Christoffel symbols

In mathematics and physics, the Christoffel symbols, named for Elwin Bruno Christoffel (1829–1900), are numerical arrays of real numbers that describe, in coordinates, the effects of parallel transport in curved surfaces and, more generally, manifolds. As such, they are coordinate-space expressions for the Levi-Civita connection derived from the metric tensor. In a broader sense, the connection coefficients of an arbitrary (not necessarily metric) affine connection in a coordinate basis are also called Christoffel symbols.[1] The Christoffel symbols may be used for performing practical calculations in differential geometry. For example, the Riemann curvature tensor can be expressed entirely in terms of the Christoffel symbols and their first partial derivatives.

At each point of the underlying n-dimensional manifold, for any local coordinate system, the Christoffel symbol is an array with three dimensions: n × n × n. Each of the n3 components is a real number.

Under linear coordinate transformations on the manifold, it behaves like a tensor, but under general coordinate transformations, it does not. In many practical problems, most components of the Christoffel symbols are equal to zero, provided the coordinate system and the metric tensor possess some common symmetries.

In general relativity, the Christoffel symbol plays the role of the gravitational force field with the corresponding gravitational potential being the metric tensor.

Preliminaries

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. Einstein summation convention is used in this article.

Definition

If xi, i = 1,2,...,n, is a local coordinate system on a manifold M, then the tangent vectors

$e_i = \frac{\partial}{\partial x^i}, \quad i=1,2,\dots,n$

define a basis of the tangent space of M at each point.

Christoffel symbols of the first kind

The Christoffel symbols of the first kind can be derived from the Christoffel symbols of the second kind and the metric,

$\Gamma_{\gamma \, \alpha \beta} = g_{\gamma \delta} \Gamma^{\delta}_{\alpha \beta} = {1 \over 2} (g_{\gamma \alpha, \beta} + g_{\beta \gamma, \alpha} - g_{\alpha \beta, \gamma}) \,.$

Christoffel symbols of the second kind (symmetric definition)

The Christoffel symbols of the second kind, using the definition symmetric in i and j,[2] $\Gamma^k_{ij}$ (sometimes Γkij ) are defined as the unique coefficients such that the equation

$\nabla_ie_j = \Gamma^k_{ij}e_k$

holds, where $\nabla_i$ is the Levi-Civita connection on M taken in the coordinate direction ei.

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

$0 = \nabla_\ell g_{ik}= \frac{\partial g_{ik}}{\partial x^\ell}- g_{mk}\Gamma^m_{i\ell} - g_{im}\Gamma^m_{k\ell} = \frac{\partial g_{ik}}{\partial x^\ell}- 2g_{m(k}\Gamma^m_{i)\ell}. \$

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

$0 = \,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 as a function of the metric tensor:

$\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,[3] since they do not transform like tensors under a change of coordinates; see below.

The Christoffel symbols are most typically defined in a coordinate basis, which is the convention followed here. However, the Christoffel symbols can also be defined in an arbitrary basis of tangent vectors ei by

$\nabla_{e_i}e_j = \Gamma^k_{ij}e_k.$

Explicitly, in terms of the metric tensor, this is[2]

$\Gamma^i_{k\ell}=\frac{1}{2}g^{im} \left( g_{mk,\ell} + g_{m\ell,k} - g_{k\ell,m} + c_{mk\ell}+c_{m\ell k} + c_{k\ell m} \right)\,$

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. The standard unit vectors in spherical and cylindrical coordinates furnish an example of a basis with non-vanishing commutation coefficients.

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

Christoffel symbols of the second kind (asymmetric definition)

A different definition of Christoffel symbols of the second kind is Misner et al.'s 1973 definition, which is asymmetric in i and j:[2]

$\Gamma^k_{ij} := {\widehat{\mathbf{u}}}_k \cdot \left( \nabla_j {\widehat{\mathbf{u}}}_i \right) .$

Relationship to index-free notation

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).\$

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:

$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.

Covariant derivatives of tensors

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}. \$

Change of variable

Under a change of variable from $(x^1,\dots,x^n)\$ to $(y^1,\dots,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 system. Note that the Christoffel symbol does not transform as a tensor, but rather as an object in the jet bundle.

In fact, at each point, there exist coordinate systems in which the Christoffel symbols vanish at the point.[4] These are called (geodesic) normal coordinates, and are often used in Riemannian geometry.

Applications to general relativity

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.

Notes

1. ^ See, for instance, (Spivak 1999) and (Choquet-Bruhat & DeWitt-Morette 1977)
2. ^ a b c http://mathworld.wolfram.com/ChristoffelSymboloftheSecondKind.html.
3. ^ See, for example, (Kreyszig 1991), page 141
4. ^ This is assuming that the connection is symmetric (e.g., the Levi-Civita connection). If the connection has torsion, then only the symmetric part of the Christoffel symbol can be made to vanish.

References

• Abraham, Ralph; Marsden, Jerrold E. (1978), Foundations of Mechanics, Benjamin/Cummings Publishing, pp. See chapter 2, paragraph 2.7.1, ISBN 0-8053-0102-X
• Choquet-Bruhat, Yvonne; DeWitt-Morette, Cécile (1977), Analysis, Manifolds and Physics, Amsterdam: Elsevier, ISBN 978-0720404944
• Landau, Lev Davidovich; Lifshitz, Evgeny Mikhailovich (1951), The Classical Theory of Fields, Course of Theoretical Physics, Volume 2 (Fourth Revised English ed.), Pergamon Press, pp. See chapter 10, paragraphs 85, 86 and 87, ISBN 0-08-025072-6
• Misner, Charles W.; Thorne, Kip S.; Wheeler, John Archibald (1970), Gravitation, W.H. Freeman, pp. See chapter 8, paragraph 8.5, ISBN 0-7167-0344-0
• Spivak, Michael (1999), A Comprehensive introduction to differential geometry, Volume 2, Publish or Perish, ISBN 0-914098-71-3

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