![]() ![]() ![]() GRQuery T2b, g, NullTetradĪpply a spatial rotation to the null tetrad T. T2b ≔ NullTetradTransformation T, null rotation, I b, n assuming b :: real T2a ≔ NullTetradTransformation T, null rotation, a, n assuming a :: real GRQuery T1b, g, NullTetradĪpply a null rotation about the "n" axis to the null tetrad T. T1b ≔ NullTetradTransformation T, null rotation, I b, l assuming b :: real T1a ≔ NullTetradTransformation T, null rotation, a, l assuming a :: real T := D_u, D_v, D_x + I D_y, D_x − I D_yĪpply a null rotation to the null tetrad T about the "l" axis. L, N, M, barM ≔ D_u, D_v, evalDG D_x + I D_y, evalDG D_x − I D_y G := du dv + dv du − 1 2 dx dx − 1 2 dy dy G ≔ evalDG 2 du &s dv − 1 2 dx &t dx + dy &t dy Here we define the metric and a null tetrad.ĭGsetup u, v, x, y, S This is the easiest setting to see the effects the 4 basic Lorentz transformations. With DifferentialGeometry : with Tensor :įor the first 4 examples we work with coordinates u, v, x, y and an off-diagonal form for the metric. It can always be used in the long form DifferentialGeometry:-Tensor:-NullTetradTransformation. 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. The command NullTetradTransformation(NullTetrad, TransType, &theta, axis) returns the new null tetrad &lsqb L ', N ', M ', M ‾ ' &rsqb obtained from NullTetrad = &lsqb L, N, M, M ‾ &rsqb through the application of one of the above Lorentz transformations. L ' = &theta L, N ' = 1 &theta N, M ' = M, M ‾ ' = M. L ' = L, N ' = N, M ' = e i&theta M, M ‾ ' = e − i&theta M ‾. A spatial rotation in the M − M ‾ plane ( &theta real): L ' = L + &theta M + &theta ‾ M ‾ + θ θ ‾ N, N ' = N, M ' = M + &theta ‾ L, M ‾ ' = M ‾ + &theta N. A null rotation about the N axis ( θ complex) L ' = L, N ' = N + &theta M + &theta ‾ M ‾ + &theta &theta ‾ L, M ' = M + &theta ‾ L, M ‾ ' = M ‾ + &theta L. A null rotation about the L axis ( &theta complex): Every Lorentz transformation can be expressed as the composition of the following 4 basic Lorentz transformations.ġ. In particular, the vectors L, N, M, M ‾ are all null vectors.Ī Lorentz transformation is a (linear) change of frame which transforms a null tetrad L, N, M, M ‾ into another null tetrad L ', N ', M ', M ‾ '. G L, N = 1, g M, M ‾ = − 1 ,Īnd all other inner products vanish. ![]() A list of 4 vectors L, N, M, M ‾ defines a null tetrad if L and N are real, M ‾ is the complex conjugate of M , Let g be a metric on a 4-dimensional manifold with signature 1, − 1, − 1, − 1. ![]() TransType - a string, "null rotation", "spatial rotation", or "boost", describing the transformation typeĪxis -( optional) a string, specifies the axis of rotation as "l" (or "L" ) or "m" (or "M" ) in the case where TransType = "null rotation" NullTetrad - a list of 4 vectors defining a null tetrad NullTetradTransformation( NullTetrad, TransType, &theta, axis ) Tensor - apply a Lorentz transformation to a null tetrad ![]()
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