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(σ, φ)

     NullVector( σ, φ, ψ)

Parameters

   σ         - a spin-tensor defining a solder form on a 4-dimensional spacetime

   φ, ψ       - rank 1 spinors

 

Description

Examples

See Also

Description

• 

Let g be a metric on a 4-dimensional manifold with signature1, 1, 1, 1.   A null vector X satisfies gX, X = 0.

• 

Let σ be a solder form for the metric g, that is, σ is a rank 3 spin-tensor such that gij = σi AA'σjAA' . 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 Xi = σiAA'φA φA'

• With three arguments, the NullVector command returns the (complex) vector with components Xi = σiAA'φAψA' .

• 

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 M  with spacetime coordinates t, x, y,  z and fiber coordinates z1, z2, w1,w2.

DGsetup([t, x, y, z], [z1, z2, w1, w2], M);

frame name: M

(2.1)

 

Define a spacetime metric g on M with signature 1, 1, 1, 1.

M > 

g := evalDG(dt &t dt - dx &t dx - dy &t dy - dz &t dz);

g:=dtdtdxdxdydydzdz

(2.2)

 

Define an orthonormal tetrad F on M with respect to the metric g. Use the command SolderForm to create a solder form σ.

M > 

F := [D_t, D_x, D_y, D_z];

F:=D_t,D_x,D_y,D_z

(2.3)
M > 

sigma := SolderForm(F);

σ:=122dtD_z1D_w1+122dtD_z2D_w2+122dxD_z1D_w2+122dxD_z2D_w112I2dyD_z1D_w2+12I2dyD_z2D_w1+122dzD_z1D_w1122dzD_z2D_w2

(2.4)

 

Define rank 1 spinors φ1, φ2 and φ3.

M > 

phi1 := D_z1;

φ1:=D_z1

(2.5)
M > 

phi2 := evalDG(a*D_z1 + b*D_z2);

φ2:=aD_z1+bD_z2

(2.6)
M > 

phi3 := D_w2;

φ3:=D_w2

(2.7)

 

Use the command NullVector to find the corrresponding null vectors X, Y, Z.

M > 

X := NullVector(sigma, phi1);

X:=122D_t+122D_z

(2.8)
M > 

Y := NullVector(sigma, phi2) assuming a::real, b::real;

Y:=122b2+122a2D_t+2abD_x+122b2+122a2D_z

(2.9)
M > 

Z:= NullVector(sigma, phi1, phi3);

Z:=122D_x+12I2D_y

(2.10)

 

We can use the command TensorInnerProduct to check that the vectors X, Y, Z 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)

See Also

DifferentialGeometry, Tensor, NullTetrad,  PrincipalNullDirections, SolderForm,   TensorInnerProduct