calculate the Newman-Penrose curvature scalars associated to the trace-free Ricci tensor and the Weyl tensor - Maple Programming Help

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Tensor[NPCurvatureScalars] - calculate the Newman-Penrose curvature scalars associated to the trace-free Ricci tensor and the Weyl tensor

Calling Sequences

     NPCurvatureScalars(NTetrad, g, R, option1)

     NPCurvatureScalars(Fr, g, R, option1)

     NPCurvatureScalars(SpinCoeff, NTetrad, option1, option2)

     NPCurvatureScalars( W)

Parameters

   NTetrad   - a list of 4 vectors defining a null tetrad

   g         - (optional) a metric tensor g

   R         - (optional) the curvature tensor of g or the Ricci tensor of g

   Fr        - the name of an initialized anholonomic frame, created from a null tetrad

   SpinCoeff - a table, the Newman Penrose spin coefficients

   W         - a symmetric rank 4, covariant spinor, the Weyl spinor defined by the Weyl tensor.

   option1   - (optional keyword argument) output = O, where O is a list of strings: "RicciScalars", "AllRicciScalars", "WeylScalars", "AllWeylScalars"

   option2   - (optional keyword argument) keyword argument conjugatecoordinates = CC, where CC is a list of lists specifying conjugate coordinates

 

Description

Examples

See Also

Description

• 

The Newman-Penrose curvature scalars are components of the trace-free Ricci tensor and Weyl tensor with respect to a null tetrad [L, N, M, barM] for a Lorentz metric g on a 4-dimensional manifold.

• 

If S is the trace-free Ricci tensor and R is the trace of the Ricci tensor, then the Newman-Penrose Ricci scalars are:

 

Λ=124R

Φ00=SL,L

Φ01=SL,M

Φ02=SM,M

Φ11=SL,N

Φ12=SN,M

Φ22=SN,N

Φ10=Φ01=SL,M

Φ20=Φ02=SM,M

Φ21=Φ12=SN,M

The scalars Φ00, Φ11, Φ22 are real. The others are complex, in general.

 

• 

If W is the Weyl tensor, then the Weyl scalars are:

 

Ψ0=WL,M,L,M

Ψ1=WL,N,L,M

Ψ2=WL,M,M,N

Ψ3=WL,N,M,N

Ψ4=WN,M,N,M

Ψ0=WL,M,L,M

Ψ1=WL,N,L,M

Ψ2=WL,M,M,N

Ψ3=WL,N,M,N

Ψ4=WN,M,N,M

 

All of the Weyl scalars are complex, in general.

• 

The minus sign appears in these formulas because there is a sign difference in the definition of the curvature between what is used in the Newman-Penrose formalism and what is used in the DifferentialGeometry CurvatureTensor command.

• 

The command NPCurvatureScalars returns a pair of tables NPRicci, NPWeyl. The table NPRicci contains the values of the NP Ricci scalars and has indices "Phi00", "Phi01", "Phi02", "Phi11","Phi12", "Phi22". The table NPWeyl contains the values of the NP Weyl scalars and has indices "Psi0", "Psi1", "Psi2", "Psi3", "Psi4".

• 

With output = ["RicciScalars"] only the table of Ricci scalars is computed.  With output = ["AllRicciScalars"] the values of the conjugate Ricci scalars are included. With output = ["WeylScalars"] only the table of Weyl scalars is computed. With output = ["AllWeylScalars"] the values of the conjugate Weyl scalars are included.

• 

If the curvature tensor R of the metric g has been previously calculated, then passing R to the command NPCurvatureScalars will avoid the need to recalculate it.  For this calling sequence, any rank 4 tensor R with indices ["cov_bas", "cov_bas", "cov_bas", "cov_bas"] can be used.

• 

If the Ricci tensor S of the metric g has been previously calculated and only the Ricci scalars are to be computed, then passing the value of S to the command NPCurvatureScalars will avoid the need to recalculate it. For this calling sequence, any symmetric, covariant rank 2 tensor S can be used.

• 

With either of the first two calling sequences, the Ricci and Weyl scalars are computed directly from the above formulas.

• 

With the third calling sequence the Ricci and Weyl scalars are computed from the NP Ricci identities using the Ricci and Weyl scalars. These identities require knowledge of the null tetrad.  If the current frame is a coordinate frame, then the null tetrad must be specified as the second argument.  If the current frame is an anholonomic frame, then that frame is taken to be the null tetrad.

• 

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

Examples

withDifferentialGeometry:withTools:withTensor:

 

Define a manifold Q with coordinates t,x,y,z.

Q > 

DGsetupt,x,y,z,Q

frame name: Q

(2.1)

 

Define a metric g.

Q > 

gevalDG1x2dt&tdtdx&tdxdy&tdydz&tdz

g1x2dtdtdxdxdydydzdz

(2.2)

 

Here is the null tetrad we shall use for our examples. It is not the simplest null tetrad for the metric g, but it serves well for illustration.

Q > 

NTetradevalDG12212xD_t+12212D_z,3212xD_t+2212D_x+212D_y+2212D_z,1+12I212xD_t+12212D_x+12I212D_y+1+12I212D_z,112I212xD_t+12212D_x12I212D_y+112I212D_z

NTetrad2x2D_t+22D_z,32xD_t+22D_x+2D_y+22D_z,1+I22xD_t+22D_x+I22D_y+1+I22D_z,1I22xD_t+22D_xI22D_y+1I22D_z

(2.3)

 

Use GRQuery to check that NTetrad is a null tetrad for the metric g.

Q > 

GRQueryNTetrad,g,NullTetrad

true

(2.4)

 

Example 1.

In this example we compute the NP Ricci scalars and the NP Weyl scalars directly from the null tetrad NTetrad.

Q > 

RicciScalars,WeylScalarsNPCurvatureScalarsNTetrad

RicciScalars,WeylScalarstableLambda=16x2,Phi12=4+3Ix2,Phi22=10x2,Phi02=1+2Ix2,Phi11=52x2,Phi00=12x2,Phi01=1+I2x2,tablePsi4=−68Ix2,Psi3=5Ix2,Psi1=1I2x2,Psi0=12x2,Psi2=432Ix2

(2.5)

 

The individual values for the Ricci scalars are easily accessed as table entries:

Q > 

RicciScalarsPhi02

1+2Ix2

(2.6)
Q > 

WeylScalarsPsi3

5Ix2

(2.7)

 

Example 2.

Use the keyword argument output to find the complete set of NP Ricci scalars (some of which are just complex conjugates of others).

Q > 

NPCurvatureScalarsNTetrad,output=AllRicciScalars

tablePhi10=1I2x2,Lambda=16x2,Phi12=4+3Ix2,Phi22=10x2,Phi02=1+2Ix2,Phi11=52x2,Phi21=43Ix2,Phi00=12x2,Phi01=1+I2x2,Phi20=12Ix2

(2.8)

 

Example 3.

Calculate the curvature tensor C for the metric g and use it to calculate the NP Weyl scalars.

Q > 

CCurvatureTensorg

C2x2D_tdxdtdx+2x2D_tdxdxdt2x4D_xdtdtdx+2x4D_xdtdxdt

(2.9)
Q > 

NPCurvatureScalarsNTetrad,g,C,output=WeylScalars

tablePsi4=−68Ix2,Psi3=5Ix2,Psi1=1I2x2,Psi0=12x2,Psi2=432Ix2

(2.10)

 

Example 4.

Calculate the NP Ricci scalars for a given Ricci tensor.

Q > 

RevalDGadt&sdx+bdy&tdy

Ra2dtdx+a2dxdt+bdydy

(2.11)
Q > 

NPCurvatureScalarsNTetrad,g,R,output=RicciScalars

tableLambda=b24,Phi12=Ib2+7xa4+Ixa2,Phi22=b+6xa,Phi02=b4+xa2+Ixa4,Phi11=b8+xa2,Phi00=0,Phi01=xa8

(2.12)

 

Example 5.

Let us check the computed values of the NP Ricci scalars in Example 1 against the definition given in the Description. For this we need the trace-free Ricci tensor S and the individual vectors L,N,M,M of the null tetrad.

 

Q > 

CCurvatureTensorg

C2x2D_tdxdtdx+2x2D_tdxdxdt2x4D_xdtdtdx+2x4D_xdtdxdt

(2.13)
Q > 

S12&multevalDGRicciTensorC14gRicciScalarg,C

S12x4dtdt+12x2dxdx12x2dydy12x2dzdz

(2.14)
Q > 

L,N,M,barMopNTetrad:

 

1. Φ00=SL,L

Q > 

RicciScalarsPhi00=HookL,L,S

12x2=12x2

(2.15)

 

2. Φ01=SL,M

Q > 

RicciScalarsPhi01=HookL,M,S

1+I2x2=1+I2x2

(2.16)

 

3. Φ02=SM,M

Q > 

RicciScalarsPhi01=HookL,M,S

1+12Ix2=1+12Ix2

(2.17)

 

4. Φ11=SL,N

Q > 

RicciScalarsPhi11=HookL,N,S

52x2=52x2

(2.18)

 

5. Φ12=SN,M

Q > 

RicciScalarsPhi12=HookN,M,S

4+3Ix2=4+3Ix2

(2.19)

 

6. Φ22=SN,N

Q > 

RicciScalarsPhi22=HookN,N,S

10x2=10x2

(2.20)

 

Example 6.

When working with the NP formalism, it is usually advantageous to work with the anholonomic frame defined by the null tetrad. To create anholonomic frames in DifferentialGeometry, see FrameData.

Q > 

FDFrameDataNTetrad,NP

FDE1,E2=62E1x2E2x+2I2E3x+2+I2E4x,E1,E3=32E12x2E24x+12I42E3x+12+I42E4x,E1,E4=32E12x2E24x+12I42E3x+12+I42E4x,E2,E3=3+6I2E1x+12+I2E2x2+3I22E3x5I2E42x,E2,E4=36I2E1x+12I2E2x+5I2E32x+−2+3I22E4x,E3,E4=3I2E1xI2E22x+12+I2E3x+12+I2E4x

(2.21)
Q > 

DGsetupFD

frame name: NP

(2.22)

 

The basis of tangent vectors is now denoted by E1,E2,E3,E4 with dual basis Θ1,Θ2,Θ3,Θ4. 

Note that in the null tetrad frame, the metric has the following standard form.

NP > 

hevalDG2Θ1&sΘ22Θ3&sΘ4

hΘ1Θ2+Θ2Θ1Θ3Θ4Θ4Θ3

(2.23)

Compute the Weyl Scalars using the anholonomic frame.

NP > 

NPCurvatureScalarsNP,output=WeylScalars

tablePsi4=−68Ix2,Psi3=5Ix2,Psi1=1I2x2,Psi0=12x2,Psi2=432Ix2

(2.24)

 

In this setting the NP Weyl scalars can be read-off directly from the components of the Weyl tensor using the DGinfo command. First compute the Weyl tensor from the metric h in the anholonomic frame.

NP > 

WWeylTensorh:

 

1. Ψ0=WL,M,L,M

NP > 

WeylScalarsPsi0=HookE1,E3,E1,E3,W

12x2=12x2

(2.25)

 

or, equivalently,

NP > 

WeylScalarsPsi0=DGinfoW,CoefficientList,1,3,1,31

12x2=12x2

(2.26)

 

2. Ψ1=WL,N,L,M

NP > 

WeylScalarsPsi1=DGinfoW,CoefficientList,1,2,1,31

1I2x2=1I2x2

(2.27)

 

3. Ψ2=WL,M,M,N

NP > 

WeylScalarsPsi2=DGinfoW,CoefficientList,1,3,4,21

432Ix2=432Ix2

(2.28)

 

4. Ψ3=WL,N,M,N

NP > 

WeylScalarsPsi3=DGinfoW,CoefficientList,1,2,4,21

5Ix2=5Ix2

(2.29)

 

5. Ψ4=WN,M,N,M

NP > 

WeylScalarsPsi4=DGinfoW,CoefficientList,2,4,2,41

−68Ix2=−68Ix2

(2.30)

 

Example 7.

The NP curvature scalars can also be computed from the NP spin coefficients using the NP Ricci identities. First compute the spin coefficients.

NP > 

SpinCoeffNPSpinCoefficientsNTetrad

SpinCoefftablelambda=5I22x,sigma=12+I42x,gamma=32x3I22x,tau=322x,kappa=24x,pi=12I2x,rho=12I42x,epsilon=22xI24x,alpha=34I2x,nu=36I2x,beta=524x,mu=23I22x

(2.31)

 

In the coordinate frame Q, the null tetrad must be specified as the 2nd argument.

Q > 

NPCurvatureScalarsSpinCoeff,NTetrad,output=RicciScalars

tableLambda=16x2,Phi12=4+3Ix2,Phi22=10x2,Phi02=1+2Ix2,Phi11=52x2,Phi00=12x2,Phi01=1+I2x2

(2.32)
Q > 

ChangeFrameNP

Q

(2.33)

 

In the anholonomic frame NP, defined by the null tetrad NTetrad, no such 2nd argument is needed.

NP > 

NPCurvatureScalarsSpinCoeff,output=RicciScalars

tableLambda=16x2,Phi12=4+3Ix2,Phi22=10x2,Phi02=1+2Ix2,Phi11=52x2,Phi00=12x2,Phi01=1+I2x2

(2.34)

 

Example 8.

The NP Weyl curvature scalars can also be computed from the Weyl spinor.

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

frame name: Spin

(2.35)

 

To demonstrate this calling sequence, it is helpful to first define a basis for the symmetric, covariant rank 4 spinors.

Spin > 

W0evalDGdz1&tdz1&tdz1&tdz1

W0dz1dz1dz1dz1

(2.36)
Spin > 

W1evalDGdz1&tdz1&tdz1&tdz2+dz1&tdz1&tdz2&tdz1+dz1&tdz2&tdz1&tdz1+dz2&tdz1&tdz1&tdz1

W1dz1dz1dz1dz2+dz1dz