DifferentialGeometry/Tensor/NullTetrad

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

Tensor[OrthonormalTetrad] - construct an orthonormal tetrad from a null tetrad

Calling Sequences

NullTetrad(OrthTetrad)

NullTetrad( $σ$ $,$ SpinBasis)

OrthonormalTetrad(NullTetrad)

Parameters

OrthTetrad - a list of 4 vectors defining an orthonormal tetrad with respect to a metric g with signature $[1, - 1, - 1, - 1]$

$σ$ - 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 $[1, - 1, - 1, - 1]$

Description

Examples

See Also

Description

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

$g (E_{t}, E_{t}) = 1, g (E_{x}, E_{x}) = g (E_{y}, E_{y}) = g (E_{z}, E_{z}) = - 1,$

and all other inner products vanish. A list of 4 vectors $[L, N, M, \overline{M}]$ defines a null tetrad if $L$ and $N$ are real, $\overline{M}$ is the complex conjugate of $M$ ,

$g (L, N) = 1$ , $g (M, \overline{M}) = - 1,$

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

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

$L = \frac{1}{\sqrt{2}} (E_{t} + E_{z}),$ $N = \frac{1}{\sqrt{2}} (E_{t} - E_{z}), M = \frac{1}{\sqrt{2}} (E_{x} + {iE}_{y}), \overline{M} = \frac{1}{\sqrt{2}} (E_{x} - {iE}_{y})$ .

•

Let sigma be a solder form (index type ["con", " cov", "cov"]), with components $σ_{AA'}^{i},$ for the metric $g$ . Let $ο^{A}$ and $ι^{B}$ be rank 1, unprimed spinors with $ε_{AB} ο^{A} ι^{B} = 1$ . Let $\overline{ο}$ and $\overline{ι}$ be their conjugates (see ConjugateSpinor). Then the following vectors

$L^{i} = σ_{AA'}^{i} ο^{A}$ ${\overline{ο}}^{A'}, N^{i} = σ_{AA'}^{i} ι^{A}$ ${\overline{ι}}^{A'}, M^{i} = σ_{AA'}^{i} ο^{A} {\overline{ι}}^{A'}, {\overline{M}}^{i} = σ_{AA'}^{i} ι^{A} {\overline{ο}}^{A'}$

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

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

$E_{t} =$ $\frac{1}{\sqrt{2}} (L + N),$ $E_{x} = \frac{1}{\sqrt{2}} (M + \overline{M}), E_{y} = \frac{1}{i \sqrt{2}} (M - \overline{M}), E_{z} = \frac{1}{\sqrt{2}} (L - N)$

•	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

>	$with (DifferentialGeometry) &colon;$ $with (Tensor) &colon;$

Example 1.

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

>	$DGsetup ([t, x, y, z], M)$

$frame name: M$

(2.1)

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

M >	$g ≔ evalDG (dt &t dt - dx &t dx - dy &t dy - dz &t dz)$

$g :=_DG ([[tensor, M, [[cov_bas, cov_bas], []]], [[[1, 1], 1], [[2, 2], −1], [[3, 3], −1], [[4, 4], −1]]]),_DG ([[tensor, M, [[cov_bas, cov_bas], []]], [[[1, 1], 1], [[2, 2], −1], [[3, 3], −1], [[4, 4], −1]]])$

(2.2)

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

M >	$F ≔ [D_t, D_x, D_y, D_z]$

$F := [_DG ([[vector, M, []], [[[1], 1]]]),_DG ([[vector, M, []], [[[1], 1]]]),_DG ([[vector, M, []], [[[2], 1]]]),_DG ([[vector, M, []], [[[2], 1]]]),_DG ([[vector, M, []], [[[3], 1]]]),_DG ([[vector, M, []], [[[3], 1]]]),_DG ([[vector, M, []], [[[4], 1]]]),_DG ([[vector, M, []], [[[4], 1]]])]$

(2.3)

M >	$GRQuery (F, g, OrthonormalTetrad)$

$true$

(2.4)

Use the orthonormal tetrad F to construct a null tetrad NT.

M >	$NT ≔ NullTetrad (F)$

$NT := [_DG ([[vector, M, []], [[[1], \frac{\sqrt{2}}{2}], [[4], \frac{\sqrt{2}}{2}]]]),_DG ([[vector, M, []], [[[1], \frac{\sqrt{2}}{2}], [[4], \frac{\sqrt{2}}{2}]]]),_DG ([[vector, M, []], [[[1], \frac{\sqrt{2}}{2}], [[4], - \frac{\sqrt{2}}{2}]]]),_DG ([[vector, M, []], [[[1], \frac{\sqrt{2}}{2}], [[4], - \frac{\sqrt{2}}{2}]]]),_DG ([[vector, M, []], [[[2], \frac{\sqrt{2}}{2}], [[3], \frac{I}{2} \sqrt{2}]]]),_DG ([[vector, M, []], [[[2], \frac{\sqrt{2}}{2}], [[3], \frac{I}{2} \sqrt{2}]]]),_DG ([[vector, M, []], [[[2], \frac{\sqrt{2}}{2}], [[3], - \frac{I}{2} \sqrt{2}]]]),_DG ([[vector, M, []], [[[2], \frac{\sqrt{2}}{2}], [[3], - \frac{I}{2} \sqrt{2}]]])]$

(2.5)

Verify this result using the command GRQuery.

M >	$GRQuery (NT, g, NullTetrad)$

$true$

(2.6)

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

M >	$TensorInnerProduct (g, NT, NT)$

Example 2.

We use spinors to create a null tetrad. First create a vector bundle $E \to M$ with base coordinates $(t, x, y, z)$ and fiber coordinates $(z1, z2, w1, w2)$ .

>	$DGsetup ([t, x, y, z], [z1, z2, w1, w2], E)$

$frame name: E$

(2.7)

Define a spacetime metric $g2$ on $M$ with signature $(1, - 1, - 1, - 1)$ .

E >	$g2 ≔ evalDG (dt &t dt - dx &t dx - dy &t dy - dz &t dz)$

$g2 :=_DG ([[tensor, E, [[cov_bas, cov_bas], []]], [[[1, 1], 1], [[2, 2], −1], [[3, 3], −1], [[4, 4], −1]]]),_DG ([[tensor, E, [[cov_bas, cov_bas], []]], [[[1, 1], 1], [[2, 2], −1], [[3, 3], −1], [[4, 4], −1]]])$

(2.8)

Define an orthonormal frame $F2$ on $M$ with respect to the metric $g2$ .

E >	$F2 ≔ [D_t, D_x, D_y, D_z]$

$F2 := [_DG ([[vector, E, []], [[[1], 1]]]),_DG ([[vector, E, []], [[[1], 1]]]),_DG ([[vector, E, []], [[[2], 1]]]),_DG ([[vector, E, []], [[[2], 1]]]),_DG ([[vector, E, []], [[[3], 1]]]),_DG ([[vector, E, []], [[[3], 1]]]),_DG ([[vector, E, []], [[[4], 1]]]),_DG ([[vector, E, []], [[[4], 1]]])]$

(2.9)

Compute the solder form $σ$ defined by the orthonormal frame $F2$ .

E >	$σ ≔ SolderForm (F2, indextype = [con, cov, cov])$

$σ :=_DG ([[tensor, E, [[con_bas, cov_vrt, cov_vrt], []]], [[[1, 5, 7], \frac{\sqrt{2}}{2}], [[1, 6, 8], \frac{\sqrt{2}}{2}], [[2, 5, 8], \frac{\sqrt{2}}{2}], [[2, 6, 7], \frac{\sqrt{2}}{2}], [[3, 5, 8], \frac{I}{2} \sqrt{2}], [[3, 6, 7], - \frac{I}{2} \sqrt{2}], [[4, 5, 7], \frac{\sqrt{2}}{2}], [[4, 6, 8], - \frac{\sqrt{2}}{2}]]]),_DG ([[tensor, E, [[con_bas, cov_vrt, cov_vrt], []]], [[[1, 5, 7], \frac{\sqrt{2}}{2}], [[1, 6, 8], \frac{\sqrt{2}}{2}], [[2, 5, 8], \frac{\sqrt{2}}{2}], [[2, 6, 7], \frac{\sqrt{2}}{2}], [[3, 5, 8], \frac{I}{2} \sqrt{2}], [[3, 6, 7], - \frac{I}{2} \sqrt{2}], [[4, 5, 7], \frac{\sqrt{2}}{2}], [[4, 6, 8], - \frac{\sqrt{2}}{2}]]])$

(2.10)

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

E >	$ο ≔ evalDG (D_z1 + 2 D_z2)$

$ο :=_DG ([[vector, E, []], [[[5], 1], [[6], 2]]]),_DG ([[vector, E, []], [[[5], 1], [[6], 2]]])$

(2.11)

E >	$ι ≔ evalDG (2 D_z1 + 5 D_z2)$

$ι :=_DG ([[vector, E, []], [[[5], 2], [[6], 5]]]),_DG ([[vector, E, []], [[[5], 2], [[6], 5]]])$

(2.12)

E >	$SpinorInnerProduct (ο, ι)$

$1$

(2.13)

E >	$N2 ≔ NullTetrad (σ, [ο, ι])$

$N2 := [_DG ([[vector, E, []], [[[1], \frac{5 \sqrt{2}}{2}], [[2], 2 \sqrt{2}], [[4], - \frac{3 \sqrt{2}}{2}]]]),_DG ([[vector, E, []], [[[1], \frac{5 \sqrt{2}}{2}], [[2], 2 \sqrt{2}], [[4], - \frac{3 \sqrt{2}}{2}]]]),_DG ([[vector, E, []], [[[1], \frac{29 \sqrt{2}}{2}], [[2], 10 \sqrt{2}], [[4], - \frac{21 \sqrt{2}}{2}]]]),_DG ([[vector, E, []], [[[1], \frac{29 \sqrt{2}}{2}], [[2], 10 \sqrt{2}], [[4], - \frac{21 \sqrt{2}}{2}]]]),_DG ([[vector, E, []], [[[1], 6 \sqrt{2}], [[2], \frac{9 \sqrt{2}}{2}], [[3], \frac{I}{2} \sqrt{2}], [[4], - 4 \sqrt{2}]]]),_DG ([[vector, E, []], [[[1], 6 \sqrt{2}], [[2], \frac{9 \sqrt{2}}{2}], [[3], \frac{I}{2} \sqrt{2}], [[4], - 4 \sqrt{2}]]]),_DG ([[vector, E, []], [[[1], 6 \sqrt{2}], [[2], \frac{9 \sqrt{2}}{2}], [[3], - \frac{I}{2} \sqrt{2}], [[4], - 4 \sqrt{2}]]]),_DG ([[vector, E, []], [[[1], 6 \sqrt{2}], [[2], \frac{9 \sqrt{2}}{2}], [[3], - \frac{I}{2} \sqrt{2}], [[4], - 4 \sqrt{2}]]])]$

(2.14)

E >	$TensorInnerProduct (g2, N2, N2)$

Example 3.

Convert the null tetrad $N2$ constructed in Example 2 to an orthonormal tetrad $T$ .

E >	$T ≔ OrthonormalTetrad (N2)$

$T := [_DG ([[vector, E, []], [[[1], 17], [[2], 12], [[4], −12]]]),_DG ([[vector, E, []], [[[1], 17], [[2], 12], [[4], −12]]]),_DG ([[vector, E, []], [[[1], 12], [[2], 9], [[4], −8]]]),_DG ([[vector, E, []], [[[1], 12], [[2], 9], [[4], −8]]]),_DG ([[vector, E, []], [[[3], 1]]]),_DG ([[vector, E, []], [[[3], 1]]]),_DG ([[vector, E, []], [[[1], −12], [[2], −8], [[4], 9]]]),_DG ([[vector, E, []], [[[1], −12], [[2], −8], [[4], 9]]])]$

(2.15)

Check the result.

E >	$TensorInnerProduct (g2, T, T)$

	See Also
	DifferentialGeometry, Tensor, ConjugateSpinor, DGGramSchmidt, GRQuery, SolderForm, SpinorInnerProduct, TensorInnerProduct

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