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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.

Contents

Preliminaries

Definition

Relationship to index-less notation

Covariant derivatives of tensors

Change of variable

Applications to general relativity

See also

References

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. The Christoffel symbols are defined by:
:$$
abla_ie_j=Gamma_{ij}^ke_k

Definition

The Christoffel symbols can be derived from the vanishing of the covariant derivative of the metric tensor $g\_\{ik\}$:
:$$
abla_ell g_{ik}=rac{partial g_{ik}}{partial x^ell}- g_{mk}Gamma^m {}_{iell} - g_{im}Gamma^m {}_{kell}=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
:$,g\_\{ik;ell\}\; =\; g\_\{ik,ell\}\; -\; g\_\{mk\}\; Gamma^m\; \{\}\_\{iell\}\; -\; g\_\{im\}\; Gamma^m\; \{\}\_\{kell\}.$
By permuting the indices, and resumming, one can solve explicitly for the Christoffel symbols:
:
ight) = {1 over 2} g^{im} (g_{mk,ell} + g_{mell,k} - g_{kell,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
:
rac{partial g_{mk}}{partial x^ell} +
rac{partial g_{mell}}{partial x^k} -
rac{partial g_{kell}}{partial x^m} +
c_{mkell}+c_{mell k} - c_{kell m}
ight)
where $c\_\{kell\; m\}=g\_\{mp\}\; \{c\_\{kell\}\}^p$ are the commutation coefficients of the basis; that is,
:$[e\_k,e\_ell]\; =\; c\_\{kell\}\{\}^m\; e\_m,$
where ''e''_k are the basis vectors and $[,]$ is the Lie bracket. An example of an anholonomic basis with non-vanishing commutation coefficients are the unit vectors in 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 ''k''th component of the covariant derivative of ''Y'' with respect to ''X'' is given by
:$left($
abla_X Y
ight)^k = X^i (
abla_i Y)^k = X^i left(rac{partial Y^k}{partial x^i} + Gamma^k {}_{im} Y^m
ight).
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:
:$langle\; X,Y$
angle = g(X,Y) = X^i Y_i = g_{ik}X^i Y^k = g^{ik}X_i Y_k.
Keep in mind that $g\_\{ik\}$
eq 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
:$$
abla_X Y -
abla_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
:$$
abla_ell V^m = rac{partial V^m}{partial x^ell} + Gamma^m {}_{kell} V^k.
The covariant derivative of a scalar field is just
:$$
abla_i arphi = rac{partial arphi}{partial x^i}
and the covariant derivative of a covector field $omega\_m$ is
:$$
abla_ell omega_m = rac{partial omega_m}{partial x^ell} - Gamma^k {}_{ell m} omega_k.
The symmetry of the Christoffel symbol now implies
:$$
abla_i
abla_j arphi =
abla_j
abla_i arphi
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
:$$
abla_ell A^{ik}=rac{partial A^{ik}}{partial x^ell} + Gamma^i {}_{mell} A^{mk} + Gamma^k {}_{mell} A^{im},
that is,
:$A^\{ik\}\; \{\}\_\{;ell\}\; =\; A^\{ik\}\; \{\}\_\{,ell\}\; +\; A^\{mk\}\; Gamma^i\; \{\}\_\{mell\}\; +\; A^\{im\}\; Gamma^k\; \{\}\_\{mell\}.$
If the tensor field is mixed then its covariant derivative is
:$A^i\; \{\}\_\{k;ell\}\; =\; A^i\; \{\}\_\{k,ell\}\; +\; A^\{m\}\; \{\}\_k\; Gamma^i\; \{\}\_\{mell\}\; -\; A^i\; \{\}\_m\; Gamma^m\; \{\}\_\{kell\},$
and if the tensor field is of type (0,2) then its covariant derivative is
:$A\_\{ik;ell\}\; =\; A\_\{ik,ell\}\; -\; A\_\{mk\}\; Gamma^m\; \{\}\_\{iell\}\; -\; A\_\{im\}\; Gamma^m\; \{\}\_\{kell\}.$

Change of variable

Under a change of variable from $(x^1,...,x^n)$ to $(y^1,...,y^n)$, vectors transform as
:
and so
:$overline\{Gamma^k\; \{\}\_\{ij\}\}\; =$
rac{partial x^p}{partial y^i},
rac{partial x^q}{partial y^j},
Gamma^r {}_{pq},
rac{partial y^k}{partial x^r}
+
rac{partial y^k}{partial x^m},
rac{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.

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

★ List of formulas in Riemannian geometry

★ Basic introduction to the mathematics of curved spacetime

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