On the significance of Killing tensors (Q1118826)

The authors generalize the idea of a Lie algebra of Killing vector fields on a space-time to the space of Killing tensors by using the Schouten- Nijenhuis bracket to replace the usual Lie bracket for vector fields. The closure property for such Killing tensors under this bracket operation is demonstrated by using the Jacobi-type relation that the Schouten- Nijenhuis inherits from the Poisson bracket from which it is defined. With this bracket operation and if U,V are Killing tensors of valence m and n respectively, their bracket is a Killing tensor of valence \(m+n-1\). By introducing sets of ``basic'' Killing tensors of a given valence (so that a Killing tensor of valence p can be written as a linear combination of the basic Killing tensors of valence p) the coefficients appearing in the linear expression of the bracket of U and V in terms of the basic Killing tensors of valence \(m+n-1\) are shown to satisfy a Jacobi-type identity. Some examples of metrics admitting Killing tensors, including the Kerr space-time and the Kimura class of space-times are discussed. The Kerr metric admits (apart from the usual two Killing vector fields) an irreducible Killing tensor of valence 2 whereas the Kimura metrics examined admit (apart from their four Killing vector fields) two irreducible Killing tensors of valence two.