Christoffel symbols

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In mathematics and physics, the Christoffel symbols, named after Elwin Bruno Christoffel (1829 - 1900), are expressions in spatial coordinates for the Levi-Civita connection derived from the metric tensor. Christoffel symbols are used whenever theoretical calculations involving geometry must be carried out, as they allow very complex calculations to be carried out without confusion. Conversely, the formal notation (without indices) for the Levi-Civita connection is elegant and allows theorems to be stated briefly, but is almost useless for practical calculations.

Preliminaries

The definitions given below are valid for Riemann manifolds and pseudo-Riemannian manifolds, such as those of general relativity, with careful distinction that must be made between upper and lower indices (counter- and covariant indices). Formulas work for any sign convention, unless explicitly stated otherwise.

Definition

The Christoffel symbols can be derived by canceling the covariant derivative of the metric tensor g_{i k}:

${displaystyle D_{l}g_{ik}={frac {partial g_{ik}}{partial x^{l}}}-g_{mk}Gamma _{il}^{m}-g_{im}Gamma _{kl}^{m}=0}$

By swapping the indices, and in short, it can be fixed explicitly for the connection:

${displaystyle Gamma _{kl}^{i}={frac {1}{2}}g^{im}left({frac {partial g_{mk}}{partial x^{l}}}+{frac {partial g_{ml}}{partial x^{k}}}-{frac {partial g_{kl}}{partial x^{m}}}right)={frac {1}{2}}g^{im}{frac {partial g_{mk}}{partial x^{l}}}+{frac {1}{2}}g^{im}{frac {partial g_{ml}}{partial x^{k}}}-{frac {1}{2}}g^{im}{frac {partial g_{kl}}{partial x^{m}}}}$

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

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

{displaystyle Gamma _{kl}^{i}={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 ${displaystyle c_{klm}=g_{mp}{c_{kl}}^{p}}$ It's the switching coefficients of the base; that is,

{displaystyle [e_{k},e_{l}]={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-trivial commutation coefficients are spherical and cylindrical coordinates.

The expressions below are valid only on a holonomic basis, unless otherwise stated.

Relation to indexless notation

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

${displaystyle left(nabla _{X}Yright)^{k}=X^{i}D_{i}Y^{k}=X^{i}left({frac {partial Y^{k}}{partial x^{i}}}+Gamma _{im}^{k}Y^{m}right)}$ .

Some old physics books occasionally write dx instead of X, and put it after the equation, rather than before. Here, the Einstein notation is used, the repeated indices establish the addition on those indices and the contraction with the metric tensor serves to raise and lower indices:

${displaystyle langle X,Yrangle =g(X,Y)=X^{i}Y_{i}=g_{ik}X^{i}Y^{k}}$ .

Be aware that ${displaystyle g_{ik}neq g^{ik}}$ and that ${displaystyle g_{k}^{i}=delta _{k}^{i}}$ Kronecker delta. The convention is that the metric tensor is the one with the lower indices; the correct way to get g^{i k} of g_{i k} is to solve the linear equation ${displaystyle g^{ij}g_{jk}=delta _{k}^{i}}$ .

The statement that the connection is free of torsion to know that

${displaystyle nabla _{X}Y-nabla _{Y}X=[X,Y]}$

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

${displaystyle Gamma _{jk}^{i}=Gamma _{kj}^{i}}$ .

The article on covariant derivative provides further discussion of the correspondence between notations with and without indices. By collapsing bound indices, we get

${displaystyle Gamma _{ki}^{i}={frac {1}{2}}g^{im}{frac {partial g_{im}}{partial x_{k}}}={frac {1}{2g}}{frac {partial g}{partial x_{k}}}={frac {partial log {sqrt {|g}}}{partial x_{k}}}}$

Where |g| is the absolute value of the metric tensor determinant g_{i k}. Similarly,

${displaystyle g^{kl}Gamma _{kl}^{i}={frac {-1}{sqrt {|g}}};{frac {partial {sqrt {|g|}},g^{ik}}{partial x^{k}}}}$

The covariant derivative of a vector V^m is

${displaystyle D_{l}V^{m}={frac {partial V^{m}}{partial x^{l}}}+Gamma _{kl}^{m}V^{k}}$

The covariant divergence is

${displaystyle D_{m}V^{m}={frac {partial V^{m}}{partial x^{m}}}+V^{k}{frac {partial log {sqrt {|g}}}{partial x^{k}}}={frac {1}{sqrt {|g}}}{frac {partial (V^{m}{sqrt {|g|}})}{partial x^{m}}}}$ .

The covariant derivative of a tensor A^{i k} is

${displaystyle D_{l}A^{ik}={frac {partial A^{ik}}{partial x^{l}}}+Gamma _{ml}^{i}A^{mk}+Gamma _{ml}^{k}A^{im}}$

If the tensor is antisymmetric, then its divergence simplifies to

${displaystyle D_{k}A^{ik}={frac {1}{sqrt {|g}}}{frac {partial (A^{ik}{sqrt {|g|}})}{partial x^{k}}}}$ .

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:

${displaystyle D^{i}phi =g^{ik}{frac {partial phi }{partial x^{k}}}}$

The Laplacian of a scalar potential is given by

${displaystyle Delta phi ={frac {1}{sqrt {|g}}}{frac {partial }{partial x^{i}}}left(g^{ik}{sqrt {|g|}}{frac {partial phi }{partial x^{k}}}right)}$ .

The Laplacian is the covariant divergence of the gradient, that is Δφ = D_i Dⁱφ.

Riemann curvature

The Riemann curvature tensor is given in terms of the Christoffel symbols and their derivatives by:

${displaystyle R_{iklm}={frac {1}{2}}left({frac {partial ^{2}g_{im}}{partial x^{k}partial x^{l}}}+{frac {partial ^{2}g_{kl}}{partial x^{i}partial x^{m}}}-{frac {partial ^{2}g_{il}}{partial x^{k}partial x^{m}}}-{frac {partial ^{2}g_{km}}{partial x^{i}partial x^{l}}}right)+g_{np}left(Gamma _{kl}^{n}Gamma _{im}^{p}-Gamma _{km}^{n}Gamma _{il}^{p}right)}$ .

The symmetries of the tensor are:

${displaystyle R_{iklm}=R_{lmik},}$ and ${displaystyle R_{iklm}=-R_{kilm}=-R_{ikml},}$ .

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

The sum of the cyclic permutation is

${displaystyle R_{iklm}+R_{imkl}+R_{ilmk}=0,}$

The identity of Bianchi is

${displaystyle D_{m}R_{ikl}^{n}+D_{l}R_{imk}^{n}+D_{k}R_{ilm}^{n}=0}$

Ricci curvature

The Ricci tensor is given by an indices contraction of the Riemann curvature tensor, explicitly contracting indices:

{displaystyle R_{ik}=g^{lm}R_{limk},}

Using the formulas from the previous section and applying contraction, it can be seen that the Ricci tensor is given in terms of the Christoffel symbols by:

{displaystyle R_{ik}={frac {partial Gamma _{ik}^{l}}{partial x^{l}}}-{frac {partial Gamma _{il}^{l}}{partial x^{k}}}+Gamma _{ik}^{l}Gamma _{lm}^{m}-Gamma _{il}^{m}Gamma _{km}^{l}}

{displaystyle R_{ii}={frac {left(partial Gamma _{ii}^{l}right)}{partial x^{l}}}-{frac {left(partial Gamma _{il}^{l}right)}{partial x^{i}}}+Gamma _{ii}^{l}Gamma _{lm}^{m}-Gamma _{il}^{m}Gamma _{im}^{l}}

{displaystyle R_{ii}={frac {left(partial Gamma _{ii}^{l}right)}{partial x^{l}}}-{frac {left(partial Gamma _{il}^{l}right)}{partial x^{i}}}+Gamma _{ii}^{l}Gamma _{ll}^{l}-Gamma _{ii}^{i}Gamma _{ii}^{i}}

{displaystyle R_{ik}=-Gamma _{ii}^{k}Gamma _{kk}^{i}}

This tensor is symmetrical: ${displaystyle R_{ik}=R_{ki}}$ . If a last contraction is made on the two indexes of the Ricci tensor, you get the climbing curvature that is given by:

{displaystyle R=g^{ik}R_{ik},}

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

{displaystyle D_{l}R_{m}^{l}={frac {1}{2}}{frac {partial R}{partial x^{m}}}}

Weyl tensor

The Weyl tensor is given by

${displaystyle C_{iklm}=R_{iklm}+{frac {1}{2}}left(-R_{il}g_{km}+R_{im}g_{kl}+R_{kl}g_{im}-R_{km}g_{il}right)+{frac {1}{6}}Rleft(g_{il}g_{km}-g_{im}g_{kl}right)}$

Change of variables

Low variable change ${displaystyle (x^{1},...,x^{n})}$ a ${displaystyle (y^{1},...,y^{n})}$ , vectors are transformed as

{displaystyle {frac {partial }{partial y^{i}}}={frac {partial x^{k}}{partial y^{i}}}{frac {partial }{partial x^{k}}}}

and therefore

{displaystyle {overline {Gamma _{ij}^{k}}}={frac {partial x^{p}}{partial y^{i}}},{frac {partial x^{q}}{partial y^{j}}},Gamma _{pq}^{r},{frac {partial y^{k}}{partial x^{r}}}+{frac {partial y^{k}}{partial x^{r}}},{frac {partial ^{2}x^{r}}{partial y^{i}partial y^{j}}}}

where the overline denotes Christoffel symbols in the framework coordinates ${displaystyle (y^{1},...,y^{n})}$ .

Christoffel Symbols under General Coordinate Transformations (GCT's)

Christoffel symbols do not transform like tensors under general coordinate transformations (GCT's). However, the variation of the Christoffel symbols are tensors (δΓ).

Demonstration
${displaystyle {begin{aligned}{bar {Gamma }}_{ jk}^{i}&={dfrac {1}{2}}{bar {g}}^{il}left({dfrac {partial {bar {g}}_{jl}}{partial {bar {x}}^{k}}}+{dfrac {partial {bar {g}}_{lk}}{partial {bar {x}}^{j}}}-{dfrac {partial {bar {g}}_{kj}}{partial {bar {x}}^{l}}}right)end{aligned}}}$ (1) We develop the first member of the equation and then we only change indexes properly to get the second and third term ${displaystyle {begin{aligned}{dfrac {partial {bar {g}}_{jl}}{partial {bar {x}}^{k}}}&={dfrac {partial }{partial {bar {x}}^{k}}}left({dfrac {partial x^{s}}{partial {bar {x}}^{j}}}{dfrac {partial x^{t}}{partial {bar {x}}^{l}}}g_{st}right)\&={dfrac {partial }{partial {bar {x}}^{k}}}left({dfrac {partial x^{s}}{partial {bar {x}}^{j}}}{dfrac {partial x^{t}}{partial {bar {x}}^{l}}}right)g_{st}+{dfrac {partial x^{s}}{partial {bar {x}}^{j}}}{dfrac {partial x^{t}}{partial {bar {x}}^{l}}}{dfrac {partial g_{st}}{partial {bar {x}}^{k}}}\&={dfrac {partial }{partial {bar {x}}^{k}}}left({dfrac {partial x^{s}}{partial {bar {x}}^{j}}}right){dfrac {partial x^{t}}{partial {bar {x}}^{l}}}g_{st}+{dfrac {partial x^{s}}{partial {bar {x}}^{j}}}{dfrac {partial }{partial {bar {x}}^{k}}}left({dfrac {partial x^{t}}{partial {bar {x}}^{l}}}right)g_{st}+{dfrac {partial x^{s}}{partial {bar {x}}^{j}}}{dfrac {partial x^{t}}{partial {bar {x}}^{l}}}{dfrac {partial x^{r}}{partial {bar {x}}^{k}}}{dfrac {partial g_{st}}{partial x^{r}}}\&={dfrac {partial ^{2}x^{s}}{partial {bar {x}}^{k}partial {bar {x}}^{j}}}{dfrac {partial x^{t}}{partial {bar {x}}^{l}}}g_{st}+{dfrac {partial x^{s}}{partial {bar {x}}^{j}}}{dfrac {partial ^{2}x^{t}}{partial {bar {x}}^{k}partial {bar {x}}^{l}}}g_{st}+{dfrac {partial x^{s}}{partial {bar {x}}^{j}}}{dfrac {partial x^{t}}{partial {bar {x}}^{l}}}{dfrac {partial x^{r}}{partial {bar {x}}^{k}}}{dfrac {partial g_{st}}{partial x^{r}}}end{aligned}}}$

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