Killing tensors and quadratic geodesic integrals

Assume ${\mathrm{A}\mathrm{C}}_{\omega }$ for the tangent-bundle smooth structure of the Lorentzian conventions.

Let $(\mathcal{M},g,N)$ be a smooth time-oriented Lorentzian four-manifold with its metric-selected connection $\mathrm{\nabla }$. A smooth symmetric covariant $2$-tensor field $K_{ab}$ is a rank-two Killing tensor when

${\mathrm{\nabla }}_{(a}{K}_{bc)}:=\frac{1}{3}({\mathrm{\nabla }}_a{K}_{bc}+{\mathrm{\nabla }}_b{K}_{ca}+{\mathrm{\nabla }}_c{K}_{ab})=0.$

For a smooth curve $\gamma :I\to \mathcal{M}$ with open interval domain and affine parameter $\lambda$, write $v=\dot{\gamma }$. If ${\mathrm{\nabla }}_{v}v=0$, the curve is an affinely parametrized geodesic, and $K_{ab}{v}^a{v}^b$ is its associated quadratic geodesic integral. It is called reducible if it is a finite constant linear combination of $g_{ab}$ and symmetric products ${\xi }^{\mathrm{♭}}\odot {\eta }^{\mathrm{♭}}$, where $\xi ,\eta$ are Killing fields, ${\xi }^{\mathrm{♭}}:=g(\xi ,\cdot )$, and $\alpha \odot \beta :=(\alpha \otimes \beta +\beta \otimes \alpha )\mathrm{/}2$.

Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model

Smooth covariant 2-tensor fields on a smooth manifold

Existence and uniqueness of the metric-selected connection of a nondegenerate metric, and its extension to tensor fields

Killing fields of a Lorentzian metric

The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)

This definition supplies the mechanism by which a Killing tensor yields a quadratic geodesic integral. It does not exhibit a particular tensor for Kerr and does not assert that any tensor is irreducible.

The smooth tangent-bundle and tensor-derivative conventions assume only countable choice ${\mathrm{A}\mathrm{C}}_{\omega }$; no stronger choice principle is used.