The Mathematica package "KY_upperbound" computes the Killing curvatures for Killing-Yano tensors, as well as closed conformal Killing-Yano tensors, for a given metric. The package also solves the curvature conditions obtained from the Killing curvatures.
The package can be downloaded at the bottom of this page.
The functions included in the package are:
Input paramters: For a given metric g in n dimensions, we denote the metric components in a frame {e^a} of T^*M by g_{ab},
In a local coordinate system x^\mu, the basis e^a can be expressed as
Then, we read the metric components g_{ab} and the vielbein e^a_\mu in the coordinates x^\mu as
Generate basis: Using the frame {e^a}, the basis {E^A} of \Lambda^kT^*M for 1 < k < n are automatically generated in the programme. For instance, the basis of \Lambda^2T^*M in four dimensions are given by
The basis of \Lambda^2T^*M in five dimensions are given by
The basis of \Lambda^3T^*M in four dimensions are given by
Following the same rule, the basis {E^A} of \Lambda^kT^*M in arbitrary dimensions are generated.
The programme also generates the dual basis {X_a} such that
Compute Killing curvatures: The Killing curvatures for rank-p Killing-Yano tensors are given as N x N matrices, where N is the rank of the vector bundle E^p(M)=\Lambda^pT^*M + \Lambda^{p+1}T^*M, for any vectors X_i and X_j. The matrices are given by the function
For instance, when we consider rank-2 Killing-Yano tensors in four dimensions (p=2, n=4), the Killing curvatures give linear maps a section s of E^2(M) to a section s^\prime of E^2(M) for any vectors X_i and X_j. The rank of the vector bundle E^2(M) is 10. When the sections s is denoted by
the curvature conditions are given by
The curvature conditions are given for every pair of vectors X_i and X_j. The number depends on the dimension of the spacetime, which is given by n(n-1)/2.
Alternatively, for rank-p closed conformal Killing-Yano tensors, one can use the function
which returns L x L matrices, where L is the rank of the vector bundle E^p(M)=\Lambda^pT^*M + \Lambda^{p-1}T^*M, for vectors X_i and X_j.
Solve kernel: The function
solves the kernel of the curvature conditions on rank-p Killing-Yano tensors for
Similarly, one can use for rank-p closed conformal Killing-Yano tensors the function
The Kerr metric in four dimensions is given by
where
For the Kerr metric, for example, we can choose an orthonormal frame as
where
Then, we input the metric g, the vierbein e and the coordinates x as
The files are available in zipped format.
The files and individual components are shown in the following list.