nLab
Killing tensor

Context

Riemannian geometry

Differential geometry

differential geometry

synthetic differential geometry

Axiomatics

Models

Concepts

Theorems

Applications

Contents

Definition

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

KSym kΓ(TX) K \in Sym^k \Gamma(T X)

which is covariantly constant in that

(μK α 1,,α k)=0. \nabla_{(\mu} K_{\alpha_1, \cdots, \alpha_k)} = 0 \,.

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

Properties

For every Killing tensor KK on (X,g)(X,g) the dynamics of the relativistic particle on XX has a further conserved quantity. In the canonical case K=gK = 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.

Revised on April 2, 2015 13:52:01 by Tim Porter (2.31.48.51)