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Tensor[NullTetradTransformation] - apply a Lorentz transformation to a null tetrad
Calling Sequences
NullTetradTransformation(NullTetrad, TransType, theta, axis)
Parameters
NullTetrad - a list of 4 vectors defining a null tetrad
TransType - a string, "null rotation", "spatial rotation", or "boost", describing the transformation type
theta - the transformation parameter
axis - (optional) a string, specifies the axis of rotation as "l"(or "L") or "m"(or"M") in the case whereTransType = "null rotation"
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Description
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Let g be a metric on a 4-dimensional manifold with signature [1, -1, -1, -1]. A list of 4 vectors [L, N, M, barM] defines a (complex) null tetrad if barM is the complex conjugate of M,
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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.
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A Lorentz transformation is a (linear) change of frame which transforms a null tetrad [L, N, M, barM] into another null tetrad [L', N', M', barM']. Every Lorentz transformation can be expressed as the composition of the following 4 basic Lorentz transformations.
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1. A null rotation about the L axis (theta complex):
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L' = L, N' = N + theta*M + conj(theta)*barM + theta*conj(theta)*L, M' = M + conj(theta)*L, barM' = barM + theta*L.
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2. A null rotation about the N axis (theta complex)
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L' = L + theta*M + conj(theta)*barM + theta*conj(theta)*N, N' = N, M' = M + conj(theta)*M, barM' = barM + theta*N.
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3. A spatial rotation in the [M, barM] plane (theta real):
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L' = L, N' = N, M' = (cos(theta) + I*sin(theta))*M, barM' = (cos(theta) - I*sin(theta))*barM.
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4. A boost (theta real and non-zero):
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L' = theta*L, N' = 1/theta*N, M' = M, barM' = barM.
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The command NullTetradTransformation(NullTetrad, TransType, theta, axis) returns the new null tetrad [L', N', M', barM'] obtained from NullTetrad = [L, N, M, barM] through the application of one of the above Lorentz transformations.
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This command is part of the DifferentialGeometry:-Tensor package, and so can be used in the form NullTetradTransformation(...) only after executing the commands with(DifferentialGeometry); with(Tensor); in that order. It can always be used in the long form DifferentialGeometry:-Tensor:-NullTetradTransformation.
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Examples
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For the first 4 examples we work with coordinates [u, v, x, y] and an off-diagonal form for the metric. This is the easiest setting to see the effects the 4 basic Lorentz transformations.
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![DGsetup([u, v, x, y], S)](/support/helpjp/helpview.aspx?si=5661/file05898/math146.png)
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| (2.2) |
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| (2.3) |
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![T := [L, N, M, barM]](/support/helpjp/helpview.aspx?si=5661/file05898/math167.png)
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![T := [D_u, D_v, D_x+D_y*I, D_x-I*D_y]](/support/helpjp/helpview.aspx?si=5661/file05898/math170.png)
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Example 1.
Apply a null rotation to the null tetrad T about the "l" axis. Check that the result is a null tetrad.
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![T1a := `assuming`([NullTetradTransformation(T, "null rotation", a, "l")], [a::real])](/support/helpjp/helpview.aspx?si=5661/file05898/math191.png)
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![T1a := [D_u, a^2*D_u+D_v+2*a*D_x, a*D_u+D_x+D_y*I, a*D_u+D_x-I*D_y]](/support/helpjp/helpview.aspx?si=5661/file05898/math194.png)
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| (2.7) |
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![T1b := `assuming`([NullTetradTransformation(T, "null rotation", I*b, "l")], [b::real])](/support/helpjp/helpview.aspx?si=5661/file05898/math205.png)
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![T1b := [D_u, b^2*D_u+D_v-2*b*D_y, -I*b*D_u+D_x+D_y*I, b*D_u*I+D_x-I*D_y]](/support/helpjp/helpview.aspx?si=5661/file05898/math208.png)
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Example 2.
Apply a null rotation about the "n" axis to the null tetrad T. Check that the result is a null tetrad.
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![T2a := `assuming`([NullTetradTransformation(T, "null rotation", a, "n")], [a::real])](/support/helpjp/helpview.aspx?si=5661/file05898/math229.png)
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![T2a := [D_u+a^2*D_v+2*a*D_x, D_v, a*D_v+D_x+D_y*I, a*D_v+D_x-I*D_y]](/support/helpjp/helpview.aspx?si=5661/file05898/math232.png)
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![T2b := `assuming`([NullTetradTransformation(T, "null rotation", I*b, "n")], [b::real])](/support/helpjp/helpview.aspx?si=5661/file05898/math243.png)
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![T2b := [D_u+b^2*D_v-2*b*D_y, D_v, -I*b*D_v+D_x+D_y*I, b*D_v*I+D_x-I*D_y]](/support/helpjp/helpview.aspx?si=5661/file05898/math246.png)
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Example 3.
Apply a spatial rotation to the null tetrad T. Check that the result is a null tetrad.
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![T3 := `assuming`([NullTetradTransformation(T, "spatial rotation", theta, "n")], [theta::real])](/support/helpjp/helpview.aspx?si=5661/file05898/math265.png)
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![T3 := [D_u, D_v, (cos(theta)+sin(theta)*I)*D_x+(cos(theta)*I-sin(theta))*D_y, (cos(theta)-I*sin(theta))*D_x+(-I*cos(theta)-sin(theta))*D_y]](/support/helpjp/helpview.aspx?si=5661/file05898/math268.png)
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Example 4.
Apply a boost to the null tetrad T. Check that the result is a null tetrad.
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![T4 := `assuming`([NullTetradTransformation(T, "spatial rotation", theta, "n")], [theta::real])](/support/helpjp/helpview.aspx?si=5661/file05898/math287.png)
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![T4 := [D_u, D_v, (cos(theta)+sin(theta)*I)*D_x+(cos(theta)*I-sin(theta))*D_y, (cos(theta)-I*sin(theta))*D_x+(-I*cos(theta)-sin(theta))*D_y]](/support/helpjp/helpview.aspx?si=5661/file05898/math290.png)
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Example 5.
In this example we show how the use of a null tetrad transformation can be use to simplify the NP Weyl scalars. First we define our manifold.
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![DGsetup([t, x, y, z], S)](/support/helpjp/helpview.aspx?si=5661/file05898/math311.png)
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| (2.18) |
Define a null tetrad T1. (By decreeing this to be a null tetrad we implicitly define the spacetime metric.)
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![T1 := evalDG([(1/2)*2^(1/2)*D_t+(1/2)*2^(1/2)*D_z, (1/2)*2^(1/2)*D_t-(1/2)*2^(1/2)*D_z, (1/2)*2^(1/2)*z^2*D_x+I*2^(1/2)*x^2*D_y*(1/2), (1/2)*2^(1/2)*z^2*D_x-I*2^(1/2)*x^2*D_y*(1/2)])](/support/helpjp/helpview.aspx?si=5661/file05898/math324.png)
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![T1 := [2^(1/2)*D_t/2+2^(1/2)*D_z/2, 2^(1/2)*D_t/2-2^(1/2)*D_z/2, 2^(1/2)*z^2*D_x/2+((1/2)*I)*2^(1/2)*x^2*D_y, 2^(1/2)*z^2*D_x/2-((1/2)*I)*2^(1/2)*x^2*D_y]](/support/helpjp/helpview.aspx?si=5661/file05898/math327.png)
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Apply a null rotation to T1.
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![T2 := `assuming`([NullTetradTransformation(T1, "null rotation", a, "l")], [a::real])](/support/helpjp/helpview.aspx?si=5661/file05898/math337.png)
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![T2 := [2^(1/2)*D_t/2+2^(1/2)*D_z/2, (a^2*2^(1/2)/2+2^(1/2)/2)*D_t+a*2^(1/2)*z^2*D_x+(a^2*2^(1/2)/2-2^(1/2)/2)*D_z, a*2^(1/2)*D_t/2+2^(1/2)*z^2*D_x/2+((1/2)*I)*2^(1/2)*x^2*D_y+a*2^(1/2)*D_z/2, a*2^(1/2)*D_t/2+2^(1/2)*z^2*D_x/2-((1/2)*I)*2^(1/2)*x^2*D_y+a*2^(1/2)*D_z/2]](/support/helpjp/helpview.aspx?si=5661/file05898/math340.png)
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Calculate the NP Weyl scalars for the null tetrad T2.
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![NPCurvatureScalars(T2, output = ["WeylScalars"])](/support/helpjp/helpview.aspx?si=5661/file05898/math350.png)
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![TABLE(["Psi4" = -(8*x*a^3*z^3-8*x*a*z^3+3*a^4*x^2+6*a^2*x^2-12*a^2*z^6+3*x^2)/(2*z^2*x^2), "Psi1" = -(2*z^3+3*a*x)/(2*z^2*x), "Psi3" = -(6*z^3*a^2*x-2*z^3*x-6*z^6*a+3*a^3*x^2+3*a*x^2)/(2*z^2*x^2), "Psi2" = -(-2*z^6+x^2+3*a^2*x^2+4*x*a*z^3)/(2*z^2*x^2), "Psi0" = -3/(2*z^2)])](/support/helpjp/helpview.aspx?si=5661/file05898/math353.png)
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We can make Psi1 = 0 by choosing a = -2/3z^3/x.
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![T3 := [2^(1/2)*D_t/2+2^(1/2)*D_z/2, (2*z^6*2^(1/2)/(9*x^2)+2^(1/2)/2)*D_t-2*z^5*2^(1/2)*D_x/(3*x)+(2*z^6*2^(1/2)/(9*x^2)-2^(1/2)/2)*D_z, -z^3*2^(1/2)*D_t/(3*x)+2^(1/2)*z^2*D_x/2+((1/2)*I)*2^(1/2)*x^2*D_y-z^3*2^(1/2)*D_z/(3*x), -z^3*2^(1/2)*D_t/(3*x)+2^(1/2)*z^2*D_x/2-((1/2)*I)*2^(1/2)*x^2*D_y-z^3*2^(1/2)*D_z/(3*x)]](/support/helpjp/helpview.aspx?si=5661/file05898/math368.png)
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Recalculate the NP Weyl scalars and note that Psi1 = 0.
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![NPCurvatureScalars(T3, output = ["WeylScalars"])](/support/helpjp/helpview.aspx?si=5661/file05898/math378.png)
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![TABLE(["Psi4" = -(-64*z^12+72*z^6*x^2+27*x^4)/(18*x^4*z^2), "Psi1" = 0, "Psi3" = 2*z*(-13*z^6+9*x^2)/(9*x^3), "Psi2" = -(-10*z^6+3*x^2)/(6*z^2*x^2), "Psi0" = -3/(2*z^2)])](/support/helpjp/helpview.aspx?si=5661/file05898/math381.png)
| (2.23) |
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