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Example 1.
First create a vector bundle over with base coordinates and fiber coordinates . It is understood that are complex conjugates of .
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Define a spacetime metric on with signature .
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Define an orthonormal frame on with respect to the metric . Verify the frame is orthonormal using the command GRQuery.
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| (2.3) |
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Calculate the solder form from the frame .
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Let us obtain this result directly from the definition. First we define the Pauli matrices.
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Define the corresponding rank 2 Hermitian spinors.
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| (2.7) |
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| (2.8) |
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| (2.9) |
Define the dual coframe to .
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| (2.10) |
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This coincides with .
Example 2.
The solder form satisfies two important identities. The first identity involves contracting a pair of solder forms over their spinor indices:
The second identity involves contracting a pair of solder forms over their tensor indices:
Let us check the first identity using the solder form from Example 1. First calculate the covariant form of the solder form, using the orthonormal frame of the previous example.
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| (2.13) |
Note that this coincides with the result of using RaiseLowerSpinorIndices to lower the spinor indices of using the epsilon spinor.
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The contraction of and sigmaCov over their spinor indices gives the metric .
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| (2.15) |
The same result can be obtained using SpinorInnerProduct.
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| (2.16) |
To check the second identity calculate the contravariant form of .
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| (2.17) |
Note that this coincides with the result of using RaiseLowerIndices to raise the tensor index of using the inverse of the metric .
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The contraction of and sigmaCon over their tensor indices gives a product of epsilon spinors (EpsilonSpinor).
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| (2.19) |
Rearrange the indices so that the spinor indices are first, the barred spinor indices second.
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| (2.20) |
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| (2.21) |
Example 3.
Here we compute a solder form for the Gödel spacetime. (See (12.26) in Stephani Kramer et al.) First create a vector bundle over with base coordinates and fiber coordinates .
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Define the Gödel metric on . (Note that we have adjusted the metric to conform to the signature convention used by the spinor formalism in DifferentialGeometry
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| (2.23) |
Use DGGramSchmidt to calculate an orthonormal frame for the metric .
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| (2.24) |
Use SolderForm to compute the solder form from the orthonormal frame .
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Example 4.
For any metric of Lorentz signature , a compatible solder form can be constructed.
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Define a spacetime metric .
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Use the command DGGramSchmidt to find an orthonormal frame.
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| (2.28) |
Calculate the solder form from .
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| (2.29) |
Use SpinorInnerProduct to check that is compatible with the metric .
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| (2.30) |