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Tensor[AdaptedNullTetrad] - find a null tetrad which transforms the Newman-Penrose Weyl scalars to a standard form

Calling Sequences

     AdaptedNullTetrad(NT, PT, options )

     AdaptedNullTetrad(NT, PT, W, options )

     AdaptedNullTetrad(NT, PT, NP , options )

Parameters

    NT      - a null tetrad for the spacetime metric g

    PT      -  the Petrov type of g

    W       - (optional) the Weyl tensor of g

    NP      - (optional) the Newman-Penrose Weyl scalars

    options - one or more of the keyword arguments method and output

 

 

 

Description

Examples

Description

• 

 The Newman-Penrose Weyl scalars are a set of 5 complex scalars, labeled Ψ0, Ψ1, Ψ2, Ψ3, Ψ4 , and defined by certain components of the Weyl tensor with respect to a given null tetrad in a four dimensional spacetime of signature [1, -1, -1, -1]. Under local Lorentz transformations, the Newman-Penrose Weyl scalars transform among themselves in a natural way. Depending upon the Petrov type of the spacetime it is possible to transform the Newman-Penrose Weyl scalars to one of following normal forms. Below, η and χ are complex scalars. See NPCurvatureScalars, NullTetradTransformation.

Type I. Ψ0= 32 η χ , Ψ1 = 0, Ψ2=12η2  χ, Ψ3 =0, Ψ4 = 32η χ .

Type II. Ψ0= 0, Ψ1= 0, Ψ2 =η, Ψ3=0,Ψ4 = 6 η.

Type III. Ψ0 = 0, Ψ1=0, Ψ2 =0, Ψ3=1, Ψ4 = 0.

Type D. Ψ0= 0, Ψ1 = 0, Ψ2 =η, Ψ3 = 0, Ψ4= 0.

Type N. Ψ0= 0, Ψ1 =0, Ψ2 = 0, Ψ3 = 0, Ψ4 = 1.

Type O. Ψ0= 0, Ψ1 = 0, Ψ2 =0, Ψ3 =0, Ψ4 = 0.

See Penrose and Rindle Vol. 2, Section 8.3.

• 

Null tetrads for which the Newman-Penrose Weyl scalars are in the above normal form are called adapted null tetrads. Calculations are often simplified by using an adapted null tetrad.

• 

The command AdaptedNullTetrad returns a null tetrad which will put the Newman-Penrose Weyl scalars in the above normal form.

• 

The procedure AdaptedNullTetrad first calculates the Weyl spinor and calls the procedure AdaptedSpinorDyad to find a spinor dyad which transforms the Weyl spinor to normal form. The adapted null tetrad is then constructed from the spinor dyad.

• 

The command AdaptedNullTetrad is part of the DifferentialGeometry:-Tensor package. It can be used in the form AdaptedNullTetrad(...) only after executing the commands with(DifferentialGeometry) and with(Tensor), but can always be used by executing DifferentialGeometry:-Tensor:-AdaptedNullTetrad(...).

Examples

withDifferentialGeometry:withTensor:

 

Set the global environment variable _EnvExplicit to true to insure that the adapted null tetrads are free of RootOf expressions.

_EnvExplicittrue:

 

Example 1. Type I

We calculate an adapted null tetrad for a type I spacetime. First define the coordinates to be used and then define the metric.

 

DGsetupt,x,y,z,M

frame name: M

(2.1)
M > 

g1evalDGdt &t dtt2dx &t dxx2dy &t dydz &t dz

g1:=dtdtt2dxdxx2dydydzdz

(2.2)

 

Here is an initial null tetrad.

NT1evalDGD_t+D_z,1D_tD_z2,12D_x2t+1I2D_y2x,12D_x2t1I2D_y2x

NT1:=D_t+D_z,12D_t12D_z,22tD_x+I22xD_y,22tD_xI22xD_y

(2.3)

 

We check that this is indeed a null tetrad for the given metric using GRQuery.

M > 

GRQueryNT1,g1,NullTetrad

true

(2.4)

 

Compute the Newman-Penrose coefficients and check that the Petrov type is I. The coefficients are not in normal form for type I (for example, Ψ1  0), so NT1 is not an adapted null tetrad.

M > 

NP1NPCurvatureScalarsNT1,output=WeylScalars

NP1:=tablePsi1=142xt2,Psi0=0,Psi2=0,Psi4=0,Psi3=182xt2

(2.5)
M > 

PetrovTypeNP1

I

(2.6)

 

Calculate an adapted null tetrad and simplify.

newNT1combineAdaptedNullTetradNT1,I,symbolic

newNT1:=22D_t22tD_x,22D_t+22tD_x,12+I2xD_y+12I2D_z,12I2xD_y+12+I2D_z

(2.7)

 

Calculate the Newman-Penrose coefficients for the new null tetrad. We obtain the correct normal form (with χ = 2)since Ψ1= Ψ3 = 0 and Ψ0= Ψ4.

M > 

newNP1NPCurvatureScalarsnewNT1,output=WeylScalars

newNP1:=tablePsi1=0,Psi0=12It2x,Psi2=0,Psi4=12It2x,Psi3=0

(2.8)

 

Example 2. Type II

We calculate an adapted null tetrad for a type II spacetime. First define the coordinates to be used and then define the metric.

 

DGsetupr,u,x,y,M

frame name: M

(2.9)
M > 

g2evalDG2r2dx &t dx+dy &t dy2x3+2du &s dr32x+2mrdu &t du

g2:=drdu+dudr23xr+mrdudur24x3dxdxr24x3dydy

(2.10)

 

Here is an initial null tetrad.

M > 

NT2evalDGD_r,3xr+mD_rr+D_u,I2x32D_xr+2x32D_yr,I2x32D_xr+2x32D_yr

NT2:=D_r,3xr+mrD_r+D_u,I2x32rD_x+2x32rD_y,I2x32rD_x+2x32rD_y

(2.11)

 

We check that this is indeed a null tetrad for the given metric.

M > 

GRQueryNT2,g2,NullTetrad

true

(2.12)

 

Compute the Newman-Penrose coefficients and check that the Petrov type is II. The coefficients are not in normal form for type II (for example, Ψ3  0), so NT2 is not an adapted null tetrad.

M > 

NP2NPCurvatureScalarsNT2,output=WeylScalars

NP2:=tablePsi1=0,Psi0=0,Psi2=mr3,Psi4=18x2r2,Psi3=3I2x3/2r2

(2.13)
M > 

PetrovTypeNP2

II

(2.14)

 

Calculate an adapted null tetrad. We use the third calling sequence so that the Weyl tensor, or equivalently, the Newman-Penrose Weyl scalars need not be computed. Moreover, all computations are then algebraic and we can use Maple's assuming feature to simplify all intermediate calculations.

newNT2AdaptedNullTetradNT2&comma;II&comma;NP2assuming0<x&comma;0<y&comma;0<r&comma;3m2xr<0&comma;m<0

newNT2:=r2xr3mxmD_r&comma;2r3x3+3m2rx+m3r322xr3mxmD_rmr2xr3mxD_u4x2r2xr3mD_x&comma;I2x32rmD_rI2x32rD_x2x32rD_y&comma;I2x32rmD_r+I2x32rD_x2x32rD_y

(2.15)

 

Calculate the Newman-Penrose coefficients for the new null tetrad. We obtain the correct normal form (with &eta; &equals; mr3 ) since &Psi;0&equals; &Psi;1 &equals; &Psi;3 &equals; 0 and &Psi;2 &equals; &eta;&comma; &Psi;4 &equals; 6 &eta;.

M > 

newNP2NPCurvatureScalarsnewNT2&comma;output&equals;WeylScalars

newNP2:=tablePsi1&equals;0&comma;Psi0&equals;0&comma;Psi2&equals;mr3&comma;Psi4&equals;6mr3&comma;Psi3&equals;0

(2.16)

 

Example 3. Type III

We calculate an adapted null tetrad for a type III spacetime. First define the coordinates to be used and then define the metric.

DGsetupr&comma;u&comma;x&comma;y&comma;M

frame name: M

(2.17)
M > 

g3evalDGr2dx &t dx&plus;dy &t dyx3&plus;2du &s dr3xdu &t du2

g3:=drdu+dudr3x2dudur2x3dxdxr2x3dydy

(2.18)

 

Here is an initial null tetrad.

NT3evalDG3x8&plus;12D_r&plus;1D_u2&plus;12x32D_y2r&comma;3x8&plus;12D_r&plus;1D_u212x32D_y2r&comma;3x8&plus;12D_r1D_u2&plus;1I2x32D_x2r&comma;3x8&plus;12D_r1D_u21I2x32D_x2r

NT3:=3x8+12D_r+12D_u+2x322rD_y&comma;3x8+12D_r+12D_u2x322rD_y&comma;3x812D_r12D_u+I22x32rD_x&comma;3x812D_r12D_uI22x32rD_x

(2.19)

 

We check that this is indeed a null tetrad for the given metric.

M > 

GRQueryNT3&comma;g3&comma;NullTetrad

true

(2.20)

 

Compute the Newman-Penrose coefficients and check that the Petrov type is III. The coefficients are not in normal form for type III (for example, &Psi;1  0), so NT3 is not an adapted null tetrad.

M > 

NP3NPCurvatureScalarsNT3&comma;output&equals;WeylScalars

NP3:=tablePsi1&equals;3322I2&plus;3xx3&sol;2r2&comma;Psi0&equals;332x4I2x&plus;3xr2&comma;Psi2&equals;932x2r2&comma;Psi4&equals;332x4I2x3xr2&comma;Psi3&equals;3322I23xx3&sol;2r2

(2.21)

PetrovTypeNP3

III

(2.22)

 

Calculate an adapted null tetrad.

newNT3AdaptedNullTetradNT3&comma;III&comma;NP3assuming0<x

newNT3:=32x328r2D_r&comma;112r28xD_r+42r23x32D_u+rx2D_x&comma;32x8D_r+2x322rD_x+I22x32rD_y&comma;32x8D_r+2x322rD_xI22x32rD_y

(2.23)

 

Calculate the Newman-Penrose coefficients for the new null tetrad. We obtain the correct normal form since &Psi;0 &equals; &Psi;1&equals; &Psi;2 &equals; &Psi;4 &equals;0 and &Psi;3&equals; 1 .

M > 

NPCurvatureScalarsnewNT3&comma;output&equals;WeylScalars

tablePsi1&equals;0&comma;Psi0&equals;0&comma;Psi2&equals;0&comma;Psi4&equals;0&comma;Psi3&equals;1

(2.24)

 

 

Example 4. Type D

We calculate an adapted null tetrad for a type D spacetime. First define the coordinates to be used and then define the metric.

DGsetupt&comma;x&comma;y&comma;z&comma;M

frame name: M

(2.25)
M > 

g4evalDGdx &t dxdy &t dy1&ExponentialE;2xdz &t dz2&plus;dt&plus;&ExponentialE;xdz &s dt&plus;&ExponentialE;xdz

g4:=dtdt+&ExponentialE;xdtdzdxdxdydy+&ExponentialE;xdzdt+&ExponentialE;2x2dzdz

(2.26)

 

Here is an initial null tetrad.

NT4evalDG1221D_t2&plus;&ExponentialE;xD_z&comma;121&plus;2D_t2&ExponentialE;xD_z&comma;12D_x2&plus;1I2D_y2&comma;12D_x21I2D_y2

NT4:=2212D_t+&ExponentialE;xD_z&comma;22+12D_t&ExponentialE;xD_z&comma;22D_x+I22D_y&comma;22D_xI22D_y

(2.27)

 

We check that this is indeed a null tetrad for the given metric.

M > 

GRQueryNT4&comma;g4&comma;NullTetrad

true

(2.28)

 

Compute the Newman-Penrose coefficients and check that the Petrov type is D. The coefficients are not in normal form for type D (for example, &Psi;0  0), so NT4 is not an adapted null tetrad.

M > 

NP4NPCurvatureScalarsNT4&comma;output&equals;WeylScalars

NP4:=tablePsi1&equals;0&comma;Psi0&equals;14&comma;Psi2&equals;112&comma;Psi4&equals;14&comma;Psi3&equals;0

(2.29)
M > 

PetrovTypeNP4

D

(2.30)

 

Calculate an adapted null tetrad.

newNT4AdaptedNullTetradNT4&comma;D

newNT4:=2D_t2D_y&comma;24D_t+24D_y&comma;ID_t+22D_x+I&ExponentialE;xD_z&comma;ID_t+22D_xI&ExponentialE;xD_z

(2.31)

 

Calculate the Newman-Penrose coefficients for the new null tetrad. We obtain the correct normal form since &Psi;0&equals; &Psi;1 &equals; &Psi;3 &equals;&Psi;4 = 0.

M > 

newNPNPCurvatureScalarsnewNT4&comma;output&equals;WeylScalars

newNP:=tablePsi1&equals;0&comma;Psi0&equals;0&comma;Psi2&equals;16&comma;Psi4&equals;0&comma;Psi3&equals;0

(2.32)

 

Example 5. Type N

We calculate an adapted null tetrad for a type N spacetime. First define the coordinates to be used and then define the metric.

DGsetupu&comma;x&comma;y&comma;z&comma;M

frame name: M

(2.33)
M > 

g5evalDG&ExponentialE;2zdu &t dx&plus;&ExponentialE;2zdx &t du&plus;&ExponentialE;zdx &t dx&ExponentialE;2zdy &t dydz &t dz

g5:=&ExponentialE;2zdudx+&ExponentialE;2zdxdu+&ExponentialE;zdxdx&ExponentialE;2zdydydzdz

(2.34)

 

Here is the initial null tetrad.

NT5evalDG1&ExponentialE;3z2&ExponentialE;zD_u4&plus;1&ExponentialE;zD_x2&plus;12D_z2&comma;1&ExponentialE;3z2&ExponentialE;zD_u4&plus;1&ExponentialE;zD_x212D_z2&comma;1&ExponentialE;3z&plus;2&ExponentialE;zD_u41&ExponentialE;zD_x2&plus;1I2&ExponentialE;zD_y2&comma;1&ExponentialE;3z&plus;2&ExponentialE;zD_u41&ExponentialE;zD_x21I2&ExponentialE;zD_y2

NT5:=&ExponentialE;3z2&ExponentialE;z4D_u+&ExponentialE;z2D_x+22D_z&comma;&ExponentialE;3z2&ExponentialE;z4D_u+&ExponentialE;z2D_x22D_z&comma;&ExponentialE;3z+2&ExponentialE;z4D_u&ExponentialE;z2D_x+I22&ExponentialE;zD_y&comma;&ExponentialE;3z+2&ExponentialE;z4D_u&ExponentialE;z2D_xI22&ExponentialE;zD_y

(2.35)

 

We check that this is indeed a null tetrad for the given metric.

M > 

GRQueryNT5&comma;g5&comma;NullTetrad

true

(2.36)

 

Compute the Newman-Penrose coefficients and check that the Petrov type is N. The coefficients are not in normal form for type N (for example, &Psi;1  0), so NT5 is not an adapted null tetrad.

M > 

NP5NPCurvatureScalarsNT5&comma;output&equals;WeylScalars

NP5:=tablePsi1&equals;38&ExponentialE;3z&comma;Psi0&equals;38&ExponentialE;3z&comma;Psi2&equals;38&ExponentialE;3z&comma;Psi4&equals;38&ExponentialE;3z&comma;Psi3&equals;38&ExponentialE;3z

(2.37)
M > 

PetrovTypeNP5

N

(2.38)

 

Calculate an adapted null tetrad.

newNT5AdaptedNullTetradNT5&comma;N&comma;NP5assuming0<z

newNT5:=&ExponentialE;5z262D_u&comma;6&ExponentialE;3z2&ExponentialE;z26D_u+&ExponentialE;z263D_x+23&ExponentialE;3z23D_z&comma;&ExponentialE;zD_u+I22&ExponentialE;zD_y22D_z&comma;&ExponentialE;zD_uI22&ExponentialE;zD_y22D_z

(2.39)

 

Calculate the Newman-Penrose coefficients for the new null tetrad. We obtain the correct normal form since &Psi;0&equals; &Psi;1 &equals; &Psi;2 &equals;&Psi;3 = 0 and &Psi;4 &equals;1.

M > 

newNP5NPCurvatureScalarsnewNT5&comma;output&equals;WeylScalars

newNP5:=tablePsi1&equals;0&comma;Psi0&equals;0&comma;Psi2&equals;0&comma;Psi4&equals;1&comma;Psi3&equals;0

(2.40)

See Also

DifferentialGeometry

Tensor

AdaptedSpinorDyad,

AdaptedNullTetrad

FactorWeylSpinor

NPCurvatureScalars

NullVector

PetrovType

WeylSpinor

 


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