Tensor[NullTetrad] - construct a null tetrad from an orthonormal tetrad or from a solder form and a spinor basis

Calling Sequences

NullTetrad($,$SpinBasis)

Parameters

OrthTetrad   - a list of 4 vectors defining an orthonormal tetrad with respect to a metric g with signature

$\mathrm{σ}$            - a solder form with index type ["con", " cov", "cov"]

SpinBasis    - a list of 2 rank 1 spinors, with spinor inner product = 1

NullTetrad   - a list of 4 vectors defining a null tetrad with respect to a Lorentzian metric $g$ with signature $\left[1,-1,-1,-1\right]$

Description

 • Let $g$ be a metric on a 4-dimensional manifold with signature $\left[1,-1,-1,-1\right]$. A list of 4 vectors $\left[{E}_{t},{E}_{x},{E}_{y},{E}_{z}\right]$ defines an orthonormal tetrad if

and all other inner products vanish. 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.

 • Given an orthonormal tetrad OrthTetrad = $\left[{E}_{t},{E}_{x},{E}_{y},{E}_{z}\right]$, the command NullTetrad(OrthTetrad) constructs the null tetrad given by

$L=\frac{1}{\sqrt{2}}\left({E}_{t}+{E}_{z}\right),$  .

 • Let sigma be a solder form (index type ["con", " cov", "cov"]), with components for the metric $g$. Let and ${\mathrm{\iota }}^{B}$ be rank 1, unprimed spinors with ${\mathrm{ε}}_{\mathrm{AB}}{\mathrm{ο}}^{A}{\mathrm{ι}}^{B}=1$. Let $\stackrel{‾}{\mathrm{ο}}$ and $\stackrel{‾}{\mathrm{ι}}$ be their conjugates (see ConjugateSpinor).  Then the following vectors

${L}^{i}={\mathrm{σ}}_{\mathrm{AA}'}^{i}{\mathrm{ο}}^{A}$

define a null tetrad. This null tetrad is computed with the second calling sequence NullTetrad(sigma, [${\mathbf{ο}}$, ${\mathbf{ι}}$]).

 • Given a null tetrad NullTetrad =$[L,N,M,{\stackrel{‾}{M]}}^{}$, the command OrthonormalTetrad(NullTetrad) constructs the orthonormal tetrad defined by

${E}_{t}=$$\frac{1}{\sqrt{2}}\left(L+N\right),$

 • The command DGGramSchmidt can also be used to construct an orthonormal tetrad.
 • The command GRQuery can be used to check that a given tetrad is a null tetrad or an orthonormal tetrad.
 • These commands are part of the DifferentialGeometry:-Tensor package, and so can be used in the form NullTetrad(...) or OrthonormalTetrad(...) only after executing the commands with(DifferentialGeometry); with(Tensor) in that order. They can always be used in the long form DifferentialGeometry:-Tensor:-NullTetrad or DifferentialGeometry:-Tensor:-OrthonormalTetrad.

Examples

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

Example 1.

First create manifold $M$ with coordinates $\left(t,x,y,z\right)$.

 > $\mathrm{DGsetup}\left(\left[t,x,y,z\right],M\right)$
 ${\mathrm{frame name: M}}$ (2.1)

Define a spacetime metric $g$ on $M$ with signature $\left(1,-1,-1,-1\right)$.

 M > $g≔\mathrm{evalDG}\left(\mathrm{dt}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{dt}-\mathrm{dx}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{dx}-\mathrm{dy}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{dy}-\mathrm{dz}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{dz}\right)$
 ${g}{:=}{\mathrm{dt}}{}{\mathrm{dt}}{-}{\mathrm{dx}}{}{\mathrm{dx}}{-}{\mathrm{dy}}{}{\mathrm{dy}}{-}{\mathrm{dz}}{}{\mathrm{dz}}$ (2.2)

Define an orthonormal tetrad F on $M$ with respect to the metric $g$. Verify using the command GRQuery.

 M > $F≔\left[\mathrm{D_t},\mathrm{D_x},\mathrm{D_y},\mathrm{D_z}\right]$
 ${F}{:=}\left[{\mathrm{D_t}}{,}{\mathrm{D_x}}{,}{\mathrm{D_y}}{,}{\mathrm{D_z}}\right]$ (2.3)
 M > $\mathrm{GRQuery}\left(F,g,"OrthonormalTetrad"\right)$
 ${\mathrm{true}}$ (2.4)

 M > $\mathrm{NT}≔\mathrm{NullTetrad}\left(F\right)$
 ${\mathrm{NT}}{:=}\left[\frac{\sqrt{{2}}}{{2}}{}{\mathrm{D_t}}{+}\frac{\sqrt{{2}}}{{2}}{}{\mathrm{D_z}}{,}\frac{\sqrt{{2}}}{{2}}{}{\mathrm{D_t}}{-}\frac{\sqrt{{2}}}{{2}}{}{\mathrm{D_z}}{,}\frac{\sqrt{{2}}}{{2}}{}{\mathrm{D_x}}{+}\frac{{I}}{{2}}{}\sqrt{{2}}{}{\mathrm{D_y}}{,}\frac{\sqrt{{2}}}{{2}}{}{\mathrm{D_x}}{-}\frac{{I}}{{2}}{}\sqrt{{2}}{}{\mathrm{D_y}}\right]$ (2.5)

Verify this result using the command GRQuery.

 M > $\mathrm{GRQuery}\left(\mathrm{NT},g,"NullTetrad"\right)$
 ${\mathrm{true}}$ (2.6)

It is a simple matter to check directly, using the TensorInnerProduct command, that NT is a null tetrad,

 M > $\mathrm{TensorInnerProduct}\left(g,\mathrm{NT},\mathrm{NT}\right)$

Example 2.

We use spinors to create a null tetrad. First create a vector bundle $E\to M$ with base coordinates $\left(t,x,y,z\right)$ and fiber coordinates .

 > $\mathrm{DGsetup}\left(\left[t,x,y,z\right],\left[\mathrm{z1},\mathrm{z2},\mathrm{w1},\mathrm{w2}\right],E\right)$
 ${\mathrm{frame name: E}}$ (2.7)

Define a spacetime metric $\mathrm{g2}$ on $M$ with signature $\left(1,-1,-1,-1\right)$.

 E > $\mathrm{g2}≔\mathrm{evalDG}\left(\mathrm{dt}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{dt}-\mathrm{dx}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{dx}-\mathrm{dy}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{dy}-\mathrm{dz}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{dz}\right)$
 ${\mathrm{g2}}{:=}{\mathrm{dt}}{}{\mathrm{dt}}{-}{\mathrm{dx}}{}{\mathrm{dx}}{-}{\mathrm{dy}}{}{\mathrm{dy}}{-}{\mathrm{dz}}{}{\mathrm{dz}}$ (2.8)

Define an orthonormal frame $\mathrm{F2}$ on $M$ with respect to the metric $\mathrm{g2}$.

 E > $\mathrm{F2}≔\left[\mathrm{D_t},\mathrm{D_x},\mathrm{D_y},\mathrm{D_z}\right]$
 ${\mathrm{F2}}{:=}\left[{\mathrm{D_t}}{,}{\mathrm{D_x}}{,}{\mathrm{D_y}}{,}{\mathrm{D_z}}\right]$ (2.9)

Compute the solder form $\mathrm{σ}$ defined by the orthonormal frame $\mathrm{F2}$.

 E > $\mathrm{\sigma }≔\mathrm{SolderForm}\left(\mathrm{F2},\mathrm{indextype}=\left["con","cov","cov"\right]\right)$
 ${\mathrm{σ}}{:=}\frac{\sqrt{{2}}}{{2}}{}{\mathrm{D_t}}{}{\mathrm{dz1}}{}{\mathrm{dw1}}{+}\frac{\sqrt{{2}}}{{2}}{}{\mathrm{D_t}}{}{\mathrm{dz2}}{}{\mathrm{dw2}}{+}\frac{\sqrt{{2}}}{{2}}{}{\mathrm{D_x}}{}{\mathrm{dz1}}{}{\mathrm{dw2}}{+}\frac{\sqrt{{2}}}{{2}}{}{\mathrm{D_x}}{}{\mathrm{dz2}}{}{\mathrm{dw1}}{+}\frac{{I}}{{2}}{}\sqrt{{2}}{}{\mathrm{D_y}}{}{\mathrm{dz1}}{}{\mathrm{dw2}}{-}\frac{{I}}{{2}}{}\sqrt{{2}}{}{\mathrm{D_y}}{}{\mathrm{dz2}}{}{\mathrm{dw1}}{+}\frac{\sqrt{{2}}}{{2}}{}{\mathrm{D_z}}{}{\mathrm{dz1}}{}{\mathrm{dw1}}{-}\frac{\sqrt{{2}}}{{2}}{}{\mathrm{D_z}}{}{\mathrm{dz2}}{}{\mathrm{dw2}}$ (2.10)

Define a pair of rank 1 spinors $\mathrm{ο}$ and $\mathrm{ι}$. Check that their spinor inner product is 1. Construct the corresponding null tetrad, $\mathrm{N2}$.

 E > $\mathrm{ο}≔\mathrm{evalDG}\left(\mathrm{D_z1}+2\mathrm{D_z2}\right)$
 ${\mathrm{ο}}{:=}{\mathrm{D_z1}}{+}{2}{}{\mathrm{D_z2}}$ (2.11)
 E > $\mathrm{\iota }≔\mathrm{evalDG}\left(2\mathrm{D_z1}+5\mathrm{D_z2}\right)$
 ${\mathrm{ι}}{:=}{2}{}{\mathrm{D_z1}}{+}{5}{}{\mathrm{D_z2}}$ (2.12)
 E > $\mathrm{SpinorInnerProduct}\left(\mathrm{ο},\mathrm{\iota }\right)$
 ${1}$ (2.13)
 E > $\mathrm{N2}≔\mathrm{NullTetrad}\left(\mathrm{\sigma },\left[\mathrm{ο},\mathrm{\iota }\right]\right)$
 ${\mathrm{N2}}{:=}\left[\frac{{5}{}\sqrt{{2}}}{{2}}{}{\mathrm{D_t}}{+}{2}{}\sqrt{{2}}{}{\mathrm{D_x}}{-}\frac{{3}{}\sqrt{{2}}}{{2}}{}{\mathrm{D_z}}{,}\frac{{29}{}\sqrt{{2}}}{{2}}{}{\mathrm{D_t}}{+}{10}{}\sqrt{{2}}{}{\mathrm{D_x}}{-}\frac{{21}{}\sqrt{{2}}}{{2}}{}{\mathrm{D_z}}{,}{6}{}\sqrt{{2}}{}{\mathrm{D_t}}{+}\frac{{9}{}\sqrt{{2}}}{{2}}{}{\mathrm{D_x}}{+}\frac{{I}}{{2}}{}\sqrt{{2}}{}{\mathrm{D_y}}{-}{4}{}\sqrt{{2}}{}{\mathrm{D_z}}{,}{6}{}\sqrt{{2}}{}{\mathrm{D_t}}{+}\frac{{9}{}\sqrt{{2}}}{{2}}{}{\mathrm{D_x}}{-}\frac{{I}}{{2}}{}\sqrt{{2}}{}{\mathrm{D_y}}{-}{4}{}\sqrt{{2}}{}{\mathrm{D_z}}\right]$ (2.14)
 E > $\mathrm{TensorInnerProduct}\left(\mathrm{g2},\mathrm{N2},\mathrm{N2}\right)$

Example 3.

Convert the null tetrad $\mathrm{N2}$ constructed in Example 2 to an orthonormal tetrad $T$.

 E > $T≔\mathrm{OrthonormalTetrad}\left(\mathrm{N2}\right)$
 ${T}{:=}\left[{17}{}{\mathrm{D_t}}{+}{12}{}{\mathrm{D_x}}{-}{12}{}{\mathrm{D_z}}{,}{12}{}{\mathrm{D_t}}{+}{9}{}{\mathrm{D_x}}{-}{8}{}{\mathrm{D_z}}{,}{\mathrm{D_y}}{,}{-}{12}{}{\mathrm{D_t}}{-}{8}{}{\mathrm{D_x}}{+}{9}{}{\mathrm{D_z}}\right]$ (2.15)

Check the result.

 E > $\mathrm{TensorInnerProduct}\left(\mathrm{g2},T,T\right)$