calculate the trace-free Ricci tensor of a metric tensor - Maple Programming Help

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Tensor[TraceFreeRicciTensor] - calculate the trace-free Ricci tensor of a metric tensor

Calling Sequences

     TraceFreeRicciTensor(g)

     TraceFreeRicciTensor(g, C)

     TraceFreeRicciTensor(g, R)

Parameters

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

   C    - the curvature tensor of the metric g

   R    - the Ricci tensor of the metric g

 

Description

Examples

See Also

Description

• 

Let g be a metric tensor with associated Ricci tensor R and Ricci scalar S. The trace-free Ricci tensor P is the symmetric, rank 2 covariant tensor with components Pij=Rij  1n gij S, where n  is the dimension of the underlying manifold. It is trace-free with respect to the metric g  in the sense that gij Pij = 0, where gij are the components of the inverse metric.

• 

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

Examples

withDifferentialGeometry:withTensor:

 

Example 1.

In this example we calculate the trace-free Ricci tensor for a metric.

M1 > 

DGsetupx,y,z,M

frame name: M

(2.1)
M > 

gevalDGydx&tdx+zdy&tdy+dz&tdz

g:=ydxdx+zdydy+dzdz

(2.2)

 

Calculate the trace-free Ricci tensor for the metric directly.

M > 

PTraceFreeRicciTensorg

P:=1122y2zdxdxz2y+112y2+zdydyzy2+14dydzyz+14dzdyyz+112y22zdzdzz2y2

(2.3)

 

We check that T is trace-free by computing the inverse metric and using the ContractIndices command.

M > 

hInverseMetricg

h:=D_xD_xy+D_yD_yz+D_zD_z

(2.4)
M > 

ContractIndicesh,P,1,1,2,2

0

(2.5)

 

The same calculation can be done with the TensorInnerProduct command.

M > 

TensorInnerProductg,g,P

0

(2.6)

 

Example 2.

The third calling sequence can be applied to any rank 2 symmetric tensor to construct a trace-free, rank 2 symmetric tensor.

M > 

A1evalDGdx&sdy

A1:=dxdy+dydx

(2.7)
M > 

A2g

A2:=ydxdx+zdydy+dzdz

(2.8)
M > 

A3evalDGdx&tdx

A3:=dxdx

(2.9)

 

The tensor A1 is already trace-free, so its trace-free part is itself.

M > 

TraceFreeRicciTensorg,A1

dxdy+dydx

(2.10)

The trace-free part of the metric itself is always 0.

M > 

TraceFreeRicciTensorg,A2

0dxdx

(2.11)

 

The trace-free part of A3 is

M > 

TTraceFreeRicciTensorg,A3

T:=23dxdx13zdydyy13dzdzy

(2.12)
M > 

TensorInnerProductg,g,T

0

(2.13)

 

See Also

DifferentialGeometry, Tensor, Christoffel, Physics[Christoffel], CovariantDerivative, Physics[D_], CurvatureTensor, Physics[Riemann], SectionalCurvature, RicciScalar, Physics[Ricci], RicciSpinor, NPCurvatureScalars