DifferentialGeometry/Tensor/NullTetradTransformation - Maple Help
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Tensor[NullTetradTransformation] - apply a Lorentz transformation to a null tetrad

Calling Sequences

NullTetradTransformation(NullTetrad, TransType, ${\mathbf{θ}}$, 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

$\mathrm{θ}$          - 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

 • Let $g$ be a metric on a 4-dimensional manifold with signature  $\left[1,-1,-1,-1\right]$. A list of 4 vectors $\left[L,N,M,\stackrel{‾}{M}\right]$ defines a null tetrad if $L$ and $N$ are real, $\stackrel{‾}{M}$ is the complex conjugate of $M$,

$g\left(L,N\right)=1$,   $g\left(M,\stackrel{‾}{M}\right)=-1,$

and all other inner products vanish. In particular, the vectors $\left[L,N,M,\stackrel{‾}{M}\right]$ are all null vectors.

 • A Lorentz transformation is a (linear) change of frame which transforms a null tetrad $\left[L,N,M,\stackrel{‾}{M}\right]$ into another null tetrad $\left[L',N',M',\stackrel{‾}{M}'\right]$. Every Lorentz transformation can be expressed as the composition of the following 4 basic Lorentz transformations.
 – 1.  A null rotation about the $L$ axis ($\mathrm{θ}$ complex):

$N'=N+$ .

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

$N'=N$.

 – 3.  A spatial rotation in the $M-\stackrel{‾}{M}$ plane ($\mathrm{θ}$ real):

$\stackrel{‾}{M}'$

 – 4.  A boost ($\mathrm{θ}$ real and non-zero):

 • The command NullTetradTransformation(NullTetrad, TransType, ${\mathbf{θ}}$, 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

 > $\mathrm{with}\left(\mathrm{DifferentialGeometry}\right):$$\mathrm{with}\left(\mathrm{Tensor}\right):$

For the first 4 examples we work with coordinates $\left(u,v,x,y\right)$ 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.

 > $\mathrm{DGsetup}\left(\left[u,v,x,y\right],S\right)$
 ${\mathrm{frame name: S}}$ (2.1)
 S > $g≔\mathrm{evalDG}\left(2\mathrm{du}&s\mathrm{dv}-\frac{1\left(\mathrm{dx}&t\mathrm{dx}+\mathrm{dy}&t\mathrm{dy}\right)}{2}\right)$
 ${g}{:=}{\mathrm{du}}{}{\mathrm{dv}}{+}{\mathrm{dv}}{}{\mathrm{du}}{-}\frac{{1}}{{2}}{}{\mathrm{dx}}{}{\mathrm{dx}}{-}\frac{{1}}{{2}}{}{\mathrm{dy}}{}{\mathrm{dy}}$ (2.2)
 S > $L,N,M,\mathrm{barM}≔\mathrm{D_u},\mathrm{D_v},\mathrm{evalDG}\left(\mathrm{D_x}+I\mathrm{D_y}\right),\mathrm{evalDG}\left(\mathrm{D_x}-I\mathrm{D_y}\right)$
 ${L}{,}{N}{,}{M}{,}{\mathrm{barM}}{:=}{\mathrm{D_u}}{,}{\mathrm{D_v}}{,}{\mathrm{D_x}}{+}{I}{}{\mathrm{D_y}}{,}{\mathrm{D_x}}{-}{I}{}{\mathrm{D_y}}$ (2.3)
 S > $T≔\left[L,N,M,\mathrm{barM}\right]$
 ${T}{:=}\left[{\mathrm{D_u}}{,}{\mathrm{D_v}}{,}{\mathrm{D_x}}{+}{I}{}{\mathrm{D_y}}{,}{\mathrm{D_x}}{-}{I}{}{\mathrm{D_y}}\right]$ (2.4)
 S > $\mathrm{GRQuery}\left(T,g,"NullTetrad"\right)$
 ${\mathrm{true}}$ (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 > $\mathrm{T1a}≔\mathrm{NullTetradTransformation}\left(T,"null rotation",a,"l"\right)\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}assuming\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}a::\mathrm{real}$
 ${\mathrm{T1a}}{:=}\left[{\mathrm{D_u}}{,}{{a}}^{{2}}{}{\mathrm{D_u}}{+}{\mathrm{D_v}}{+}{2}{}{a}{}{\mathrm{D_x}}{,}{a}{}{\mathrm{D_u}}{+}{\mathrm{D_x}}{+}{I}{}{\mathrm{D_y}}{,}{a}{}{\mathrm{D_u}}{+}{\mathrm{D_x}}{-}{I}{}{\mathrm{D_y}}\right]$ (2.6)
 S > $\mathrm{GRQuery}\left(\mathrm{T1a},g,"NullTetrad"\right)$
 ${\mathrm{true}}$ (2.7)
 S > $\mathrm{T1b}≔\mathrm{NullTetradTransformation}\left(T,"null rotation",Ib,"l"\right)\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}assuming\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}b::\mathrm{real}$
 ${\mathrm{T1b}}{:=}\left[{\mathrm{D_u}}{,}{{b}}^{{2}}{}{\mathrm{D_u}}{+}{\mathrm{D_v}}{-}{2}{}{b}{}{\mathrm{D_y}}{,}{-}{I}{}{b}{}{\mathrm{D_u}}{+}{\mathrm{D_x}}{+}{I}{}{\mathrm{D_y}}{,}{I}{}{b}{}{\mathrm{D_u}}{+}{\mathrm{D_x}}{-}{I}{}{\mathrm{D_y}}\right]$ (2.8)
 S > $\mathrm{GRQuery}\left(\mathrm{T1b},g,"NullTetrad"\right)$
 ${\mathrm{true}}$ (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 > $\mathrm{T2a}≔\mathrm{NullTetradTransformation}\left(T,"null rotation",a,"n"\right)\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}assuming\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}a::\mathrm{real}$
 ${\mathrm{T2a}}{:=}\left[{\mathrm{D_u}}{+}{{a}}^{{2}}{}{\mathrm{D_v}}{+}{2}{}{a}{}{\mathrm{D_x}}{,}{\mathrm{D_v}}{,}{a}{}{\mathrm{D_v}}{+}{\mathrm{D_x}}{+}{I}{}{\mathrm{D_y}}{,}{a}{}{\mathrm{D_v}}{+}{\mathrm{D_x}}{-}{I}{}{\mathrm{D_y}}\right]$ (2.10)
 S > $\mathrm{GRQuery}\left(\mathrm{T2a},g,"NullTetrad"\right)$
 ${\mathrm{true}}$ (2.11)
 S > $\mathrm{T2b}≔\mathrm{NullTetradTransformation}\left(T,"null rotation",Ib,"n"\right)\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}assuming\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}b::\mathrm{real}$
 ${\mathrm{T2b}}{:=}\left[{\mathrm{D_u}}{+}{{b}}^{{2}}{}{\mathrm{D_v}}{-}{2}{}{b}{}{\mathrm{D_y}}{,}{\mathrm{D_v}}{,}{-}{I}{}{b}{}{\mathrm{D_v}}{+}{\mathrm{D_x}}{+}{I}{}{\mathrm{D_y}}{,}{I}{}{b}{}{\mathrm{D_v}}{+}{\mathrm{D_x}}{-}{I}{}{\mathrm{D_y}}\right]$ (2.12)
 S > $\mathrm{GRQuery}\left(\mathrm{T2b},g,"NullTetrad"\right)$
 ${\mathrm{true}}$ (2.13)

Example 3.

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

 S > $\mathrm{T3}≔\mathrm{NullTetradTransformation}\left(T,"spatial rotation",\mathrm{θ},"n"\right)\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}assuming\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\mathrm{θ}::\mathrm{real}$
 ${\mathrm{T3}}{:=}\left[{\mathrm{D_u}}{,}{\mathrm{D_v}}{,}\left({\mathrm{cos}}{}\left({\mathrm{θ}}\right){+}{I}{}{\mathrm{sin}}{}\left({\mathrm{θ}}\right)\right){}{\mathrm{D_x}}{+}\left({I}{}{\mathrm{cos}}{}\left({\mathrm{θ}}\right){-}{\mathrm{sin}}{}\left({\mathrm{θ}}\right)\right){}{\mathrm{D_y}}{,}\left({\mathrm{cos}}{}\left({\mathrm{θ}}\right){-}{I}{}{\mathrm{sin}}{}\left({\mathrm{θ}}\right)\right){}{\mathrm{D_x}}{-}\left({I}{}{\mathrm{cos}}{}\left({\mathrm{θ}}\right){+}{\mathrm{sin}}{}\left({\mathrm{θ}}\right)\right){}{\mathrm{D_y}}\right]$ (2.14)
 S > $\mathrm{GRQuery}\left(\mathrm{T3},g,"NullTetrad"\right)$
 ${\mathrm{true}}$ (2.15)

Example 4.

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

 S > $\mathrm{T4}≔\mathrm{NullTetradTransformation}\left(T,"spatial rotation",\mathrm{θ},"n"\right)\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}assuming\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\mathrm{θ}::\mathrm{real}$
 ${\mathrm{T4}}{:=}\left[{\mathrm{D_u}}{,}{\mathrm{D_v}}{,}\left({\mathrm{cos}}{}\left({\mathrm{θ}}\right){+}{I}{}{\mathrm{sin}}{}\left({\mathrm{θ}}\right)\right){}{\mathrm{D_x}}{+}\left({I}{}{\mathrm{cos}}{}\left({\mathrm{θ}}\right){-}{\mathrm{sin}}{}\left({\mathrm{θ}}\right)\right){}{\mathrm{D_y}}{,}\left({\mathrm{cos}}{}\left({\mathrm{θ}}\right){-}{I}{}{\mathrm{sin}}{}\left({\mathrm{θ}}\right)\right){}{\mathrm{D_x}}{-}\left({I}{}{\mathrm{cos}}{}\left({\mathrm{θ}}\right){+}{\mathrm{sin}}{}\left({\mathrm{θ}}\right)\right){}{\mathrm{D_y}}\right]$ (2.16)
 S > $\mathrm{GRQuery}\left(\mathrm{T4},g,"NullTetrad"\right)$
 ${\mathrm{true}}$ (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 > $\mathrm{DGsetup}\left(\left[t,x,y,z\right],S\right)$
 ${\mathrm{frame name: S}}$ (2.18)

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

 S > $\mathrm{T1}≔\mathrm{evalDG}\left(\left[\frac{1{2}^{\frac{1}{2}}\mathrm{D_t}}{2}+\frac{1{2}^{\frac{1}{2}}\mathrm{D_z}}{2},\frac{1{2}^{\frac{1}{2}}\mathrm{D_t}}{2}-\frac{1{2}^{\frac{1}{2}}\mathrm{D_z}}{2},\frac{1{2}^{\frac{1}{2}}{z}^{2}\mathrm{D_x}}{2}+\frac{1I{2}^{\frac{1}{2}}{x}^{2}\mathrm{D_y}}{2},\frac{1{2}^{\frac{1}{2}}{z}^{2}\mathrm{D_x}}{2}-\frac{1I{2}^{\frac{1}{2}}{x}^{2}\mathrm{D_y}}{2}\right]\right)$
 ${\mathrm{T1}}{:=}\left[\frac{{1}}{{2}}{}\sqrt{{2}}{}{\mathrm{D_t}}{+}\frac{{1}}{{2}}{}\sqrt{{2}}{}{\mathrm{D_z}}{,}\frac{{1}}{{2}}{}\sqrt{{2}}{}{\mathrm{D_t}}{-}\frac{{1}}{{2}}{}\sqrt{{2}}{}{\mathrm{D_z}}{,}\frac{{1}}{{2}}{}\sqrt{{2}}{}{{z}}^{{2}}{}{\mathrm{D_x}}{+}\frac{{1}}{{2}}{}{I}{}\sqrt{{2}}{}{{x}}^{{2}}{}{\mathrm{D_y}}{,}\frac{{1}}{{2}}{}\sqrt{{2}}{}{{z}}^{{2}}{}{\mathrm{D_x}}{-}\frac{{1}}{{2}}{}{I}{}\sqrt{{2}}{}{{x}}^{{2}}{}{\mathrm{D_y}}\right]$ (2.19)

Apply a null rotation with parameter $\mathrm{θ}=a$ to T1.

 S > $\mathrm{T2}≔\mathrm{NullTetradTransformation}\left(\mathrm{T1},"null rotation",a,"l"\right)\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}assuming\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}a::\mathrm{real}$
 ${\mathrm{T2}}{:=}\left[\frac{{1}}{{2}}{}\sqrt{{2}}{}{\mathrm{D_t}}{+}\frac{{1}}{{2}}{}\sqrt{{2}}{}{\mathrm{D_z}}{,}\left(\frac{{1}}{{2}}{}{{a}}^{{2}}{}\sqrt{{2}}{+}\frac{{1}}{{2}}{}\sqrt{{2}}\right){}{\mathrm{D_t}}{+}{a}{}\sqrt{{2}}{}{{z}}^{{2}}{}{\mathrm{D_x}}{+}\left(\frac{{1}}{{2}}{}{{a}}^{{2}}{}\sqrt{{2}}{-}\frac{{1}}{{2}}{}\sqrt{{2}}\right){}{\mathrm{D_z}}{,}\frac{{1}}{{2}}{}{a}{}\sqrt{{2}}{}{\mathrm{D_t}}{+}\frac{{1}}{{2}}{}\sqrt{{2}}{}{{z}}^{{2}}{}{\mathrm{D_x}}{+}\frac{{1}}{{2}}{}{I}{}\sqrt{{2}}{}{{x}}^{{2}}{}{\mathrm{D_y}}{+}\frac{{1}}{{2}}{}{a}{}\sqrt{{2}}{}{\mathrm{D_z}}{,}\frac{{1}}{{2}}{}{a}{}\sqrt{{2}}{}{\mathrm{D_t}}{+}\frac{{1}}{{2}}{}\sqrt{{2}}{}{{z}}^{{2}}{}{\mathrm{D_x}}{-}\frac{{1}}{{2}}{}{I}{}\sqrt{{2}}{}{{x}}^{{2}}{}{\mathrm{D_y}}{+}\frac{{1}}{{2}}{}{a}{}\sqrt{{2}}{}{\mathrm{D_z}}\right]$ (2.20)

Calculate the NP Weyl scalars for the null tetrad T2.

 S > $\mathrm{NPCurvatureScalars}\left(\mathrm{T2},\mathrm{output}=\left["WeylScalars"\right]\right)$
 ${\mathrm{table}}\left(\left[{"Psi3"}{=}{-}\frac{{1}}{{2}}{}\frac{{6}{}{{z}}^{{3}}{}{{a}}^{{2}}{}{x}{-}{2}{}{{z}}^{{3}}{}{x}{-}{6}{}{{z}}^{{6}}{}{a}{+}{3}{}{{a}}^{{3}}{}{{x}}^{{2}}{+}{3}{}{a}{}{{x}}^{{2}}}{{{z}}^{{2}}{}{{x}}^{{2}}}{,}{"Psi1"}{=}{-}\frac{{1}}{{2}}{}\frac{{2}{}{{z}}^{{3}}{+}{3}{}{a}{}{x}}{{{z}}^{{2}}{}{x}}{,}{"Psi2"}{=}{-}\frac{{1}}{{2}}{}\frac{{-}{2}{}{{z}}^{{6}}{+}{{x}}^{{2}}{+}{3}{}{{a}}^{{2}}{}{{x}}^{{2}}{+}{4}{}{x}{}{a}{}{{z}}^{{3}}}{{{z}}^{{2}}{}{{x}}^{{2}}}{,}{"Psi0"}{=}{-}\frac{{3}}{{2}{}{{z}}^{{2}}}{,}{"Psi4"}{=}{-}\frac{{1}}{{2}}{}\frac{{8}{}{x}{}{{a}}^{{3}}{}{{z}}^{{3}}{-}{8}{}{x}{}{a}{}{{z}}^{{3}}{+}{3}{}{{a}}^{{4}}{}{{x}}^{{2}}{+}{6}{}{{a}}^{{2}}{}{{x}}^{{2}}{-}{12}{}{{a}}^{{2}}{}{{z}}^{{6}}{+}{3}{}{{x}}^{{2}}}{{{z}}^{{2}}{}{{x}}^{{2}}}\right]\right)$ (2.21)

We can make Psi1 = 0 by choosing

 S > $\mathrm{T3}≔\genfrac{}{}{0}{}{\mathrm{T2}}{\phantom{a=-\frac{2{z}^{3}}{3x}}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}|\phantom{\rule[-0.0ex]{0.1em}{0.0ex}}\genfrac{}{}{0}{}{\phantom{\mathrm{T2}}}{a=-\frac{2{z}^{3}}{3x}}$
 ${\mathrm{T3}}{:=}\left[\frac{{1}}{{2}}{}\sqrt{{2}}{}{\mathrm{D_t}}{+}\frac{{1}}{{2}}{}\sqrt{{2}}{}{\mathrm{D_z}}{,}\left(\frac{{2}}{{9}}{}\frac{{{z}}^{{6}}{}\sqrt{{2}}}{{{x}}^{{2}}}{+}\frac{{1}}{{2}}{}\sqrt{{2}}\right){}{\mathrm{D_t}}{-}\frac{{2}}{{3}}{}\frac{{{z}}^{{5}}{}\sqrt{{2}}{}{\mathrm{D_x}}}{{x}}{+}\left(\frac{{2}}{{9}}{}\frac{{{z}}^{{6}}{}\sqrt{{2}}}{{{x}}^{{2}}}{-}\frac{{1}}{{2}}{}\sqrt{{2}}\right){}{\mathrm{D_z}}{,}{-}\frac{{1}}{{3}}{}\frac{{{z}}^{{3}}{}\sqrt{{2}}{}{\mathrm{D_t}}}{{x}}{+}\frac{{1}}{{2}}{}\sqrt{{2}}{}{{z}}^{{2}}{}{\mathrm{D_x}}{+}\frac{{1}}{{2}}{}{I}{}\sqrt{{2}}{}{{x}}^{{2}}{}{\mathrm{D_y}}{-}\frac{{1}}{{3}}{}\frac{{{z}}^{{3}}{}\sqrt{{2}}{}{\mathrm{D_z}}}{{x}}{,}{-}\frac{{1}}{{3}}{}\frac{{{z}}^{{3}}{}\sqrt{{2}}{}{\mathrm{D_t}}}{{x}}{+}\frac{{1}}{{2}}{}\sqrt{{2}}{}{{z}}^{{2}}{}{\mathrm{D_x}}{-}\frac{{1}}{{2}}{}{I}{}\sqrt{{2}}{}{{x}}^{{2}}{}{\mathrm{D_y}}{-}\frac{{1}}{{3}}{}\frac{{{z}}^{{3}}{}\sqrt{{2}}{}{\mathrm{D_z}}}{{x}}\right]$ (2.22)

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

 S > $\mathrm{NPCurvatureScalars}\left(\mathrm{T3},\mathrm{output}=\left["WeylScalars"\right]\right)$
 ${\mathrm{table}}\left(\left[{"Psi3"}{=}\frac{{2}}{{9}}{}\frac{{z}{}\left({-}{13}{}{{z}}^{{6}}{+}{9}{}{{x}}^{{2}}\right)}{{{x}}^{{3}}}{,}{"Psi1"}{=}{0}{,}{"Psi2"}{=}{-}\frac{{1}}{{6}}{}\frac{{-}{10}{}{{z}}^{{6}}{+}{3}{}{{x}}^{{2}}}{{{z}}^{{2}}{}{{x}}^{{2}}}{,}{"Psi0"}{=}{-}\frac{{3}}{{2}{}{{z}}^{{2}}}{,}{"Psi4"}{=}{-}\frac{{1}}{{18}}{}\frac{{-}{64}{}{{z}}^{{12}}{+}{72}{}{{z}}^{{6}}{}{{x}}^{{2}}{+}{27}{}{{x}}^{{4}}}{{{z}}^{{2}}{}{{x}}^{{4}}}\right]\right)$ (2.23)
 See Also