Tensor[Christoffel] - find the Christoffel symbols of the first or second kind for a metric tensor

Calling Sequences

     Christoffel(g, h, keyword)

Parameters

   g       - a metric tensor on the tangent bundle of a manifold

   h       - (optional) the inverse of the metric g

   keyword - (optional) a keyword string, either "FirstKind" or "SecondKind"

Description

Examples

See Also

Description

The Christoffel symbol of the second kind for a metric g is the unique torsion-free connection such that the associated covariant derivative operator ∇ satisfies ∇g=0. It can be represented as a 3-index set of coefficients:

C  ijk = 12gklgil,j + gjl,i − gij,l,

where gij  and gij are the components of the metric and its inverse, respectively, and where a comma indicates a partial derivative.

The Christoffel symbol of the first kind is the non-tensorial quantity obtained from the Christoffel symbol of the second kind by lowering its upper index with the metric:

Clij = gklCijk=12gil,j + gjl,i − gij,l,

The default value for the keyword is "SecondKind", that is, the calling sequence Christoffel(g) computes the Christoffel symbol of the second kind.

The inverse of the metric can be computed using InverseMetric.

This command is part of the DifferentialGeometry:-Tensor package, and so can be used in the form Christoffel(...) only after executing the command with(DifferentialGeometry) and with(Tensor) in that order.  It can always be used in the long form DifferentialGeometry:-Tensor:-Christoffel.

Examples

with⁡DifferentialGeometry:with⁡Tensor:

Example 1.

First create a 2 dimensional manifold M and define a metric g1 on the tangent space of M.

DGsetup⁡x,y,M

frame name: M

g1≔evalDG⁡1y2⁢dx&tdx+dy&tdy

g1:=dx⁢dxy2+dy⁢dyy2

Calculate the Christoffel symbols of the first and second kind for g1.

C1≔Christoffel⁡g1,FirstKind

C1:=−dx⁢dx⁢dyy3−dx⁢dy⁢dxy3+dy⁢dx⁢dxy3−dy⁢dy⁢dyy3

C2≔Christoffel⁡g1,SecondKind

C2:=−D_x⁢dx⁢dyy−D_x⁢dy⁢dxy+D_y⁢dx⁢dxy−D_y⁢dy⁢dyy

CovariantDerivative⁡g1,C2

0⁢dx⁢dx⁢dx

TorsionTensor⁡C2

0⁢D_x⁢dx⁢dx

Example 2.

Define an anholonomic frame on M and use this frame to calculate the Christoffel symbol for a metric on the tangent space of M.

FR≔FrameData⁡dx1+x2+y2,dy1+x2+y2,M1

FR:=d⁢Θ1=2⁢y⁢Θ1⁢⋀⁢Θ2,d⁢Θ2=−2⁢x⁢Θ1⁢⋀⁢Θ2

DGsetup⁡FR,E,σ

frame name: M1

g2≔evalDG⁡σ1&tσ1+σ2&tσ2

g2:=σ1⁢σ1+σ2⁢σ2

C≔Christoffel⁡g2

C:=−2⁢y⁢E1⁢σ2⁢σ1+2⁢x⁢E1⁢σ2⁢σ2+2⁢y⁢E2⁢σ1⁢σ1−2⁢x⁢E2⁢σ1⁢σ2

CovariantDerivative⁡g2,C

0⁢σ1⁢σ1⁢σ1

TorsionTensor⁡C

0⁢E1⁢σ1⁢σ1

See Also

DifferentialGeometry, Tensor, CovariantDerivative, Physics[D_], CurvatureTensor, Physics[Riemann], DirectionalCovariantDerivative, GeodesicEquations, ParallelTransportEquations, TorsionTensor