>
|
|
Example 1.
Let be a metric on a 4-dimensional manifold with signature . A list of 4 vectors defines an orthonormal tetrad if
and all other inner products vanish. The command GRQuery, with the keyword "OrthonormalTetrad", can be used to check that a list of 4 vectors defines an orthonormal tetrad.
First create manifold with coordinates .
M >
|
|
Define a spacetime metric on .
M >
|
|
| (2.2) |
Define a tetrad on . Verify that is an orthonormal tetrad with respect to the metric .
M >
|
|
| (2.3) |
M >
|
|
Note that the same vectors, listed in a different order, do not necessarily define an orthonormal tetrad.
M >
|
|
| (2.5) |
M >
|
|
Example 2.
A list of 4 vectors defines a (complex) null tetrad if is the complex conjugate of ,
, ,
and all other inner products vanish. In particular, the vectors are all null vectors. The command GRQuery, with the keyword "NullTetrad", can be used to check that a list of 4 vectors defines a null tetrad.
M >
|
|
| (2.7) |
M >
|
|
Example 3.
To check that a given frame or co-frame is orthonormal in other dimensions or with different metric signatures, the keywords "OrthonormalFrame", "OrthonormalCoframe" are used.
First create a 3-manifold with coordinates .
M >
|
|
Define a Riemannian metric on .
M >
|
|
| (2.10) |
Define a frame on with respect to the metric . Verify that is an orthonormal frame.
M >
|
|
| (2.11) |
>
|
|
Define a co-frame with respect to the metric . Verify that is an orthonormal co-frame.
M >
|
|
| (2.13) |
>
|
|
One can use an optional 3rd argument, a square matrix , to specify the orthogonality relations to be verified - if , then GRQuery(F, g, A, "OrthonormalFrame") returns true if . For example:
M >
|
|
| (2.15) |
M >
|
|
| (2.16) |
M >
|
|
M >
|
|
Example 4.
The keyword argument "PrincipalNullDirection" will test to see if a given vector is a principal null direction for a given metric. The Weyl tensor of the metric is a required argument.
M >
|
|
M >
|
|
| (2.19) |
M >
|
|
The metric g4 is of Petrov type D and therefore admits two independent principal null directions.
M >
|
|
| (2.22) |
M >
|
|
Example 5.
The keyword argument "RecurrentTensor" will test to see if a given tensor is a recurrent tensor with respect to a given metric or connection. If true, then the associated eigen-form is also returned.
M >
|
|
M >
|
|
| (2.25) |
M >
|
|
| (2.26) |
M >
|
|
| (2.27) |