calculate the Plebanski tensor from a trace-free rank 2 symmetric tensor, the Plebanski spinor from a symmetric (2, 2) spinor, the Plebanski Newman-Penrose coefficients from a table of Newman-Penrose Ricci coefficients - Maple Programming Help

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Tensor[PlebanskiTensor] - calculate the Plebanski tensor from a trace-free rank 2 symmetric tensor, the Plebanski spinor from a symmetric (2, 2) spinor, the Plebanski Newman-Penrose coefficients from a table of Newman-Penrose Ricci coefficients

Calling Sequences

     PlebanskiTensor(g, A)

     PlebanskiTensor(S)

     PlebanskiTensor(T)

Parameters

  g     - the metric tensor on a 4 dimensional manifold M

  S     -  a symmetric, trace-free, covariant rank 2 tensor Sij on M or a rank 4 spinor SABA'B' on M

  T     - a table with indices Phi00, Phi01,  Phi02, Phi10, Phi11, Phi12, Phi20, Phi21, Phi22, Lambda

  

 

Description

Examples

See Also

Description

• 

The Plebanski tensor P is the rank 4 covariant tensor constructed the metric tensor g and a symmetric covariant trace-free rank two tensor S by the formula

P    cd ab = 4 S[c[a  Sd]b]   +  4 δ[c[a Sd]e Sb]e  23 δ[c[a δd] b] Sef Sef,    Pabcd  =  ga igbj P  cd ij,   Sca  = gai Sic .

The tensor Pabcd  has all the algebraic properties of the Weyl tensor. It is skew-symmetric in the indices ab and cd , satisfies the cyclic identity onbcd, and is trace-free with respect to the metric g.

• 

 The 2 component spinors PABCD  and SABA'B' corresponding to the tensors Pabcd  and Sab are related by

PABCD= S(AB       A'B' SCD)A'B'.

• 

The Newman Penrose coefficients {Ψ0 , Ψ1 , Ψ2 , Ψ3 , Ψ4 } for Pabcd are given in terms of the Newman-Penrose coefficients Φ00,  Φ01,Φ02,Φ10,Φ11,Φ12,Φ20,Φ21,Φ22  for Sab by

         Ψ0 =2Φ00Φ02  Φ012,    Ψ1 = Φ00 Φ12 + Φ02Φ10 2 Φ01Φ11,    Ψ2 = 13Φ00Φ22 2 Φ01Φ21 + Φ02Φ20 + 4 Φ10 Φ12  4 Φ112,    Ψ3=   Φ10Φ22  2Φ11Φ21  + Φ12Φ20 ,     Ψ4 = 2Φ20Φ22  Φ212 .  

• 

The command PlebanskiTensor calculates the Plebanski tensor P for a given tensor S. If a tensorial form of S is given, the tensorial form of P is returned (first calling sequence); if the spinor components of S are given, the spinor components of P are returned (second calling sequence); and if the Newman-Penrose components of S are given, then the Newman-Penrose components of P are returned (third calling sequence).

• 

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

Examples

withDifferentialGeometry:withTensor:

 

Example 1.

First create a spinor bundle with space-time coordinates (t,  x,  y,  z)  and spinor coordinates z1,  z2 ,  w1,  w2.  (Spinors are not needed for this first example but will be used in Example 2.)

DGsetupt,x,y,z,z1,z2,w1,w2,M

frame name: M

(2.1)

 

Define a metric tensor g.

M > 

gevalDGx2dt&tdtt2dx&tdxdy&tdydz&tdz

g:=_DGtensor,M,cov_bas,cov_bas,,1,1,x2,2,2,t2,3,3,−1,4,4,−1,_DGtensor,M,cov_bas,cov_bas,,1,1,x2,2,2,t2,3,3,−1,4,4,−1

(2.2)

 

Define a symmetric, trace-free, rank 2 tensor.

M > 

SevalDGdx&sdy

S:=_DGtensor,M,cov_bas,cov_bas,,2,3,12,3,2,12,_DGtensor,M,cov_bas,cov_bas,,2,3,12,3,2,12

(2.3)

 

Compute the Plebanski tensor of S.

M > 

PPlebanskiTensorg,S

P:=_DGtensor,M,cov_bas,cov_bas,cov_bas,cov_bas,,1,2,1,2,x212,1,2,2,1,x212,1,3,1,3,x212t2,1,3,3,1,x212t2,1,4,1,4,x26t2,1,4,4,1,x26t2,2,1,1,2,x212,2,1,2,1,x212,2,3,2,3,16,2,3,3,2,16,2,4,2,4,112,2,4,4,2,112,3,1,1,3,x212t2,3,1,3,1,x212t2,3,2,2,3,16,3,2,3,2,16,3,4,3,4,112t2,3,4,4,3,112t2,4,1,1,4,x26t2,4,1,4,1,x26t2,4,2,2,4,112,4,2,4,2,112,4,3,3,4,112t2,4,3,4,3,112t2,_DGtensor,M,cov_bas,cov_bas,cov_bas,cov_bas,,1,2,1,2,x212,1,2,2,1,x212,1,3,1,3,x212t2,1,3,3,1,x212t2,1,4,1,4,x26t2,1,4,4,1,x26t2,2,1,1,2,x212,2,1,2,1,x212,2,3,2,3,16,2,3,3,2,16,2,4,2,4,112,2,4,4,2,112,3,1,1,3,x212t2,3,1,3,1,x212t2,3,2,2,3,16,3,2,3,2,16,3,4,3,4,112t2,3,4,4,3,112t2,4,1,1,4,x26t2,4,1,4,1,x26t2,4,2,2,4,112,4,2,4,2,112,4,3,3,4,112t2,4,3,4,3,112t2

(2.4)

 

We check that the tensor P has the same algebraic properties as the Weyl tensor.  We use the command SymmetrizeIndices to show that  P is skew-symmetric on its 1st and 2nd indices

M > 

SymmetrizeIndicesP,1,2,Symmetric

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

(2.5)

 

The Plebanski tensor is skew-symmetric on its 3rd and 4th indices

M > 

SymmetrizeIndicesP,3,4,Symmetric

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

(2.6)

 

The Plebanski tensor satisfies the cyclic identity on its first 3 indices.

M > 

SymmetrizeIndicesP,1,2,3,SkewSymmetric

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

(2.7)

 

The Plebanski tensor is also trace-free on its 1st and 3rd indices.  To check this we use the commands InverseMetric and ContractIndices  to evaluate  gacPabcd.

M > 

hInverseMetricg

h:=_DGtensor,M,con_bas,con_bas,,1,1,1x2,2,2,1t2,3,3,−1,4,4,−1,_DGtensor,M,con_bas,con_bas,,1,1,1x2,2,2,1t2,3,3,−1,4,4,−1

(2.8)
M > 

ContractIndicesh,P,1,1,2,3

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

(2.9)

 

Example 2.

In this example we will convert the tensor S to a spinor φ and compute the spinor form of the Plebanski tensor. We start by defining an orthonormal tetrad for the metric g and using this tetrad and the command SolderForm to construct a solder form σ for the metric g.

M > 

otevalDG1xD_t,1tD_x,D_y,D_z

ot:=_DGvector,M,,1,1x,_DGvector,M,,1,1x,_DGvector,M,,2,1t,_DGvector,M,,2,1t,_DGvector,M,,3,1,_DGvector,M,,3,1,_DGvector,M,,4,1,_DGvector,M,,4,1

(2.10)
M > 

σSolderFormot

σ:=_DGtensor,M,cov_bas,con_vrt,con_vrt,,1,5,7,x22,1,6,8,x22,2,5,8,t22,2,6,7,t22,3,5,8,I22,3,6,7,I22,4,5,7,22,4,6,8,22,_DGtensor,M,cov_bas,con_vrt,con_vrt,,1,5,7,x22,1,6,8,x22,2,5,8,t22,2,6,7,t22,3,5,8,I22,3,6,7,I22,4,5,7,22,4,6,8,22

(2.11)

 

The command RicciSpinor gives the spinor form of S.

M > 

φRicciSpinorσ,S

φ:=_DGtensor,M,cov_vrt,cov_vrt,cov_vrt,cov_vrt,,5,5,8,8,I4t,6,6,7,7,I4t,_DGtensor,M,cov_vrt,cov_vrt,cov_vrt,cov_vrt,,5,5,8,8,I4t,6,6,7,7,I4t

(2.12)

 

We calculate the Plebanski tensor in its spinor form.

M > 

ψPlebanskiTensorφ

ψ:=_DGtensor,M,cov_vrt,cov_vrt,cov_vrt,cov_vrt,,5,5,6,6,112t2,5,6,5,6,112t2,5,6,6,5,112t2,6,5,5,6,112t2,6,5,6,5,112t2,6,6,5,5,112t2,_DGtensor,M,cov_vrt,cov_vrt,cov_vrt,cov_vrt,,5,5,6,6,112t2,5,6,5,6,112t2,5,6,6,5,112t2,6,5,5,6,112t2,6,5,6,5,112t2,6,6,5,5,112t2

(2.13)

 

We can check the consistency of this result using the command WeylSpinor to calculate the spinor form of P.  

M > 

ψ1WeylSpinorσ,P

ψ1:=_DGtensor,M,cov_vrt,cov_vrt,cov_vrt,cov_vrt,,5,5,6,6,112t2,5,6,5,6,112t2,5,6,6,5,112t2,6,5,5,6,112t2,6,5,6,5,112t2,6,6,5,5,112t2,_DGtensor,M,cov_vrt,cov_vrt,cov_vrt,cov_vrt,,5,5,6,6,112t2,5,6,5,6,112t2,5,6,6,5,112t2,6,5,5,6,112t2,6,5,6,5,112t2,6,6,5,5,112t2

(2.14)
M > 

ψ&minusψ

_DGtensor,M,cov_vrt,cov_vrt,cov_vrt,cov_vrt,,5,5,5,5,0,_DGtensor,M,cov_vrt,cov_vrt,cov_vrt,cov_vrt,,5,5,5,5,0

(2.15)

 

Example 3.

In this example we will calculate the Newman-Penrose coefficients of the tensor S  from the Newman-Penrose coefficients of the Plebanski tensor P. For these calculations we need the NullTetrad determined by the orthonormal tetrad ot.

M > 

ntNullTetradot

nt:=_DGvector,M,,1,22x,4,22,_DGvector,M,,1,22x,4,22,_DGvector,M,,1,22x,4,22,_DGvector,M,,1,22x,4,22,_DGvector,M,,2,22t