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 are used whenever practical calculations involving geometry must be performed, as they allow very complex calculations to be performed without confusion. Unfortunately, they are ugly-looking, and require careful attention to detail. By contrast, the index-less, formal notation for the Levi-Civita connection is quite beautiful, and allows theorems to be stated in an elegant way, but is next to useless for practical calculations.

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- and co-variant indices). The formulas hold for either sign convention, unless otherwise noted.

Definition

The Christoffel symbols can be derived from the vanishing of the covariant derivative of the metric tensor g_ik:

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

Note that although the symbols have three indices on them, they are not tensors. They do not transform like tensors. Rather, they are the components of an object on the second tangent bundle, a spray. See below for the transformation properties of the Christoffel symbols under a change of coordinate basis.

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

where are the commutation coefficients of the basis; that is,

where e_k are the basis vectors and [,] is the Lie bracket. An example of an anholonomic basis with non-vanishing commutation coefficients are spherical and cylinderical coordinates.

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

Relationship to index-less notation

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

.

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:

.

Keep in mind that and that , the Kronecker delta. The convention is that the 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 equation .

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

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

.

The article on covariant derivatives provides addtional discussion of the correspondence between index-free and indexed notation.

Relations

Contracting indices together, one gets

where |g| is the absolute value of the determinant of the metric tensor g_ik.

Similarly,

The covariant derivative of a vector V^m is

The covariant divergence is

.

The covariant derivative of a tensor A^ik is

.

If the tensor is antisymmetric, then its divergence simplifies to

.

The contravariant derivative of a scalar field φ is called the gradient of φ. That is, the gradient is the differential with the index raised:

The Laplacian of a scalar potential is given by

.

The laplacian is the covariant divergence of the gradient, that is Δφ = D_iDⁱφ.

Riemann curvature

The Riemann curvature tensor is given by

.

The symmetries of the tensor are

R_iklm = R_lmik and R_iklm = - R_kilm = - R_ikml.

That is, it is symmetric in the exchange of the first and last pair of indices, and antisymmetric in the flipping of a pair.

The cyclic permutation sum is

R_iklm + R_imkl + R_ilmk = 0

The Bianchi identity is

Ricci curvature

The Ricci tensor is given by

This tensor is symmetric: R_ik = R_ki. It is obtained from the Riemann curvature by contracting indices:

R_ik = g^lmR_limk

The scalar curvature is given by

R = g^ikR_ik.

The covariant derivative of the scalar curvature follows from the Bianchi identity:

.

Weyl tensor

The Weyl tensor is given by

.

Change of Variable

Under a change of variable from (x¹,...,xⁿ) to (y¹,...,yⁿ), vectors transform as

and so

where the overline denotes the Christoffel symbols in the y coordinate frame.

References

• Lev Davidovich Landau and Evgeny Mikhailovich Lifshitz, The Classical Theory of Fields, Fourth Revised English Edition, Course of Theoretical Physics, Volume 2, (1951) Pergamon Press, Oxford; ISBN 0-08-025072-6. See chapter 10, paragraphs 85,86 and 87.
• Ralph Abraham and Jerrold E. Marsden, Foundations of Mechanics, (1978) Benjamin/Cummings Publishing, London; ISBN 0-8053-0102-X. See chapter 2, paragraph 2.7.1
• Charles W. Misner, Kip S. Thorne, John Archibald Wheeler, Gravitation, (1970) W.H. Freeman, New York; ISBN 0-7167-0344-0. See chapter 8, paragraph 8.5