Christoffel symbols

In mathematics and physics, the Christoffel symbols are an array of numbers describing a metric connection. ^[1] The metric connection is a specialization of the affine connection to surfaces or other manifolds endowed with a metric, allowing distances to be measured on that surface. In differential geometry, an affine connection can be defined without any reference to a metric, and many additional concepts follow: parallel transport, covariant derivatives, geodesics, etc. also do not require the concept of a metric.^[2][3] However, when a metric is available, these concepts can be directly tied to the "shape" of the manifold itself; that shape is determined by how the tangent space is attached to the cotangent space by the metric tensor.^[4] Abstractly, one would say that the manifold has an associated (orthonormal) frame bundle, with each "frame" being a possible choice of a coordinate frame. An invariant metric implies that the structure group of the frame bundle is the orthogonal group SO(m,n). As a result, such a manifold is necessarily a (pseudo-)Riemannian manifold.^[5][6] The Christoffel symbols provide a concrete representation of the connection of (pseudo-)Riemannian geometry in terms of coordinates on the manifold. Additional concepts, such as parallel transport, geodesics, etc. can then be expressed in terms of Christoffel symbols.

In general, there are an infinite number of metric connections for a given metric tensor; however, there is one, unique connection, the Levi-Civita connection, that is free of any torsion. It is very common in physics and general relativity to work almost exclusively with the Levi-Civita connection, by working in coordinate frames (called holonomic coordinates) where the torsion vanishes.

At each point of the underlying n-dimensional manifold, for any local coordinate system around that point, the Christoffel symbols are denoted Γⁱ_jk for i, j, k = 1, 2, ..., n. Each entry of this n × n × n array is a real number. Under linear coordinate transformations on the manifold, the Christoffel symbols transform like the components of a tensor, but under general coordinate transformations (diffeomorphisms) they do not. Most of the algebraic properties of the Christoffel symbols follow from their relationship to the affine connection; only a few follow from the fact that the structure group is the orthogonal group SO(m,n) (or the Lorentz group SO(3,1) for general relativity).

Christoffel symbols are used for performing practical calculations. For example, the Riemann curvature tensor can be expressed entirely in terms of the Christoffel symbols and their first partial derivatives. In general relativity, the connection plays the role of the gravitational force field with the corresponding gravitational potential being the metric tensor. When the coordinate system and the metric tensor share some symmetry, many of the Γⁱ_jk are zero.

The Christoffel symbols are named for Elwin Bruno Christoffel (1829–1900).^[7]

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. The connection coefficients of the Levi-Civita connection (or pseudo-Riemannian connection) expressed in a coordinate basis are called the Christoffel symbols.

Definition

Given a local coordinate system xⁱ, i = 1, 2, ..., n on a n-manifold M with metric tensor $g$, the tangent vectors

$\mathrm{e}_i = \frac{\partial}{\partial x^i}=\partial_i , \quad i=1,2,\dots,n$

define a local coordinate basis of the tangent space to M at each point of its domain.

Christoffel symbols of the first kind

The Christoffel symbols of the first kind can be derived either from the Christoffel symbols of the second kind and the metric,^[8]

$\Gamma_{cab} = g_{cd} \Gamma^{d}{}_{ab}\,,$

or from the metric alone,^[8]

$\Gamma_{cab} =\frac12 \left(\frac{\partial g_{ca}}{\partial x^b} + \frac{\partial g_{cb}}{\partial x^a} - \frac{\partial g_{ab}}{\partial x^c} \right) = \frac12\, (g_{ca, b} + g_{cb, a} - g_{ab, c}) = \frac12\, \left(\partial_{b}g_{ca} + \partial_{a}g_{cb} - \partial_{c}g_{ab}\right) \,.$

As an alternative notation one also finds^[7][9][10]

$\Gamma_{cab} = [ab, c].$

It is worth noting that $[ab, c] = [ba, c]$.^[11]

Christoffel symbols of the second kind (symmetric definition)

The Christoffel symbols of the second kind are the connection coefficients—in a coordinate basis—of the Levi-Civita connection, and since this connection has zero torsion, then in this basis the connection coefficients are symmetric, i.e., $\Gamma^k{}_{ij}=\Gamma^k{}_{ji}\,$.^[9] For this reason a torsion-free connection is often called 'symmetric'.

In other words, the Christoffel symbols of the second kind^[12][9] $\Gamma^k{}_{ij}$ (sometimes $\Gamma^{k}_{ij}$ or $\{\begin{smallmatrix} k\\ ij \end{smallmatrix}\}$)^[7][9] are defined as the unique coefficients such that the equation

$\nabla_i \mathrm{e}_j = \Gamma^k{}_{ij}\mathrm{e}_k$

holds, where $\nabla_i$ is the Levi-Civita connection on M taken in the coordinate direction $\mathrm{e}_{i}$, i.e., $\nabla_i\equiv \nabla_{\mathrm{e}_i}$ and where $\mathrm{e}_i=\partial_i$ is a local coordinate (holonomic) basis.

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

$0 = \,g_{ik;\ell} = g_{ik,\ell} - g_{mk} \Gamma^m{}_{i\ell} - g_{im} \Gamma^m{}_{k\ell} .$

Using that the symbols are symmetric in the lower two indices, one can solve explicitly for the Christoffel symbols as a function of the metric tensor by permuting the indices and resumming:^[11]

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

where $(g^{jk})$ is the 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,^[13] since they do not transform like tensors under a change of coordinates; see below.

Connection coefficients in a non holonomic basis

The Christoffel symbols are most typically defined in a coordinate basis, which is the convention followed here. In other words, the name Christoffel symbols is reserved only for coordinate (i.e., holonomic) frames. However, the connection coefficients can also be defined in an arbitrary (i.e., non holonomic) basis of tangent vectors $\mathbf{u}_i$ by

$\nabla_{\mathbf{u}_i}\mathbf{u}_j = \omega^k{}_{ij}\mathbf{u}_k.$

Explicitly, in terms of the metric tensor, this is^[12]

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

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

$[\mathbf{u}_k,\mathbf{u}_\ell] = c_{k\ell}{}^m \mathbf{u}_m\,\$

where $\mathbf{u}_k$ 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 difference between the connection in such a frame, and the Levi-Civita connection is known as the contorsion tensor.

Ricci rotation coefficients (asymmetric definition)

When we choose the basis $\mathrm{X}_i\equiv \mathbf{u}_i$ orthonormal: $g_{ab} \equiv \eta_{ab} = \langle X_a, X_b\rangle$ then $g_{mk,\ell}\equiv\eta_{mk,\ell}=0$. This implies that

$\omega^i{}_{k\ell}=\frac{1}{2}\eta^{im} \left( c_{mk\ell}+c_{m\ell k} - c_{k\ell m} \right)$

and the connection coefficients become antisymmetric in the first two indices:

$\omega_{abc} = - \omega_{bac}\, ,$

where $\omega_{abc} = \eta_{ad}\omega^d{}_{bc}$.

In this case, the connection coefficients $\omega^a{}_{bc}$ are called the Ricci rotation coefficients.^[14][15]

Equivalently, one can define Ricci rotation coefficients as follows:^[12]

$\omega^k{}_{ij} := {{\mathbf{u}}}^k \cdot \left( \nabla_j {{\mathbf{u}}}_i \right)\, ,$

where $\mathbf{u}_i$ is an orthonormal non holonomic basis and $\mathbf{u}^k=\eta^{k\ell}\mathbf{u}_{\ell}$ its co-basis.

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—in a coordinate basis—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 notation 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{}_{m \ell} \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}. \$

Contravariant derivatives of tensors

To find the contravariant derivative of a vector field, we must first transform it into a covariant derivative using the metric tensor

$\nabla^\ell V^m = g^{i\ell} \nabla_i V^m= g^{i\ell} \partial_i V^m + g^{i\ell} \Gamma^m_{ki} V^k=\partial^\ell V^m + g^{i\ell} \Gamma^m_{ki} V^k$

Change of variable

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

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

and so

${\bar\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. More precisely, the Christoffel symbols can be considered as functions on the jet bundle of the frame bundle of M, independent of any local coordinate system. Choosing a local coordinate system determines a local section of this bundle, which can then be used to pull back the Christoffel symbols to functions on M, though of course these functions then depend on the choice of local coordinate system.

At each point, there exist coordinate systems in which the Christoffel symbols vanish at the point.^[16] 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.

See also

• Basic introduction to the mathematics of curved spacetime
• Proofs involving Christoffel symbols
• Differentiable manifold
• List of formulas in Riemannian geometry
• Ricci calculus
• Riemann–Christoffel tensor
• Gauss–Codazzi equations
• Example computation of Christoffel symbols

Notes

1. See, for instance, (Spivak 1999) and (Choquet-Bruhat & DeWitt-Morette 1977)
2. Ronald Adler, Maurice Bazin, Menahem Schiffer, Introduction to General Relativity (1965) McGraw-Hill Book Company ISBN 0-07-000423-4 (See section 2.1)
3. Charles W. Misner, Kip S. Thorne, John Archibald Wheeler, Gravitation (1973) W. H,. Freeman ISBN 0-7167-0334-3 (See chapters 8-11)
4. Misner, Thorne, Wheeler, op. cit. (See chapter 13)
5. Jurgen Jost, Riemannian Geometry and Geometric Analysis, (2002) Springer-Verlag ISBN 3-540-42627-2
6. David Bleeker, Gauge Theory and Variational Principles (1991) Addison-Wesely Publishing Company ISBN 0-201-10096-7
7. Christoffel, E.B. (1869), "Ueber die Transformation der homogenen Differentialausdrücke zweiten Grades", Jour. für die reine und angewandte Mathematik, B. 70: 46–70
8. Ludvigsen, Malcolm (1999), General Relativity: A Geometrical Approach, p. 88
9. Chatterjee, U.; Chatterjee, N. (2010). Vector and Tensor Analysis. p. 480.
10. Struik, D.J. (1961). Lectures on Classical Differential Geometry (first published in 1988 Dover ed.). p. 114.
11. Bishop, R.L.; Goldberg (1968), Tensor Analysis on Manifolds, p. 241
12. http://mathworld.wolfram.com/ChristoffelSymboloftheSecondKind.html.
13. See, for example, (Kreyszig 1991), page 141
14. G. Ricci-Curbastro (1896). "Dei sistemi di congruenze ortogonali in una varietà qualunque". Mem. Acc. Lincei 2 (5): 276–322.
15. H. Levy (1925). "Ricci's coefficients of rotation". Bull. Amer. Math. Soc. 31 (3-4): 142–145. doi:10.1090/s0002-9904-1925-03996-8.
16. 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, London: Benjamin/Cummings Publishing, pp. See chapter 2, paragraph 2.7.1, ISBN 0-8053-0102-X
• Bishop, R.L.; Goldberg, S.I. (1968), Tensor Analysis on Manifolds (First Dover 1980 ed.), The Macmillan Company, ISBN 0-486-64039-6
• Choquet-Bruhat, Yvonne; DeWitt-Morette, Cécile (1977), Analysis, Manifolds and Physics, Amsterdam: Elsevier, ISBN 978-0-7204-0494-4
• Landau, Lev Davidovich; Lifshitz, Evgeny Mikhailovich (1951), The Classical Theory of Fields, Course of Theoretical Physics, Volume 2 (Fourth Revised English ed.), Oxford: Pergamon Press, pp. See chapter 10, paragraphs 85, 86 and 87, ISBN 0-08-025072-6
• Kreyszig, Erwin (1991), Differential Geometry, Dover Publications, ISBN 978-0-486-66721-8
• Misner, Charles W.; Thorne, Kip S.; Wheeler, John Archibald (1970), Gravitation, New York: W.H. Freeman, pp. See chapter 8, paragraph 8.5, ISBN 0-7167-0344-0
• Ludvigsen, Malcolm (1999), General Relativity: A Geometrical Approach, Cambridge University Press, ISBN 0-521-63019-3
• Spivak, Michael (1999), A Comprehensive introduction to differential geometry, Volume 2, Publish or Perish, ISBN 0-914098-71-3
• Chatterjee, U.; Chatterjee, N. (2010). Vector & Tensor Analysis. Academic Publishers. ISBN 978-93-8059-905-2.
• Struik, D.J. (1961). Lectures on Classical Differential Geometry (first published in 1988 Dover ed.). Dover. ISBN 0-486-65609-8.
• P.Grinfeld (2014). Introduction to Tensor Analysis and the Calculus of Moving Surfaces. Springer. ISBN 1-4614-7866-9.