find a null tetrad which transforms the Newman-Penrose Weyl scalars to a standard form

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Tensor[AdaptedNullTetrad] - find a null tetrad which transforms the Newman-Penrose Weyl scalars to a standard form

Calling Sequences

AdaptedNullTetrad(NT, PT, options )

AdaptedNullTetrad(NT, PT, W, options )

AdaptedNullTetrad(NT, PT, NP , options )

Parameters

NT - a null tetrad for the spacetime metric

PT - the Petrov type of

W - (optional) the Weyl tensor of

NP - (optional) the Newman-Penrose Weyl scalars

options - one or more of the keyword arguments method and output

Description

•

The Newman-Penrose Weyl scalars are a set of 5 complex scalars, labeled , and defined by certain components of the Weyl tensor with respect to a given null tetrad in a four dimensional spacetime of signature [1, -1, -1, -1]. Under local Lorentz transformations, the Newman-Penrose Weyl scalars transform among themselves in a natural way. Depending upon the Petrov type of the spacetime it is possible to transform the Newman-Penrose Weyl scalars to one of following normal forms. Below, and are complex scalars.See NPCurvatureScalars, NullTetradTransformation.

Type I.

Type II.

Type III.

Type D.

Type N.

Type O.

See Penrose and Rindle Vol. 2, Section 8.3.

•	Null tetrads for which the Newman-Penrose Weyl scalars are in the above normal form are called adapted null tetrads. Calculations are often simplified by using an adapted null tetrad.

•	The command AdaptedNullTetrad returns a null tetrad which will put the Newman-Penrose Weyl scalars in the above normal form.

•	The procedure AdaptedNullTetrad first calculates the Weyl spinor and calls the procedure AdaptedSpinorDyad to find a spinor dyad which transforms the Weyl spinor to normal form. The adapted null tetrad is then constructed from the spinor dyad.

•

The command AdaptedNullTetrad is part of the DifferentialGeometry:-Tensor package. It can be used in the form AdaptedNullTetrad(...) only after executing the commands with(DifferentialGeometry) and with(Tensor), but can always be used by executing DifferentialGeometry:-Tensor:-AdaptedNullTetrad(...).

Examples

Set the global environment variable _EnvExplicit to true to insure that the adapted null tetrads are free of expressions.

Example 1. Type I

We calculate an adapted null tetrad for a type spacetime. First define the coordinates to be used and then define the metric.

(2.1)

M >

(2.2)

Here is an initial null tetrad.

[_DG([["vector", M, []], [[[1], 1], [[4], 1]]]), _DG([["vector", M, []], [[[1], 1/2], [[4], -1/2]]]), _DG([["vector", M, []], [[[2], (1/2)*2^(1/2)/t], [[3], ((1/2)*I)*2^(1/2)/x]]]), _DG([["vector", M, []], [[[2], (1/2)*2^(1/2)/t], [[3], -((1/2)*I)*2^(1/2)/x]]])]

(2.3)

We check that this is indeed a null tetrad for the given metric using GRQuery.

M >

(2.4)

Compute the Newman-Penrose coefficients and check that the Petrov type is I. The coefficients are not in normal form for type I (for example, ), so is not an adapted null tetrad.

M >

table( [( "Psi4" ) = 0, ( "Psi2" ) = 0, ( "Psi3" ) = -(1/8)*2^(1/2)/(t^2*x), ( "Psi0" ) = 0, ( "Psi1" ) = -(1/4)*2^(1/2)/(t^2*x) ] )

(2.5)

M >

(2.6)

Calculate an adapted null tetrad and simplify.

[_DG([["vector", M, []], [[[1], (1/2)*2^(1/2)], [[2], -(1/2)*2^(1/2)/t]]]), _DG([["vector", M, []], [[[1], (1/2)*2^(1/2)], [[2], (1/2)*2^(1/2)/t]]]), _DG([["vector", M, []], [[[3], (1/2-(1/2)*I)/x], [[4], -1/2-(1/2)*I]]]), _DG([["vector", M, []], [[[3], (1/2+(1/2)*I)/x], [[4], -1/2+(1/2)*I]]])]

(2.7)

Calculate the Newman-Penrose coefficients for the new null tetrad. We obtain the correct normal form (with since and .

M >

table( [( "Psi4" ) = ((1/2)*I)/(t^2*x), ( "Psi3" ) = 0, ( "Psi0" ) = ((1/2)*I)/(t^2*x), ( "Psi2" ) = 0, ( "Psi1" ) = 0 ] )

(2.8)

Example 2. Type II

We calculate an adapted null tetrad for a type spacetime. First define the coordinates to be used and then define the metric.

(2.9)

M >

g2 := evalDG(-2*r^2*(`&t`(dx, dx)+`&t`(dy, dy))/(2*x)^3+`&s`(du, dr)-((3*2)*x+2*m/r)*`&t`(du, du))

_DG([["tensor", M, [["cov_bas", "cov_bas"], []]], [[[1, 2], 1], [[2, 1], 1], [[2, 2], -2*(3*x*r+m)/r], [[3, 3], -(1/4)*r^2/x^3], [[4, 4], -(1/4)*r^2/x^3]]])

(2.10)

Here is an initial null tetrad.

M >

NT2 := evalDG([D_r, (3*x*r+m)*D_r/r+D_u, I*sqrt(2)*x^(3/2)*D_x/r+sqrt(2)*x^(3/2)*D_y/r, -I*sqrt(2)*x^(3/2)*D_x/r+sqrt(2)*x^(3/2)*D_y/r])

[_DG([["vector", M, []], [[[1], 1]]]), _DG([["vector", M, []], [[[1], (3*x*r+m)/r], [[2], 1]]]), _DG([["vector", M, []], [[[3], I*2^(1/2)*x^(3/2)/r], [[4], 2^(1/2)*x^(3/2)/r]]]), _DG([["vector", M, []], [[[3], -I*2^(1/2)*x^(3/2)/r], [[4], 2^(1/2)*x^(3/2)/r]]])]

(2.11)

We check that this is indeed a null tetrad for the given metric.

M >

(2.12)

Compute the Newman-Penrose coefficients and check that the Petrov type is II. The coefficients are not in normal form for type II (for example, ), so is not an adapted null tetrad.

M >

table( [( "Psi4" ) = 18*x^2/r^2, ( "Psi2" ) = -m/r^3, ( "Psi3" ) = -(3*I)*2^(1/2)*x^(3/2)/r^2, ( "Psi0" ) = 0, ( "Psi1" ) = 0 ] )

(2.13)

M >

(2.14)

Calculate an adapted null tetrad. We use the third calling sequence so that the Weyl tensor, or equivalently, the Newman-Penrose Weyl scalars need not be computed. Moreover, all computations are then algebraic and we can use Maple's assuming feature to simplify all intermediate calculations.

(2.15)

Calculate the Newman-Penrose coefficients for the new null tetrad. We obtain the correct normal form (with ) since and .

M >

(2.16)

Example 3. Type III

We calculate an adapted null tetrad for a type spacetime. First define the coordinates to be used and then define the metric.

(2.17)

M >

g3 := evalDG(-r^2*(`&t`(dx, dx)+`&t`(dy, dy))/x^3+`&s`(du, dr)-3*x*`&t`(du, du)*(1/2))

_DG([["tensor", M, [["cov_bas", "cov_bas"], []]], [[[1, 2], 1], [[2, 1], 1], [[2, 2], -(3/2)*x], [[3, 3], -r^2/x^3], [[4, 4], -r^2/x^3]]])

(2.18)

Here is an initial null tetrad.

NT3 := evalDG([(3*x*(1/8)+1/2)*D_r+(1/2)*D_u+sqrt(2)*x^(3/2)*D_y/(2*r), (3*x*(1/8)+1/2)*D_r+(1/2)*D_u-sqrt(2)*x^(3/2)*D_y/(2*r), (-3*x*(1/8)+1/2)*D_r-(1/2)*D_u+I*sqrt(2)*x^(3/2)*D_x/(2*r), (-3*x*(1/8)+1/2)*D_r-(1/2)*D_u-I*sqrt(2)*x^(3/2)*D_x/(2*r)])

[_DG([["vector", M, []], [[[1], (3/8)*x+1/2], [[2], 1/2], [[4], (1/2)*2^(1/2)*x^(3/2)/r]]]), _DG([["vector", M, []], [[[1], (3/8)*x+1/2], [[2], 1/2], [[4], -(1/2)*2^(1/2)*x^(3/2)/r]]]), _DG([["vector", M, []], [[[1], -(3/8)*x+1/2], [[2], -1/2], [[3], ((1/2)*I)*2^(1/2)*x^(3/2)/r]]]), _DG([["vector", M, []], [[[1], -(3/8)*x+1/2], [[2], -1/2], [[3], -((1/2)*I)*2^(1/2)*x^(3/2)/r]]])]

(2.19)

We check that this is indeed a null tetrad for the given metric.

M >

(2.20)

Compute the Newman-Penrose coefficients and check that the Petrov type is III. The coefficients are not in normal form for type III (for example, ), so is not an adapted null tetrad.

M >

table( [( "Psi4" ) = -(3/32)*x*((4*I)*2^(1/2)*x^(1/2)-3*x)/r^2, ( "Psi2" ) = (9/32)*x^2/r^2, ( "Psi3" ) = (3/32)*(-3*x^(1/2)+(2*I)*2^(1/2))*x^(3/2)/r^2, ( "Psi0" ) = (3/32)*x*((4*I)*2^(1/2)*x^(1/2)+3*x)/r^2, ( "Psi1" ) = -(3/32)*(3*x^(1/2)+(2*I)*2^(1/2))*x^(3/2)/r^2 ] )

(2.21)

(2.22)

Calculate an adapted null tetrad.

(2.23)

Calculate the Newman-Penrose coefficients for the new null tetrad. We obtain the correct normal form since and .

M >

(2.24)

Example 4. Type D

We calculate an adapted null tetrad for a type spacetime. First define the coordinates to be used and then define the metric.

(2.25)

M >

(2.26)

Here is an initial null tetrad.

NT4 := evalDG([-(1/2)*sqrt(2)*(sqrt(2)-1)*D_t+exp(-x)*D_z, (1/2)*sqrt(2)*(1+sqrt(2))*D_t-exp(-x)*D_z, (1/2)*sqrt(2)*D_x+I*sqrt(2)*D_y*(1/2), (1/2)*sqrt(2)*D_x-I*sqrt(2)*D_y*(1/2)])

[_DG([["vector", M, []], [[[1], -(1/2)*2^(1/2)*(2^(1/2)-1)], [[4], exp(-x)]]]), _DG([["vector", M, []], [[[1], (1/2)*2^(1/2)*(1+2^(1/2))], [[4], -exp(-x)]]]), _DG([["vector", M, []], [[[2], (1/2)*2^(1/2)], [[3], ((1/2)*I)*2^(1/2)]]]), _DG([["vector", M, []], [[[2], (1/2)*2^(1/2)], [[3], -((1/2)*I)*2^(1/2)]]])]

(2.27)

We check that this is indeed a null tetrad for the given metric.

M >

(2.28)

Compute the Newman-Penrose coefficients and check that the Petrov type is D. The coefficients are not in normal form for type D (for example, ), so is not an adapted null tetrad.

M >

(2.29)

M >

(2.30)

Calculate an adapted null tetrad.

[_DG([["vector", M, []], [[[1], 2^(1/2)], [[3], -2^(1/2)]]]), _DG([["vector", M, []], [[[1], (1/4)*2^(1/2)], [[3], (1/4)*2^(1/2)]]]), _DG([["vector", M, []], [[[1], -I], [[2], (1/2)*2^(1/2)], [[4], I*exp(-x)]]]), _DG([["vector", M, []], [[[1], I], [[2], (1/2)*2^(1/2)], [[4], -I*exp(-x)]]])]

(2.31)

Calculate the Newman-Penrose coefficients for the new null tetrad. We obtain the correct normal form since = 0.

M >

(2.32)

Example 5. Type N

We calculate an adapted null tetrad for a type spacetime. First define the coordinates to be used and then define the metric.

(2.33)

M >

(2.34)

Here is the initial null tetrad.

NT5 := evalDG([-(1/4)*(exp(3*z)-2)*exp(z)*D_u+(1/2)*exp(z)*D_x+(1/2)*sqrt(2)*D_z, -(1/4)*(exp(3*z)-2)*exp(z)*D_u+(1/2)*exp(z)*D_x-(1/2)*sqrt(2)*D_z, (1/4)*(exp(3*z)+2)*exp(z)*D_u-(1/2)*exp(z)*D_x+I*sqrt(2)*exp(z)*D_y*(1/2), (1/4)*(exp(3*z)+2)*exp(z)*D_u-(1/2)*exp(z)*D_x-I*sqrt(2)*exp(z)*D_y*(1/2)])

(2.35)

We check that this is indeed a null tetrad for the given metric.

M >

(2.36)

Compute the Newman-Penrose coefficients and check that the Petrov type is N. The coefficients are not in normal form for type N (for example, ), so is not an adapted null tetrad.