Symbols of Christoffel

In Mathematical and Physical, the symbols of Christoffel , which draw their name from the mathematician Elwin Bruno Christoffel, is an expression of the Connection of Levi-Civita derived from the metric Tenseur. The symbols of Christoffel are used in practical calculations of the geometry of space: they are concrete computational tools, but n the other hand their handling is relatively long, in particular because of the number of implied terms.

They are basic tools used within the framework of the General relativity to describe the action of the mass and energy on the curve of the Espace-temps.

On the contrary, the formal notations for the Connection of Levi-Civita allow the expression of theoretical results in an elegant way, but do not have a direct application for practical calculations.

Preliminary

The definitions given below are valid at the same time for the varieties riemanniennes and the varieties pseudo-riemanniennes, such as those used in General relativity. One uses in the same way the notation of the superscripts for the coordinates contravariantes, and inferior for the coordinates covariantes.

Definition

In a riemanienne variety or pseudo-riemanienne $M$ , there does not exist frame of reference which applies to all the variety. One can nevertheless define a reference mark of Lorentz locally (see definition of a topological Variété: one can find in each point of $M$ a homeomorphic vicinity open to open of space $\ R^3$ ).

The Dérivée covariante makes it possible to evaluate the evolution of a Champ of vectors $V$ by taking into account not only its intrinsic modifications , but also that of the frame of reference. Thus, if one take a reference mark in polar coordinates, the two vectors $e_r$ and $e_ \ theta$ are not constant and depend on the studied point. The derivative covariante makes it possible to take into account these two factors of evolution.

The symbols of Christoffel $\ Gamma^k {} _ {ji}$ then represent the evolution of the basic vectors, through their derivative covariante:

$\ nabla _ {\ vec e_i} \ vec e_j = {\ Gamma^k} _ {ji} \ vec e_k$

While using the properties of the Derived covariante, one arrives at the expression:

$\ nabla _ {\ vec U} \ vec v = u^i \ partial _ I v^j \ vec e_j + u^i v^j {\ Gamma^k} _ {ji} \ vec e_k$

The coordinates of the vector $\ nabla _ {\ vec e_ \ alpha} \ vec v$ are noted using a semicolon, according to the definition:

$\ nabla _ {\ vec e_ \ alpha} \ vec v = {v^k} _ {; \ alpha} \ vec e_k$

By replacing $\ vec u$ by $\ vec e_ {\ alpha}$ in the relation above, one obtains:

${v^k}_{;\ alpha} = \ partial _ {\ alpha} v^k + v^j {\ Gamma^k} _ {J \ alpha}$

One thus sees that indeed the evolution of the vector $\ vec v$ depends at the same time on its intrinsic evolution (partial term $\ _ {\ alpha} v^k$ ) and on that of the base, attached to the second term and in particular to $\ Gamma^k {} _ {J \ alpha}$ , symbol of Christoffel.

This result is valid for a vector $\ vec v$ which is a tensor of order 1. For a tensor of order $n$ and row $(L, m)$ , one could obtain the same thing:

{T^ {I \ dowries J}} _ {K \ dowries L; m} = {T^ {I \ dowries J}} _ {K \ dowries L, m} \ + \ {\ Gamma^i} _ {\ mathbf {N} m} {T^ {\ mathbf {N} \ dowries J}} _ {K \ dowries L} \ + \ \ dowries \ + \ {\ Gamma^j} _ {\ mathbf {N} m} {T^ {I \ dowries \ mathbf {N}}} _ {K \ dowries L} \ - \ {\ Gamma^ \ mathbf {S}} _ {km} {T^ {I \ dowries J}} _ {\ mathbf {S} \ dowries L} \ - \ {\ Gamma^ \ mathbf {S}} _ {lm} {T^ {I \ dowries J}} _ {K \ dowries \ mathbf {S}}

The indices in fat below emphasize the contributions of different the components from Christoffel. It is observed besides that the indices contravariants give place to a positive contribution of the coefficient of Christoffel, while the indices covariants with a negative contribution.

Expression compared to the metric tensor

One can express the value of the coefficients of Christoffel compared to the tensor $g_ {ik}$ , by taking of account the fact that

$\ nabla _ {\ vec e_ \ alpha} \ vec g_ {ik} = 0$

because the metric one is preserved locally: there is locally a reference mark of Lorentz in each point of space.

While applying to $g$ , tensor of order 2 and row (0,2), the equation of the coefficients of Christoffel given above (2 coordinates covariantes give 2 “negative” contributions):

$\, g_ {ik; \ ell} = g_ {ik, \ ell} - g_ {mk} \ Gamma^m {} _ {I \ ell} - g_ {im} \ Gamma^m {} _ {K \ ell}. \$

One finds then, by permuting the indices and by expressing several values of the coefficients:

$\ 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 $g^ {ij}$ is the reverse of $g_ {ij}$ , defined by using the symbol of Kronecker by $g^ {K I} g_ {I L} = \ delta^k {} _l$ .

Remark : although the symbols of Christoffel are written in the same notation that the tensors, they are not not Tenseur S. Indeed, they do not change like the tensors during a change of coordinates.

NB : To note that the majority of the authors choose to define the symbols of Christoffel in a base of holonomic Coordonnées, which is convention followed here. In nonholonomic coordinates, the symbols of Christoffel are expressed in a more complex formulation:

$\ Gamma^i {} _ {K \ ell} = \ frac {1} {2} g^ {im} \ left ($

\ frac {\ partial g_ {mk}} {\ partial x^ \ ell} + \ frac {\ partial g_ {m \ ell}} {\ partial x^k} - \ frac {\ partial g_ {K \ ell}} {\ partial x^m} + c_ {mk \ ell} +c_ {m \ ell K} - c_ {K \ ell m} \ right) \

where the $c_ {K \ ell m} =g_ {mp} c_ {K \ ell} {} ^p$ are the coefficients of commutation base, i.e.

$= c_ {K \ ell} {} ^m e_m \, \$

where $e_k$ is the basic vectors and corresponds to the Crochet of Dregs. Two basic examples nonholonomic are for example those associated with the spherical or cylindrical coordinates.

For example, the only nonconstant terms of the metric tensor in spherical Coordonnées are $g_ {\ theta \ theta} = r^2$ , $g_ {\ phi \ phi} = r^2 \ sin^2 \ theta$ , and there is $g_ {\ theta \ theta, R} = 2 r$ , $g_ {\ phi \ phi, R} = 2 r \ sin^2$ , $g_ {\ phi \ phi, \ theta} = 2 r ^2 \ cos \ theta \ sin \ theta$ . The nonnull elements of the symbol of Christoffel according to the metric tensor are thus very few:

\begin{align}

\ Gamma^ {R} _ {\ theta \ theta} & = - R \ \ \ Gamma^ {R} _ {\ phi \ phi} & = - R \ sin^2 \ theta \ \ \ Gamma^ {\ theta} _ {R \ theta} = \ Gamma^ {\ theta} _ {\ theta R} &= r^ {- 1} \ \ \ Gamma^ {\ theta} _ {\ phi \ phi} &= - \ cos \ theta \ sin \ theta \ \ \ Gamma^ {\ phi} _ {R \ phi} = \ Gamma^ {\ phi} _ {\ phi R} &= r^ {- 1} \ \ \ Gamma^ {\ phi} _ {\ phi \ theta} = \ Gamma^ {\ phi} _ {\ theta \ phi} &= - \ cot \ theta \end{align}

In the same way, the only nonconstant term of the metric tensor in cylindrical Coordonnées is $g_ {\ phi \ phi} = r^2$ , and there is $g_ {\ phi \ phi, R} = 2 r$ . The nonnull elements of the symbol of Christoffel according to the metric tensor are thus very few:

\begin{align}

\ Gamma^ {R} _ {\ phi \ phi} & = - R \ \ \ Gamma^ {\ phi} _ {R \ phi} = \ Gamma^ {\ phi} _ {\ phi R} &= \ frac {1} {R} \end{align}

Contraction

See too

Random links: