Killing fields of a Lorentzian metric

Let $(M,g,N)$ be a time-oriented smooth Lorentzian four-manifold (Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model) and let $\xi$ be a smooth vector field on $M$ (A smooth vector field is a smooth section of the tangent bundle). The Lie derivative of the metric along $\xi$, written ${\mathcal{L}}_{\xi }g$, is the smooth symmetric covariant $2$-tensor field obtained by the flow definition of The Lie derivative of a tensor field applied to $g$: for the local flow ${\mathrm{\Phi }}_t$ of $\xi$,

${\mathcal{L}}_{\xi }g:=\frac{d}{dt}{\mid }_{t=0}{\mathrm{\Phi }}_t^{\ast }g\mathrm{.}$

The field $\xi$ is a Killing field of $(M,g)$ (an isometry generator, or a Killing vector) when

${\mathcal{L}}_{\xi }g=0.$

The equivalent local characterization

${\mathcal{L}}_{\xi }g=0⟺{\mathrm{\nabla }}_a{\xi }_b+{\mathrm{\nabla }}_b{\xi }_a=0,$

where $\mathrm{\nabla }$ is the metric-selected connection of $g$ (Metric compatibility, torsion freedom and the metric-selected connection class, Existence and uniqueness of the metric-selected connection of a nondegenerate metric, and its extension to tensor fields) and ${\xi }_b:=g_{bc}{\xi }^c$ is the lowered field, is proved in Killing fields and conserved currents, and on-shell conservation of the Hilbert stress-energy tensor ↗ clause (a); the definition itself asserts neither direction. Whether a given metric has Killing fields is a property of that metric: neither existence nor uniqueness of a Killing field is asserted here, and a general metric has none beyond the trivial field $\xi =0$.

• What is defined. Only the object "Killing field of the pair $(g,\xi )$" is defined, through the Lie derivative of the metric. No conserved quantity, charge or physical interpretation is attached to a Killing field by this item; conservation along geodesics and the divergence-free current constructed from a conserved stress-energy tensor are separate results of Killing fields and conserved currents, and on-shell conservation of the Hilbert stress-energy tensor ↗.

• No global isometry is asserted. A Killing field generates a local flow of local isometries where it is defined; a global one-parameter group of isometries requires the flow of $\xi$ to be global, which is a separate hypothesis (for instance compact support of $\xi$).