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Tensor[NullTetradTransformation] - apply a Lorentz transformation to a null tetrad

Calling Sequences

     NullTetradTransformation(NullTetrad, TransType, θ, axis)

Parameters

   NullTetrad - a list of 4 vectors defining a null tetrad

   TransType  - a string, "null rotation", "spatial rotation", or "boost", describing the transformation type

   θ          - the transformation parameter

   axis       -(optional) a string, specifies the axis of rotation as "l"(or "L") or "m"(or"M") in the case where TransType = "null rotation"

 

Description

Examples

See Also

Description

• 

Let g be a metric on a 4-dimensional manifold with signature  1,1,1,1. A list of 4 vectors L,N,M,M defines a null tetrad if L and N are real, M is the complex conjugate of M,

 

gL,N=1,   gM,M=1, 

 

and all other inner products vanish. In particular, the vectors L,N,M,M are all null vectors.

• 

A Lorentz transformation is a (linear) change of frame which transforms a null tetrad L,N,M,M into another null tetrad L',N',M',M'. Every Lorentz transformation can be expressed as the composition of the following 4 basic Lorentz transformations.

– 

1.  A null rotation about the L axis (θ complex):

 L'=L,  N'=N+θ M+θ M+θθL,   M'=M+θL,   M'=M+θL.

– 

2.  A null rotation about the N axis (θ complex)

L'=L+ θ M +θM+θθN,  N'=N,   M'=M+θL,   M'=M+θN.

– 

3.  A spatial rotation in the MM plane (θ real):

L'=L,  N'=N,  M'=eiθM,    M'=eiθ M.

– 

4.  A boost (θ real and non-zero):

L'=θL,   N'=1θN,   M'=M,   M'=M. 

• 

The command NullTetradTransformation(NullTetrad, TransType, θ, axis) returns the new null tetrad [L', N', M', M'] obtained from NullTetrad = [L, N, M, M] through the application of one of the above Lorentz transformations.

• 

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

Examples

withDifferentialGeometry:withTensor:

 

For the first 4 examples we work with coordinates u,v,x,y and an off-diagonal form for the metric. This is the easiest setting to see the effects the 4 basic Lorentz transformations.  Here we define the metric and a null tetrad.

DGsetupu,v,x,y,S

frame name: S

(2.1)
S > 

gevalDG2du&sdv12dx&tdx+dy&tdy

g:=dudv+dvdu12dxdx12dydy

(2.2)
S > 

L,N,M,barMD_u,D_v,evalDGD_x+ID_y,evalDGD_xID_y

L,N,M,barM:=D_u,D_v,D_x+ID_y,D_xID_y

(2.3)
S > 

TL,N,M,barM

T:=D_u,D_v,D_x+ID_y,D_xID_y

(2.4)
S > 

GRQueryT,g,NullTetrad

true

(2.5)

 

Example 1.

Apply a null rotation to the null tetrad T about the "l" axis. Check that the result is a null tetrad.

S > 

T1aNullTetradTransformationT,null rotation,a,lassuminga::real

T1a:=D_u,a2D_u+D_v+2aD_x,aD_u+D_x+ID_y,aD_u+D_xID_y

(2.6)
S > 

GRQueryT1a,g,NullTetrad

true

(2.7)
S > 

T1bNullTetradTransformationT,null rotation,Ib,lassumingb::real

T1b:=D_u,b2D_u+D_v2bD_y,IbD_u+D_x+ID_y,IbD_u+D_xID_y

(2.8)
S > 

GRQueryT1b,g,NullTetrad

true

(2.9)

 

Example 2.

Apply a null rotation about the "n" axis to the null tetrad T.  Check that the result is a null tetrad.

S > 

T2aNullTetradTransformationT,null rotation,a,nassuminga::real

T2a:=D_u+a2D_v+2aD_x,D_v,aD_v+D_x+ID_y,aD_v+D_xID_y

(2.10)
S > 

GRQueryT2a,g,NullTetrad

true

(2.11)
S > 

T2bNullTetradTransformationT,null rotation,Ib,nassumingb::real

T2b:=D_u+b2D_v2bD_y,D_v,IbD_v+D_x+ID_y,IbD_v+D_xID_y

(2.12)
S > 

GRQueryT2b,g,NullTetrad

true

(2.13)

 

Example 3.

Apply a spatial rotation to the null tetrad T. Check that the result is a null tetrad.

S > 

T3NullTetradTransformationT,spatial rotation,θ,nassumingθ::real

T3:=D_u,D_v,cosθ+IsinθD_x+IcosθsinθD_y,cosθIsinθD_xIcosθ+sinθD_y

(2.14)
S > 

GRQueryT3,g,NullTetrad

true

(2.15)

 

Example 4.

Apply a boost to the null tetrad T. Check that the result is a null tetrad.

S > 

T4NullTetradTransformationT,spatial rotation,θ,nassumingθ::real

T4:=D_u,D_v,cosθ+IsinθD_x+IcosθsinθD_y,cosθIsinθD_xIcosθ+sinθD_y

(2.16)
S > 

GRQueryT4,g,NullTetrad

true

(2.17)

 

Example 5.

In this example we show how the use of a null tetrad transformation can be use to simplify the NP Weyl scalars. First we define our manifold.

S > 

DGsetupt,x,y,z,S

frame name: S

(2.18)

 

Define a null tetrad T1. (By decreeing this to be a null tetrad we implicitly define the spacetime metric.)

S > 

T1evalDG12212D_t+12212D_z,12212D_t12212D_z,12212z2D_x+12I212x2D_y,12212z2D_x12I212x2D_y

T1:=122D_t+122D_z,122D_t122D_z,122z2D_x+12I2x2D_y,122z2D_x12I2x2D_y

(2.19)

 

Apply a null rotation with parameter θ=a to T1.

S > 

T2NullTetradTransformationT1,null rotation,a,lassuminga::real

T2:=122D_t+122D_z,12a22+122D_t+a2z2D_x+12a22122D_z,12a2D_t+122z2D_x+12I2x2D_y+12a2D_z,12a2D_t+122z2D_x12I2x2D_y+12a2D_z

(2.20)

 

Calculate the NP Weyl scalars for the null tetrad T2.

S > 

NPCurvatureScalarsT2,output=WeylScalars

tablePsi3=126z3a2x2z3x6z6a+3a3x2+3ax2z2x2,Psi1=122z3+3axz2x,Psi2=122z6+x2+3a2x2+4xaz3z2x2,Psi0=32z2,Psi4=128xa3z38xaz3+3a4x2+6a2x212a2z6+3x2z2x2

(2.21)

 

We can make Psi1 = 0 by choosing a=23 xz3.

S > 

T3evalT2,a=23z3x

T3:=122D_t+122D_z,29z62x2+122D_t23z52D_xx+29z62x2122D_z,13z32D_tx+122z2D_x+12I2x2D_y13z32D_zx,13z32D_tx+122z2D_x12I2x2D_y13z32D_zx

(2.22)

 

Recalculate the NP Weyl scalars and note that Psi1 = 0.

S > 

NPCurvatureScalarsT3,output=WeylScalars

tablePsi3=29z13z6+9x2x3,Psi1=0,Psi2=1610z6+3x2z2x2,Psi0=32z2,Psi4=11864z12+72z6x2+27x4z2x4

(2.23)

See Also

DifferentialGeometry, Tensor,  DGGramSchmidt, GRQuery, NullTetrad, OrthonormalTetrad, NPCurvatureScalars


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