coordinates.

The expressions below are valid only in a holonomic basis, unless otherwise noted.

Let X and Y be vector fields with components X^i\ and Y^k\ . Then the kth component of the covariant derivative of Y with respect to X is given by

\left(\nabla_X Y\right)^k = X^i (\nabla_i Y)^k = X^i \left(\frac{\partial Y^k}{\partial x^i} + \Gamma^k {}_{im} Y^m\right).\

Some older physics books occasionally write dx in place of X, and place it after the equation, rather than before. Here, the Einstein notation is used, so repeated indices indicate summation over indices and contraction with the metric tensor serves to raise and lower indices:

\langle X,Y\rangle = g(X,Y) = X^i Y_i = g_{ik}X^i Y^k = g^{ik}X_i Y_k.\

Keep in mind that g_{ik}\neq g^{ik}\ and that g^i {}_k=\delta^i {}_k\ , the Kronecker delta. The convention is that the metric tensor is the one with the lower indices; the correct way to obtain g^{ik}\ from g_{ik}\ is to solve the linear equations g^{ij}g_{jk}=\delta^i {}_k\ .

The statement that the connection is torsion-free, namely that

\nabla_X Y - \nabla_Y X = [X,Y]\

is equivalent to the statement that the Christoffel symbol is symmetric in the lower two indices:

\Gamma^i {}_{jk}=\Gamma^i {}_{kj}.\

The index-less transformation properties of a tensor are given by pullbacks for covariant indices, and pushforwards for contravariant indices. The article on covariant derivatives provides additional discussion of the correspondence between index-free and indexed notation.

The covariant derivative of a vector field V^m\ is

\nabla_\ell V^m = \frac{\partial V^m}{\partial x^\ell} + \Gamma^m {}_{k\ell} V^k.\

The covariant derivative of a scalar field \varphi\ is just

\nabla_i \varphi = \frac{\partial \varphi}{\partial x^i}\

and the covariant derivative of a covector field \omega_m\ is

\nabla_\ell \omega_m = \frac{\partial \omega_m}{\partial x^\ell} - \Gamma^k {}_{\ell m} \omega_k.\

The symmetry of the Christoffel symbol now implies

\nabla_i\nabla_j \varphi = \nabla_j\nabla_i \varphi\

for any scalar field, but in general the covariant derivatives of higher order tensor fields do not commute (see curvature tensor).

The covariant derivative of a type (2,0) tensor field A^{ik}\ is

\nabla_\ell A^{ik}=\frac{\partial A^{ik}}{\partial x^\ell} + \Gamma^i {}_{m\ell} A^{mk} + \Gamma^k {}_{m\ell} A^{im}, \

that is,

A^{ik} {}_{;\ell} = A^{ik} {}_{,\ell} + A^{mk} \Gamma^i {}_{m\ell} + A^{im} \Gamma^k {}_{m\ell}. \

If the tensor field is mixed then its covariant derivative is

A^i {}_{k;\ell} = A^i {}_{k,\ell} + A^{m} {}_k \Gamma^i {}_{m\ell} - A^i {}_m \Gamma^m {}_{k\ell}, \

and if the tensor field is of type (0,2) then its covariant derivative is

A_{ik;\ell} = A_{ik,\ell} - A_{mk} \Gamma^m {}_{i\ell} - A_{im} \Gamma^m {}_{k\ell}. \

Under a change of variable from (x^1,...,x^n)\ to (y^1,...,y^n)\ , vectors transform as

\frac{\partial}{\partial y^i} = \frac{\partial x^k}{\partial y^i}\frac{\partial}{\partial x^k}\

and so

\overline{\Gamma^k {}_{ij}} = \frac{\partial x^p}{\partial y^i}\, \frac{\partial x^q}{\partial y^j}\, \Gamma^r {}_{pq}\, \frac{\partial y^k}{\partial x^r} + \frac{\partial y^k}{\partial x^m}\, \frac{\partial^2 x^m}{\partial y^i \partial y^j} \

where the overline denotes the Christoffel symbols in the y coordinate frame. Note that the Christoffel symbol does not transform as a tensor, but rather as an object in the jet bundle.

The Christoffel symbols find frequent use in Einstein's theory of general relativity, where spacetime is represented by a curved 4-dimensional Lorentz manifold with a Levi-Civita connection. The Einstein field equations - which determine the geometry of spacetime in the presence of matter - contain the Ricci tensor, and so calculating the Christoffel symbols is essential. Once the geometry is determined, the paths of particles and light beams are calculated by solving the geodesic equations in which the Christoffel symbols explicitly appear.

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