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For the first 4 examples we work with coordinates and an off-diagonal form for the metric. This is the easiest setting to see the effects the 4 basic Lorentz transformations. Here we define the metric and a null tetrad.
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| (2.2) |
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| (2.3) |
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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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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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| (2.12) |
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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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| (2.14) |
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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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| (2.16) |
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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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Define a null tetrad T1. (By decreeing this to be a null tetrad we implicitly define the spacetime metric.)
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| (2.19) |
Apply a null rotation with parameter to T1.
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| (2.20) |
Calculate the NP Weyl scalars for the null tetrad T2.
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| (2.21) |
We can make Psi1 = 0 by choosing
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| (2.22) |
Recalculate the NP Weyl scalars and note that Psi1 = 0.
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| (2.23) |