Example 1.

Let g be a metric on a 4-dimensional manifold with signature [1, -1, -1, -1]. A list of 4 vectors [E_t, E_x, E_y, E_z] defines an orthonormal tetrad if

g(E_t, E_t) = 1   and   g(E_x, E_x) = g(E_y, E_y) = g(E_z, E_z) = -1

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 M with coordinates [t, x, y, z].

Define a spacetime metric g on M.

Define a tetrad F1 on M.  Verify that F1 is an orthonormal tetrad with respect to the metric g.

Note that the same vectors, listed in a different order, do not necessarily define an orthonormal tetrad.

Example 2.

A list of 4 vectors [L, N, M, barM] defines a (complex) null tetrad if barM is the complex conjugate of M,

g(L, N) = 1   and   g(M, barM) = -1,

and all other inner products vanish.  In particular, the vectors [L, N, M, barM] 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.

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 M with coordinates [x, y, z].

Define a Riemannian metric g on M.

Define a frame F1 on M with respect to the metric g.  Verify that F is an orthonormal frame.

Define a co-frame Omega3 with respect to the metric g.  Verify that Omega is an orthonormal co-frame.

One can use an optional 3rd argument, a square matrix A, to specify the othogonality relations to be verified --- if  Fr = [E_1,  E_2, ..., E_n], then GRQuery(Fr, g, A, "OrthonormalFrame") returns true if g(E_i, E)j) = A_ij. For example:

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.

The metric g4 is of Petrov type D and therefore admits two independent principal null directions.

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.

DifferentialGeometry, Tensor, DGGramSchmidt, NullTetrad, PetrovType, PrincipalNullDirections,  OrthonormalTetrad, RecurrentTensors,  SpacetimeConventions, SpinorInnerProduct, SolderForm, TensorInnerProduct