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(sigma, 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]

sigma - 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

•	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, barM] defines a (complex) null tetrad if barM is the complex conjugate of M,

g(L, N) = 1 and g(M, barM) = -1,

and all other inner products vanish. In particular, the vectors [L, N, M, barM] 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 = (E_t + E_z)/sqrt(2), N = (E_t - E_z)/sqrt(2), M = (E_x + I*E_y)/sqrt(2), barM = (E_x - I*E_y)/sqrt(2).

•	Let sigma be a solder form (index type ["con", " cov", "cov"]) for the metric g. Let omicron and iota be rank 1, unprimed spinors with SpinorInnerProduct(omicron, iota) = 1. Let baromicron and bariota be their conjugates (see ConjugateSpinor). Then the vectors with components

L^i = sigma^i_{AA'}*omicron^A*baromicron^A', N^i = sigma^i_{AA'}*iota^A*bariota^A',

M^i = sigma^i_{AA'}*omicron^A*bariota^A', barM^i = sigma^i_{AA'}*iota^A*baromicron^A'

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

•	Given a null tetrad NullTetrad = [L, N, M, barM], the command OrthonormalTetrad(NullTetrad) constructs the orthonormal tetrad defined by

E_t = (L + N)/sqrt(2), E_x = (M + barM)/sqrt(2), E_y = - I*(M - barM)/sqrt(2), E_z = (L - N)/sqrt(2).

•	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

Example 1.

First create manifold M with coordinates [t, x, y, z].

(2.1)

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

M >

(2.2)

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

M >

(2.3)

M >

(2.4)

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

M >

(2.5)

Verify this result using the command GRQuery.

M >

(2.6)

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

M >

Matrix([[0, 1, 0, 0], [1, 0, 0, 0], [0, 0, 0, -1], [0, 0, -1, 0]])

(2.7)

Example 2.

We use spinors to create a null tetrad. First create a vector bundle over M with base coordinates [t, x, y, z] and fiber coordinates [z1, z2, w1, w2].

M >

(2.8)

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

M >

(2.9)

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

M >

(2.10)

Compute the solder form sigma defined by the orthonormal frame F2.

M >

sigma := 2^(1/2)*D_t*dz1*dw1/2+2^(1/2)*D_t*dz2*dw2/2+2^(1/2)*D_x*dz1*dw2/2+2^(1/2)*D_x*dz2*dw1/2+((1/2)*I)*2^(1/2)*D_y*dz1*dw2-((1/2)*I)*2^(1/2)*D_y*dz2*dw1+2^(1/2)*D_z*dz1*dw1/2-2^(1/2)*D_z*dz2*dw2/2

(2.11)

Define a pair of rank 1 spinors omicron and iota. Check that their inner product is 1.

M >

(2.12)

M >

(2.13)

M >

(2.14)

M >

N2 := [29*2^(1/2)*D_t/2+10*2^(1/2)*D_x-21*2^(1/2)*D_z/2, 5*2^(1/2)*D_t/2+2*2^(1/2)*D_x-3*2^(1/2)*D_z/2, 6*2^(1/2)*D_t+9*2^(1/2)*D_x/2+((1/2)*I)*2^(1/2)*D_y-4*2^(1/2)*D_z, 6*2^(1/2)*D_t+9*2^(1/2)*D_x/2-((1/2)*I)*2^(1/2)*D_y-4*2^(1/2)*D_z]

(2.15)

M >

Matrix([[0, 1, 0, 0], [1, 0, 0, 0], [0, 0, 0, -1], [0, 0, -1, 0]])

(2.16)

Example 3.

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

M >

(2.17)

Check the result.

M >

Matrix([[1, 0, 0, 0], [0, -1, 0, 0], [0, 0, -1, 0], [0, 0, 0, -1]])

(2.18)

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