Metric symmetries produce constants along affine geodesics

An isometry preserves the metric under an active transformation of the manifold. Infinitesimally its generator K obeys a vanishing Lie derivative of the metric, equivalent to the symmetrized covariant derivative of K being zero. If the metric components are independent of a coordinate in a regular chart, its coordinate basis vector generates a local symmetry. The converse is local near suitable nonzero generators; symmetry-adapted coordinates can still have limited domains or fail where a generator vanishes.

A Killing vector is not automatically timelike everywhere. A rotation generator is spacelike away from its axis, and a stationary time-translation generator may become null or spacelike across a horizon or ergoregion. Multiplying K by a constant gives another Killing vector and rescales its associated charge. Multiplying by an arbitrary position-dependent function generally does not. To interpret a time-symmetry charge as a conventional energy, its normalization and relevant region must be specified; asymptotically flat stationary geometries supply a useful normalization at infinity.

Use the static Schwarzschild exterior r>2m with m=GM/c², x⁰=ct, and f=1−2m/r. The time-translation Killing field is ∂x⁰. The dimensionless specific energy is E=−K·u=f dx⁰/ds, with the minus sign chosen to make future-directed exterior energies positive. A static observer has unit tangent f⁻¹ᐟ²∂x⁰, so the local Lorentz factor of the geodesic particle is E/√f. The conserved E stays fixed while this local value changes between radii.

The axial Killing vector ∂φ gives specific angular momentum L=r²sin²θ dφ/ds, with length units in this parameter convention. In an equatorial trajectory it becomes r² times the angular rate. A varying angular coordinate rate therefore does not violate angular-momentum conservation. Nongeodesic forces add a source term Kμaμ to charge evolution. For null geodesics the affine tangent can be rescaled, so its charge normalization must be tied to physical momentum or a specified emitter rather than a nonexistent proper-time normalization.

A time coordinate can always be written, but a time-translation Killing vector need not exist. In an expanding cosmology, a photon’s locally measured energy can redshift without contradicting a globally conserved timelike Killing charge, because the relevant global time symmetry may be absent. Covariant stress conservation alone does not guarantee a preferred global energy. For fields or extended matter, a Killing vector can help construct a conserved current, but obtaining a conserved integrated charge also requires suitable boundaries and the correct invariant flux measure.

The fields lab illustrates a spatial rotation generator on a Euclidean plane. Reset and set both controls to source strength q=0 and rotation Ω=1; at its probe (1,0) predict Fx=0,Fy=1. Reset and set q=0,Ω=2, predicting Fy=2. This continuously rescales the generator K=Ω(−y,x), which remains a rotational Killing field. Use numerical components, since arrow lengths are compressed. The picture does not compute a spacetime metric, a geodesic, or its charge, and the rotating markers are not physical particle trajectories.