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Set the global environment variable _EnvExplicit to true to insure that our results are free of expressions.
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We give examples of Weyl spinors of each Petrov type and calculate an adapted spinor dyad. We check that the spinor dyad has the desired properties.
First create the spinor bundle over a 4 dimensional spacetime.
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| (2.1) |
In order to construct the Weyl spinors for our examples, we need a basis for the vector space of symmetric rank 4 spinors. This we obtain from the GenerateSymmetricTensors command.
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| (2.2) |
Example 1. Type I
Define a rank 4 spinor
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| (2.3) |
Calculate the Newman-Penrose coefficients for with respect to the initial dyad basis .
Spin >
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| (2.4) |
Use these coefficients to find the Petrov type of
Spin >
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| (2.5) |
Compute an adapted contravariant spinor dyad for
Spin >
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| (2.6) |
Here is the covariant form of the spinor dyad.
Spin >
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| (2.7) |
We check that this answer is correct in two ways. First we can re-calculate the Newman-Penrose coefficients and confirm that they are in the correct normal form.
Spin >
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| (2.8) |
This is the correct normal form since and
Second, we can calculate a type I Weyl spinor from this spinor dyad using the command WeylSpinor and check that the result coincides with the original spinor .
Spin >
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| (2.9) |
Spin >
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| (2.10) |
Alternative dyads will be computed with the keyword argument output
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Spin >
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| (2.11) |
Example 2. Type II
Define a rank 4 spinor
Spin >
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| (2.12) |
Calculate the Newman-Penrose coefficients for with respect to the standard dyad basis .
Spin >
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| (2.13) |
Find the Petrov type of
Spin >
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| (2.14) |
Compute an adapted contravariant spinor dyad for .
Spin >
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| (2.15) |
Here is the covariant form of the adapted spinor dyad.
Spin >
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| (2.16) |
We check that this answer is correct in two ways. First we can re-caculate the Newman-Penrose coefficients and confirm that they are in the correct normal form.
Spin >
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| (2.17) |
This is the correct normal form since and
Second, we can calculate a type II Weyl spinor from this spinor dyad and check that the result coincides with the original spinor .
Spin >
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| (2.18) |
Spin >
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| (2.19) |
Example 3. Type III
Define a rank 4 spinor
Spin >
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| (2.20) |
Calculate the Newman-Penrose coefficients for with respect to the initial dyad basis .
Spin >
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| (2.21) |
Find the Petrov type of
Spin >
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| (2.22) |
Compute an adapted contravariant spinor dyad for
Spin >
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| (2.23) |
Here is the covariant form of the spinor dyad.
Spin >
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| (2.24) |
We check that this answer is correct in two ways. First we can re-caculate the Newman-Penrose coefficients and confirm that they are in the correct normal form.
Spin >
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| (2.25) |
This is the correct normal form since
Second, we can calculate a type III Weyl spinor from this spinor dyad and check that the result coincides with the original spinor .
Spin >
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| (2.26) |
Spin >
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| (2.27) |
Example 4. Type D
Define a rank 4 spinor
Spin >
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| (2.28) |
Calculate the Newman-Penrose coefficients for with respect to the initial dyad basis .
Spin >
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| (2.29) |
Find the Petrov type of
Spin >
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| (2.30) |
Compute an adapted contravariant spinor dyad for
Spin >
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| (2.31) |
Here is the covariant form of the spinor dyad.
Spin >
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| (2.32) |
We check that this answer is correct in two ways. First we can re-calculate the Newman-Penrose coefficients and confirm that they are in the correct normal form.
Spin >
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| (2.33) |
This is the correct normal form since
Second, we can calculate a type D Weyl spinor from this spinor dyad and check that the result coincides with the original spinor .
Spin >
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| (2.34) |
Spin >
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| (2.35) |
Example 5. Type N
Define a rank 4 spinor
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| (2.36) |
Calculate the Newman-Penrose coefficients for with respect to the initial dyad basis .
Spin >
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| (2.37) |
Find the Petrov type of
Spin >
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| (2.38) |
Compute an adapted contravariant spinor dyad for
Spin >
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| (2.39) |
Here is the covariant form of the spinor dyad.
Spin >
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| (2.40) |
We check that this answer is correct in two ways. First we can re-caculate the Newman-Penrose coefficients and confirm that they are in the correct normal form.
Spin >
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| (2.41) |
This is the correct normal form since
Second, we can calculate a type N Weyl spinor from this spinor dyad and check that the result coincides with the original spinor .
Spin >
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| (2.42) |
Spin >
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| (2.43) |