construct a null vector from a solder form and a rank 1 spinor - Maple Programming Help

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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

 > with(DifferentialGeometry): with(Tensor):

Example 1.

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

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

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

 M > g := evalDG(dt &t dt - dx &t dx - dy &t dy - dz &t dz);
 ${g}{:=}{\mathrm{dt}}{}{\mathrm{dt}}{-}{\mathrm{dx}}{}{\mathrm{dx}}{-}{\mathrm{dy}}{}{\mathrm{dy}}{-}{\mathrm{dz}}{}{\mathrm{dz}}$ (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 := [D_t, D_x, D_y, D_z];
 ${F}{:=}\left[{\mathrm{D_t}}{,}{\mathrm{D_x}}{,}{\mathrm{D_y}}{,}{\mathrm{D_z}}\right]$ (2.3)
 M > sigma := SolderForm(F);
 ${\mathrm{σ}}{:=}\frac{{1}}{{2}}{}\sqrt{{2}}{}{\mathrm{dt}}{}{\mathrm{D_z1}}{}{\mathrm{D_w1}}{+}\frac{{1}}{{2}}{}\sqrt{{2}}{}{\mathrm{dt}}{}{\mathrm{D_z2}}{}{\mathrm{D_w2}}{+}\frac{{1}}{{2}}{}\sqrt{{2}}{}{\mathrm{dx}}{}{\mathrm{D_z1}}{}{\mathrm{D_w2}}{+}\frac{{1}}{{2}}{}\sqrt{{2}}{}{\mathrm{dx}}{}{\mathrm{D_z2}}{}{\mathrm{D_w1}}{-}\frac{{1}}{{2}}{}{I}{}\sqrt{{2}}{}{\mathrm{dy}}{}{\mathrm{D_z1}}{}{\mathrm{D_w2}}{+}\frac{{1}}{{2}}{}{I}{}\sqrt{{2}}{}{\mathrm{dy}}{}{\mathrm{D_z2}}{}{\mathrm{D_w1}}{+}\frac{{1}}{{2}}{}\sqrt{{2}}{}{\mathrm{dz}}{}{\mathrm{D_z1}}{}{\mathrm{D_w1}}{-}\frac{{1}}{{2}}{}\sqrt{{2}}{}{\mathrm{dz}}{}{\mathrm{D_z2}}{}{\mathrm{D_w2}}$ (2.4)

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

 M > phi1 := D_z1;
 ${\mathrm{φ1}}{:=}{\mathrm{D_z1}}$ (2.5)
 M > phi2 := evalDG(a*D_z1 + b*D_z2);
 ${\mathrm{φ2}}{:=}{a}{}{\mathrm{D_z1}}{+}{b}{}{\mathrm{D_z2}}$ (2.6)
 M > phi3 := D_w2;
 ${\mathrm{φ3}}{:=}{\mathrm{D_w2}}$ (2.7)

Use the command NullVector to find the corrresponding null vectors .

 M > X := NullVector(sigma, phi1);
 ${X}{:=}\frac{{1}}{{2}}{}\sqrt{{2}}{}{\mathrm{D_t}}{+}\frac{{1}}{{2}}{}\sqrt{{2}}{}{\mathrm{D_z}}$ (2.8)
 M > Y := NullVector(sigma, phi2) assuming a::real, b::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:= NullVector(sigma, phi1, phi3);
 ${Z}{:=}\frac{{1}}{{2}}{}\sqrt{{2}}{}{\mathrm{D_x}}{+}\frac{{1}}{{2}}{}{I}{}\sqrt{{2}}{}{\mathrm{D_y}}$ (2.10)

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

 M > TensorInnerProduct(g, X, X);
 ${0}$ (2.11)
 M > TensorInnerProduct(g, Y, Y);
 ${0}$ (2.12)
 M > TensorInnerProduct(g, Z, Z);
 ${0}$ (2.13)