calculate the Einstein tensor for a metric - Maple Help

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Tensor[EinsteinTensor] - calculate the Einstein tensor for a metric

Calling Sequences

     EinsteinTensor(g, R)

Parameters

   g    - a metric tensor

   R    - (optional) the curvature tensor of the metric g

Description

• 

Let Ricg and Sg be the Ricci tensor and Ricci scalar for the metric g, respectively. The covariant form of the Einstein tensor is Eing = Ricg  12Sgg. The contravariant form is obtained by raising both indices of the covariant Einstein tensor with the metric g. In terms of components,

Gab=Rab12Sgab. 

• 

The program EinsteinTensor(g, R) returns the contravariant form of the Einstein tensor. This tensor is symmetric and its covariant divergence vanishes.

• 

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

Examples

withDifferentialGeometry:withTensor:

 

Example 1.

Create a 4 dimensional manifold M, define a metric g1, and calculate the Einstein tensor E1.

DGsetupx,y,z,w,M1

frame name: M1

(2.1)
M1 > 

g1:=evalDGdx &t dx+dx &t dy+dy &t dx+xydz &t dw+dw &t dz

g1:=dxdx+dxdy+dydx+xydzdw+xydwdz

(2.2)

 

Calculate the Christoffel symbols for the metric g1.

M1 > 

C1:=Christoffelg1:

 

Calculate the Einstein tensor for the metric g1.

M1 > 

E1:=EinsteinTensorg1

E1:=12D_xD_xy2144y+xD_xD_yy2x144y+xD_yD_xy2x+142xy+x2+2y2D_yD_yx2y2+142y+xD_zD_wy3x2+142y+xD_wD_zy3x2

(2.3)

 

Check that the covariant divergence of the Einstein tensor E1 vanishes.

M1 > 

ContractIndicesCovariantDerivativeE1,C1,2,3

0D_x

(2.4)

See Also

Physics[Einstein], DifferentialGeometry, Tensor, Christoffel, Physics[Christoffel], ContractIndices, CovariantDerivative, Physics[D_], CurvatureTensor, Physics[Riemann], SectionalCurvature, RicciScalar, RicciTensor, Physics[Ricci]


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