 NullVector - Maple Help

Tensor[NullVector] - construct a null vector from a solder form and a rank 1 spinor

Calling Sequences

NullVector(${\mathbf{σ}}$, ${\mathbf{φ}}$)

NullVector( ${\mathbf{σ}}$, ${\mathbf{φ}}$, ${\mathbf{ψ}}$)

Parameters

$\mathrm{σ}$         - a spin-tensor defining a solder form on a 4-dimensional spacetime

- rank 1 spinors Description

 • Let be a metric on a 4-dimensional manifold with signature A null vector satisfies
 • Let $\mathrm{σ}$ be a solder form for the metricthat is, $\mathrm{σ}$ is a rank 3 spin-tensor such that  The NullVector command accepts, as its first argument, a solder form with either covariant or contravariant tensor and spinor indices.
 • With two arguments, the NullVector command returns the real vector with components

• With three arguments, the NullVector command returns the (complex) vector with components

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

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

Example 1.

First create the spinor bundle  with spacetime coordinates  and fiber coordinates .

 > $\mathrm{DGsetup}\left(\left[t,x,y,z\right],\left[\mathrm{z1},\mathrm{z2},\mathrm{w1},\mathrm{w2}\right],M\right)$
 ${\mathrm{frame name: M}}$ (2.1)

Define a spacetime metric $g$ on $M$ with signature .

 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 with respect to the metric Use the command SolderForm to create a solder form $\mathrm{σ}$.

 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 > $\mathrm{\sigma }≔\mathrm{SolderForm}\left(F\right)$
 ${\mathrm{σ}}{:=}\frac{{1}}{{2}}{}\sqrt{{2}}{}{\mathrm{dt}}{}{\mathrm{D_z1}}{}{\mathrm{D_w1}}{+}\frac{{1}}{{2}}{}\sqrt{{2}}{}{\mathrm{dt}}{}{\mathrm{D_z2}}{}{\mathrm{D_w2}}{+}\frac{{1}}{{2}}{}\sqrt{{2}}{}{\mathrm{dx}}{}{\mathrm{D_z1}}{}{\mathrm{D_w2}}{+}\frac{{1}}{{2}}{}\sqrt{{2}}{}{\mathrm{dx}}{}{\mathrm{D_z2}}{}{\mathrm{D_w1}}{-}\frac{{1}}{{2}}{}{I}{}\sqrt{{2}}{}{\mathrm{dy}}{}{\mathrm{D_z1}}{}{\mathrm{D_w2}}{+}\frac{{1}}{{2}}{}{I}{}\sqrt{{2}}{}{\mathrm{dy}}{}{\mathrm{D_z2}}{}{\mathrm{D_w1}}{+}\frac{{1}}{{2}}{}\sqrt{{2}}{}{\mathrm{dz}}{}{\mathrm{D_z1}}{}{\mathrm{D_w1}}{-}\frac{{1}}{{2}}{}\sqrt{{2}}{}{\mathrm{dz}}{}{\mathrm{D_z2}}{}{\mathrm{D_w2}}$ (2.4)

Define rank 1 spinors and ${\mathrm{φ}}_{3}.$

 M > $\mathrm{φ1}≔\mathrm{D_z1}$
 ${\mathrm{φ1}}{:=}{\mathrm{D_z1}}$ (2.5)
 M > $\mathrm{φ2}≔\mathrm{evalDG}\left(a\mathrm{D_z1}+b\mathrm{D_z2}\right)$
 ${\mathrm{φ2}}{:=}{a}{}{\mathrm{D_z1}}{+}{b}{}{\mathrm{D_z2}}$ (2.6)
 M > $\mathrm{φ3}≔\mathrm{D_w2}$
 ${\mathrm{φ3}}{:=}{\mathrm{D_w2}}$ (2.7)

Use the command NullVector to find the corrresponding null vectors .

 M > $X≔\mathrm{NullVector}\left(\mathrm{\sigma },\mathrm{φ1}\right)$
 ${X}{:=}\frac{{1}}{{2}}{}\sqrt{{2}}{}{\mathrm{D_t}}{+}\frac{{1}}{{2}}{}\sqrt{{2}}{}{\mathrm{D_z}}$ (2.8)
 M > $Y≔\mathrm{NullVector}\left(\mathrm{\sigma },\mathrm{φ2}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathbf{assuming}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}a::\mathrm{real},b::\mathrm{real}$
 ${Y}{:=}\left(\frac{{1}}{{2}}{}\sqrt{{2}}{}{{b}}^{{2}}{+}\frac{{1}}{{2}}{}\sqrt{{2}}{}{{a}}^{{2}}\right){}{\mathrm{D_t}}{+}\sqrt{{2}}{}{a}{}{b}{}{\mathrm{D_x}}{+}\left({-}\frac{{1}}{{2}}{}\sqrt{{2}}{}{{b}}^{{2}}{+}\frac{{1}}{{2}}{}\sqrt{{2}}{}{{a}}^{{2}}\right){}{\mathrm{D_z}}$ (2.9)
 M > $Z≔\mathrm{NullVector}\left(\mathrm{\sigma },\mathrm{φ1},\mathrm{φ3}\right)$
 ${Z}{:=}\frac{{1}}{{2}}{}\sqrt{{2}}{}{\mathrm{D_x}}{+}\frac{{1}}{{2}}{}{I}{}\sqrt{{2}}{}{\mathrm{D_y}}$ (2.10)

We can use the command TensorInnerProduct to check that the vectors  are indeed null vectors.

 M > $\mathrm{TensorInnerProduct}\left(g,X,X\right)$
 ${0}$ (2.11)
 M > $\mathrm{TensorInnerProduct}\left(g,Y,Y\right)$
 ${0}$ (2.12)
 M > $\mathrm{TensorInnerProduct}\left(g,Z,Z\right)$
 ${0}$ (2.13) See Also