Webpage of KY_upperbound

About the Mathematica package

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.

Functions

The functions included in the package are:




Description

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

Example

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



Download

The files are available in zipped format.

The files and individual components are shown in the following list.