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Example 1.
First define a 4 dimensional manifold with coordinates .
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Define a metric on .
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| (2.2) |
Define a list of vectors S1.
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| (2.3) |
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| (2.4) |
We use the command TensorInnerProduct. to check this result.
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We repeat the same computation with method = "un-normalized". The result is free of square roots but now the vectors are not unit vectors.
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| (2.5) |
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Example 2.
We continue with the metric from Example 1 but now apply the Gram-Schmidt procedure to a list of 2-forms.
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| (2.6) |
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| (2.7) |
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Example 3.
Consider now an indefinite metric.
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| (2.8) |
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| (2.9) |
A direct application of the GramSchmidt process yields complex-valued forms.
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| (2.10) |
We can adjust the normalization of the last two 1-forms to have length -1 to obtain a real basis:
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| (2.11) |
Example 4.
Consider another indefinite metric.
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| (2.12) |
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| (2.13) |
A direct application of the GramSchmidt process fails since the first two vectors are null vectors. We can work around this problem by changing the initial basis.
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| (2.14) |
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| (2.15) |
Alternatively, we can adjust the signature option.
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| (2.16) |