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

Calling Sequences

     NullTetradTransformation(NullTetrad, TransType, , 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

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

 

,    

 

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

• 

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

– 

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

  .

– 

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

.

– 

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

   

– 

4.  A boost ( real and non-zero):

 

• 

The command NullTetradTransformation(NullTetrad, TransType, , axis) returns the new null tetrad  obtained from NullTetrad =  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

 

For the first 4 examples we work with coordinates  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.

(2.1)
S > 

(2.2)
S > 

(2.3)
S > 

(2.4)
S > 

(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 > 

(2.6)
S > 

(2.7)
S > 

(2.8)
S > 

(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 > 

(2.10)
S > 

(2.11)
S > 

(2.12)
S > 

(2.13)

 

Example 3.

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

S > 

(2.14)
S > 

(2.15)

 

Example 4.

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

S > 

(2.16)
S > 

(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 > 

(2.18)

 

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

S > 

(2.19)

 

Apply a null rotation with parameter  to T1.

S > 

(2.20)

 

Calculate the NP Weyl scalars for the null tetrad T2.

S > 

(2.21)

 

We can make Psi1 = 0 by choosing

S > 

(2.22)

 

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

S > 

(2.23)

See Also

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


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