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Tensor[BelRobinson] - calculate the Bel-Robinson tensor

Calling Sequences

     BelRobinson(g, W, indexlist)

Parameters

   g         - a metric tensor on a 4-dimensional manifold

   W         - (optional) the Weyl tensor of the metric g

   indexlist - (optional) the keyword argument indexlist = ind, where ind is a list of 4 index types "con" or "cov"

 

Description

Examples

See Also

Description

• 

The Bel-Robinson tensor Bijhk is a covariant rank 4 tensor defined in terms of the Weyl tensor Wijhk on a 4-dimensional manifold by (see, for example, Penrose and Rindler Vol. 1)

Bijhk=14WilhmWj   k  l   m12gijWlmhn+gilWmjhn+ gimWjlhnW    klm  n.

The Bel-Robinson tensor is totally symmetric: Bijhk=Bjihk=Bhjik=Bkjhi . The Bel-Robinson tensor is trace-free: gijBijhk=0. If gij is an Einstein metric, that is, Rij=Λgij (where Rij is the Ricci tensor for the metric gij and Λ is a constant), then the covariant divergence of Bel-Robinson vanishes: gil l Bijhk=0.  Here l denotes the covariant derivative with respect to the Christoffel connection for gij.

• 

The keyword argument indexlist = ind allows the user to specify the index structure for the Bel-Robinson tensor. For example, with indexlist = ["con", "con", "con", "con"], the contravariant form Bijhk is returned. The default output is the purely covariant form (as above).

• 

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

Examples

withDifferentialGeometry:withTensor:

 

Example 1.

First create a 4-dimensional manifold M and define a metric gon M. The metric shown below is a homogenous Einstein metric (see (12.34) in Stephani, Kramer et al).

DGsetupx,y,z,u,M

frame name: M

(2.1)
M > 

gevalDGexpzdx&tdx+exp2zdy&tdy+dx&sdu3Λdz&tdz

g:=_DGtensor,M,cov_bas,cov_bas,,1,1,ⅇz,1,4,ⅇ2z2,2,2,ⅇ2z,3,3,3Λ,4,1,ⅇ2z2,_DGtensor,M,cov_bas,cov_bas,,1,1,ⅇz,1,4,ⅇ2z2,2,2,ⅇ2z,3,3,3Λ,4,1,ⅇ2z2,_DGtensor,M,cov_bas,cov_bas,,1,1,ⅇz,1,4,ⅇ2z2,2,2,ⅇ2z,3,3,3Λ,4,1,ⅇ2z2,_DGtensor,M,cov_bas,cov_bas,,1,1,ⅇz,1,4,ⅇ2z2,2,2,ⅇ2z,3,3,3Λ,4,1,ⅇ2z2

(2.2)

 

Calculate the Bel-Robinson tensor for the metric g.  The result is clearly a symmetric tensor.

M > 

BBelRobinsong

B:=_DGtensor,M,cov_bas,cov_bas,cov_bas,cov_bas,,1,1,1,1,Λ2ⅇ2z4,_DGtensor,M,cov_bas,cov_bas,cov_bas,cov_bas,,1,1,1,1,Λ2ⅇ2z4,_DGtensor,M,cov_bas,cov_bas,cov_bas,cov_bas,,1,1,1,1,Λ2ⅇ2z4,_DGtensor,M,cov_bas,cov_bas,cov_bas,cov_bas,,1,1,1,1,Λ2ⅇ2z4

(2.3)

 

Use the optional keyword argument indexlist to calculate the contravariant form of the Bel-Robinson tensor.

M > 

B1BelRobinsong,indexlist=con,con,con,con

B1:=_DGtensor,M,con_bas,con_bas,con_bas,con_bas,,4,4,4,4,4ⅇ10zΛ2,_DGtensor,M,con_bas,con_bas,con_bas,con_bas,,4,4,4,4,4ⅇ10zΛ2,_DGtensor,M,con_bas,con_bas,con_bas,con_bas,,4,4,4,4,4ⅇ10zΛ2,_DGtensor,M,con_bas,con_bas,con_bas,con_bas,,4,4,4,4,4ⅇ10zΛ2

(2.4)

 

The tensor B is trace-free.

hInverseMetricg

h:=_DGtensor,M,con_bas,con_bas,,1,4,2ⅇ2z,2,2,ⅇ2z,3,3,Λ3,4,1,2ⅇ2z,4,4,4ⅇ5z,_DGtensor,M,con_bas,con_bas,,1,4,2ⅇ2z,2,2,ⅇ2z,3,3,Λ3,4,1,2ⅇ2z,4,4,4ⅇ5z,_DGtensor,M,con_bas,con_bas,,1,4,2ⅇ2z,2,2,ⅇ2z,3,3,Λ3,4,1,2ⅇ2z,4,4,4ⅇ5z,_DGtensor,M,con_bas,con_bas,,1,4,2ⅇ2z,2,2,ⅇ2z,3,3,Λ3,4,1,2ⅇ2z,4,4,4ⅇ5z

(2.5)

ContractIndicesh,B,1,1,2,2

_DGtensor,M,cov_bas,cov_bas,,1,1,0,_DGtensor,M,cov_bas,cov_bas,,1,1,0,_DGtensor,M,cov_bas,cov_bas,,1,1,0,_DGtensor,M,cov_bas,cov_bas,,1,1,0

(2.6)

 

The covariant divergence of the tensor B1 vanishes.  To check this, first calculate the Christoffel connection C for the metric g and then calculate the covariant derivative of B1.

CChristoffelg

C:=_DGconnection,M,con_bas,cov_bas,cov_bas,,1,1,3,−1,1,3,1,−1,2,2,3,−1,2,3,2,−1,3,1,1,Λⅇz6,3,1,4,Λⅇ2z6,3,2,2,Λⅇ2z3,3,4,1,Λⅇ2z6,4,1,3,3ⅇ3z,4,3,1,3ⅇ3z,4,3,4,−1,4,4,3,−1,_DGconnection,M,con_bas,cov_bas,cov_bas,,1,1,3,−1,1,3,1,−1,2,2,3,−1,2,3,2,−1,3,1,1,Λⅇz6,3,1,4,Λⅇ2z6,3,2,2,Λⅇ2z3,3,4,1,Λⅇ2z6,4,1,3,3ⅇ3z,4,3,1,3ⅇ3z,4,3,4,−1,4,4,3,−1,_DGconnection,M,con_bas,cov_bas,cov_bas,,1,1,3,−1,1,3,1,−1,2,2,3,−1,2,3,2,−1,3,1,1,Λⅇz6,3,1,4,Λⅇ2z6,3,2,2,Λⅇ2z3,3,4,1,Λⅇ2z6,4,1,3,3ⅇ3z,4,3,1,3ⅇ3z,4,3,4,−1,4,4,3,−1,_DGconnection,M,con_bas,cov_bas,cov_bas,,1,1,3,−1,1,3,1,−1,2,2,3,−1,2,3,2,−1,3,1,1,Λⅇz6,3,1,4,Λⅇ2z6,3,2,2,Λⅇ2z3,3,4,1,Λⅇ2z6,4,1,3,3ⅇ3z,4,3,1,3ⅇ3z,4,3,4,−1,4,4,3,−1

(2.7)

nablaB1CovariantDerivativeB1,C

nablaB1:=_DGtensor,M,con_bas,con_bas,con_bas,con_bas,cov_bas,,3,4,4,4,1,2Λ3ⅇ8z3,4,3,4,4,1,2Λ3ⅇ8z3,4,4,3,4,1,2Λ3ⅇ8z3,4,4,4,3,1,2Λ3ⅇ8z3,4,4,4,4,3,24ⅇ10zΛ2,_DGtensor,M,con_bas,con_bas,con_bas,con_bas,cov_bas,,3,4,4,4,1,2Λ3ⅇ8z3,4,3,4,4,1,2Λ3ⅇ8z3,4,4,3,4,1,2Λ3ⅇ8z3,4,4,4,3,1,2Λ3ⅇ8z3,4,4,4,4,3,24ⅇ10zΛ2,_DGtensor,M,con_bas,con_bas,con_bas,con_bas,cov_bas,,3,4,4,4,1,2Λ3ⅇ8z3,4,3,4,4,1,2Λ3ⅇ8z3,4,4,3,4,1,2Λ3ⅇ8z3,4,4,4,3,1,2Λ3ⅇ8z3,4,4,4,4,3,24ⅇ10zΛ2,_DGtensor,M,con_bas,con_bas,con_bas,con_bas,cov_bas,,3,4,4,4,1,2Λ3ⅇ8z3,4,3,4,4,1,2Λ3ⅇ8z3,4,4,3,4,1,2Λ3ⅇ8z3,4,4,4,3,1,2Λ3ⅇ8z3,4,4,4,4,3,24ⅇ10zΛ2

(2.8)

DivergenceContractIndicesnablaB1,1,5

Divergence:=_DGtensor&comma;M&comma;con_bas&comma;con_bas&comma;con_bas&comma;&comma;1&comma;1&comma;1&comma;0&comma;_DGtensor&comma;M&comma;con_bas&comma;con_bas&comma;con_bas&comma;&comma;1&comma;1&comma;1&comma;0&comma;_DGtensor&comma;M&comma;con_bas&comma;con_bas&comma;con_bas&comma;&comma;1&comma;1&comma;1&comma;0&comma;_DGtensor&comma;M&comma;con_bas&comma;con_bas&comma;con_bas&comma;&comma;1&comma;1&comma;1&comma;<