christoffel symbols

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-variant 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}

D_lg_{ik}=\frac{\partial g_{ik}}{\partial x^l}- g_{mk}\Gamma^m_{il} - g_{im}\Gamma^m_{kl}=0.

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

\Gamma^i_{kl}=\frac{1}{2}g^{im} \left(\frac{\partial g_{mk}}{\partial x^l} + \frac{\partial g_{ml}}{\partial x^k} - \frac{\partial g_{kl}}{\partial x^m} \right).

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

\Gamma^i_{kl}=\frac{1}{2}g^{im} \left(

\frac{\partial g_{mk}}{\partial x^l} + \frac{\partial g_{ml}}{\partial x^k} - \frac{\partial g_{kl}}{\partial x^m} + c_{mkl}+c_{mlk} - c_{klm} \right) where

c_{klm}=g_{mp} {c_{kl}}^p

are the commutation coefficients of the basis; that is,

= {c_{kl}}^m e_m

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 cylindrical 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

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 D_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

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 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

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.

Relations

Contracting indices together, one gets

\Gamma^i_{ki}=\frac{1}{2} g^{im}\frac{\partial g_{im}}{\partial x_k}=\frac{1}{2g} \frac{\partial g}{\partial x_k} = \frac{\partial \log \sqrt

/dd> where |g| is the absolute value of the determinant of the metric tensor

g_{ik}

. Similarly,

g^{kl}\Gamma^i_{kl}=\frac{-1}{\sqrt

/dd> The covariant derivative of a vector

V^m

D_l V^m = \frac{\partial V^m}{\partial x^l} + \Gamma^m_{kl} V^k.

The covariant divergence is

D_m V^m = \frac{\partial V^m}{\partial x^m} + V^k \frac{\partial \log \sqrt

/dd> The covariant derivative of a tensor

A^{ik}

D_l A^{ik}=\frac{\partial A^{ik}}{\partial x^l} + \Gamma^i_{ml} A^{mk} + \Gamma^k_{ml} A^{im}

If the tensor is antisymmetric, then its divergence simplifies to

D_k A^{ik}= \frac{1}{\sqrt{|g|}} \frac{\partial (A^{ik}\sqrt{|g

)}{\partial x^k}.

The contravariant derivative of a scalar field

\phi

is called the gradient of

\phi

. That is, the gradient is the differential with the index raised:

D^i\phi=g^{ik}\frac{\partial\phi}{\partial x^k}.

The Laplacian of a scalar potential is given by

\Delta \phi=\frac{1}{\sqrt{|g

} \frac{\partial}{\partial x^i}\left(g^{ik}\sqrt{|g

\frac{\partial\phi}{\partial x^k}\right).

The Laplacian is the covariant divergence of the gradient, that is

\Delta \phi=D_i D^i\phi

Riemann curvature

The Riemann curvature tensor is given by

R_{iklm}=\frac{1}{2}\left(

\frac{\partial^2g_{im}}{\partial x^k \partial x^l} + \frac{\partial^2g_{kl}}{\partial x^i \partial x^m} - \frac{\partial^2g_{il}}{\partial x^k \partial x^m} - \frac{\partial^2g_{km}}{\partial x^i \partial x^l} \right) +g_{np} \left( \Gamma^n_{kl} \Gamma^p_{im} - \Gamma^n_{km} \Gamma^p_{il} \right) . 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

D_m R^n_{ikl} + D_l R^n_{imk} + D_k R^n_{ilm}=0.

Ricci curvature

The Ricci tensor is given by

R_{ik}=\frac{\partial\Gamma^l_{ik}}{\partial x^l} - \frac{\partial\Gamma^l_{il}}{\partial x^k} + \Gamma^l_{ik} \Gamma^m_{lm} - \Gamma^m_{il}\Gamma^l_{km}.

This tensor is symmetric:

R_{ik}=R_{ki}

. It is obtained from the Riemann curvature by contracting indices:

R_{ik}=g^{lm}R_{limk}.

The scalar curvature is given by

R=g^{ik}R_{ik}

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

D_l R^l_m = \frac{1}{2} \frac{\partial R}{\partial x^m}

Weyl tensor

The Weyl tensor is given by

C_{iklm}=R_{iklm} + \frac{1}{2}\left(

- R_{il}g_{km} + R_{im}g_{kl} + R_{kl}g_{im} - R_{km}g_{il} \right) + \frac{1}{6} R \left( g_{il}g_{km} - g_{im}g_{kl} \right).

Change of Variable

Under a change of variable from

(x^1,...,x^n)

(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.

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

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