nLab Killing tensor

Definition

For $(X,g)$ a (pseudo-)Riemannian manifold a Killing tensor is a section of a symmetric power of the tangent bundle

$$K \in Sym^k \Gamma(T X)$$

which is covariantly constant in that

$$\nabla_{(\mu} K_{\alpha_1, \cdots, \alpha_k)} = 0 \,.$$

For $k = 1$ this reduces to the notion of Killing vector.

Properties

For every Killing tensor $K$ on $(X,g)$ the dynamics of the relativistic particle on $X$ has a further conserved quantity. In the canonical case $K = g$ this quantity is the Hamiltonian of the particle (in the case of a relativistic particle its four-velocity normalization).

The analog of this for spinning particles and superparticles are Killing-Yano tensors.

Related concepts

• Killing vector

• Killing spinor

• Killing-Yano tensor.

References

Named after Wilhelm Killing.