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An exact axial Killing vector field makes Komar angular momentum independent of the surrounding vacuum surface. Therefore gravitational waves in a configuration with exact axisymmetry carry no net angular momentum about that symmetry axis, although they can carry energy. This is specific to axial angular momentum; it is not an absence of radiation.

A bifurcate Killing horizon consists of two intersecting null Killing horizons of the same Killing vector field, which vanishes on their spacelike intersection, the bifurcation surface. In regular Kruskal coordinates the horizons are $U=0$ and $V=0$, and their intersection is $U=V=0$. The displayed field is tangent and normal to each horizon branch and zero on the intersection.

For a regular analytic stationary asymptotically flat vacuum black hole with a connected nondegenerate horizon, the stationary Killing vector field either generates the horizon or the spacetime admits an additional commuting axial Killing vector field. In the rotating case the horizon generator is $K+{\Omega }_H\Phi$, where $\Phi$ has closed rotational orbits. Together with the appropriate staticity and black-hole uniqueness theorem, this identifies an equilibrium uncharged collapse remnant with the Kerr metric. The theorem concerns an equilibrium geometry; it does not itself prove that arbitrary dynamical collapse settles to equilibrium.

On a closed Riemannian manifold, a Killing vector field $X$ satisfies ${\nabla }^{\ast }\nabla X={\mathrm{Ric}}^{♯}X$ for the nonnegative rough Laplacian. This follows by differentiating ${\nabla }_i{X}_j+{\nabla }_j{X}_i=0$, contracting and commuting covariant derivatives; the contraction also gives $\mathrm{div}X=0$. Taking the inner product with $X$ and using integration by parts gives the displayed identity. If Ricci curvature is nonpositive, the nonnegative left side must vanish, so $X$ is a parallel vector field. In particular every Killing vector field on a closed flat Riemannian manifold is a parallel vector field.

Suppose $V$ is a Killing vector field, $F$ is an electromagnetic field tensor satisfying $dF=0$ and ${\mathcal{L}}_{V}F=0$, and a particle of nonzero mass $m$ obeys the Lorentz force equation ${\nabla }_{u}u=(q/m)F^a{}_b{u}^b$. Cartan's magic formula gives $d({\iota }_{V}F)=0$, so the local potential of a closed differential one-form proof gives $d\Phi ={\iota }_{V}F$ locally. The Killing equation eliminates $u^a{u}^b{\nabla }_a{V}_b$, and the force term gives $d(V\cdot u)/d\tau =(q/m)d\Phi /d\tau$. Therefore $mV\cdot u-q\Phi$ is locally conserved. The sign assumes contraction in the first argument of $F$. A global potential additionally requires the closed one-form to have zero periods; invariance alone is insufficient without the homogeneous Maxwell equations.

On a connected $n$-dimensional pseudo-Riemannian manifold, evaluation of a Killing vector field and its first derivative is injective by Killing transport. The value has $n$ components and the Killing equation makes the derivative an antisymmetric two-tensor with $n(n-1)/2$ components. Their sum gives the bound. Flat metrics attain it through ${\xi }_a=A_a+B_{ab}{x}^b$ with $B_{ab}=-B_{ba}$. Connectedness is essential for a uniform global bound in terms of $n$: independent fields can be supported on different connected components.

A mechanical Lagrangian quadratic in velocities can be represented by geodesics of a Lorentzian metric with an extra null coordinate. For $L=\frac{1}{2}|\dot{\mathbf{x}}{|}^2-2\mathbf{A}\cdot \dot{\mathbf{x}}-U$, use $g=d{\mathbf{x}}^2+2dt(du-2\mathbf{A}\cdot d\mathbf{x}-Udt)$. The null Killing vector field ${\partial }_u$ gives the conserved momentum $p_u=t\prime$. Fixing $p_u=1$ projects the geodesic equation to the mechanical Euler-Lagrange equations. This is useful for relating mechanical forces to spacetime geometry.

For a chosen stationary time-translation Killing vector field $k$, this is the region where $k$ is spacelike and observers following its orbits cannot be timelike. In an asymptotically flat geometry, the usual choice is the nonrotating time translation at infinity. The stationary-limit boundary has $g(k,k)=0$, but need not be an event horizon. The Kerr ergoregion is a particular rotating-black-hole example: its outer boundary touches the horizon only on the axis for nonzero subextremal spin.

On Euclidean space, every Killing vector field is a constant translation generator plus a constant infinitesimal rotation generator:Here $A$ is a constant vector and $B$ is an antisymmetric matrix. The second covariant derivative of a Killing vector vanishes for the Euclidean metric, so the components are affine functions of Cartesian coordinates; the Killing equation makes the linear coefficient antisymmetric.

In the Euclidean plane with metric tensor $g=d{x}^2+d{y}^2$, two useful examples areThe notation ${\partial }_x$ denotes a coordinate basis vector; $dx$ denotes a differential one-form. The corresponding flow of a vector field maps areFor translation, all arrows have the same direction and length. For rotation, arrows are tangent to circles, have length $\sqrt{x^2+y^2}$, and vanish at the origin. The rotation flow keeps $x^2+y^2$ fixed, and its nonstationary integral curves of a vector field are circles. Both flows preserve the Euclidean metric. See Carroll's original notes, section 5, equations (5.45)-(5.46).

The circular integral curves of a vector field are symmetry orbits, whereas geodesics of the Euclidean plane are straight lines. A Killing vector field does not require its own orbits to be geodesics. The geodesic conserved quantity from a Killing vector pairs the field with the velocity of a separate geodesic; it does not require the velocity to equal the Killing vector field.

Each Killing vector field supplies a conserved quantity for freely moving test particles: it turns a geometric symmetry into a first integral of the geodesic equation. Let $\gamma (\lambda )$ be a geodesic with affine parameter $\lambda$, tangent vector $U^{\mu }=d{x}^{\mu }/d\lambda$, and ${\nabla }_{U}U=0$. Wherever a Killing vector field $K$ is defined along the geodesic,The statement applies to Riemannian manifolds and to timelike geodesics, null geodesics and spacelike geodesics in a Lorentzian manifold. It requires an affine parameter; for a massive test particle, proper time is one choice. See Tong, section 4.3.2, equation (4.32).

For a worked example in the Euclidean plane, take $\gamma (t)=(1+t,2-t)$ with affine parameter $t$ and velocity $U=(1,-1)$. The Euclidean Killing vector fields for translation and rotation giveThe rotational field along the geodesic is $K_{\mathrm{rot}}(\gamma (t))=(t-2,1+t)$: it changes from $(-2,1)$ at $t=0$ to $(-1,2)$ at $t=1$, yet its pairing with $U$ stays $-3$. With toy mass $m=1$ and $t$ interpreted as time, the first constant is the $x$ component of linear momentum, and the second is the signed $z$ component of angular momentum about the origin. Their numerical values use toy units. The geodesic is straight; the rotation flow orbits are circles.

This conservation law follows from metric compatibility and the Killing equation:The geodesic equation kills the acceleration term, and the symmetric product $U^{\mu }{U}^{\nu }$ selects the vanishing symmetric derivative. This tensor contraction is the whole proof outline; see Carroll, equation (5.43).

In a stationary spacetime, a chosen time-translation Killing vector field $K$ gives Killing energy. With metric signature $(-+++)$, $c=1$, and a massive test particle's four-momentum $p^{\mu }=m{U}^{\mu }$, the physical normalization isHere $\Phi$ generates the axial rotational symmetry when one exists. In coordinates with $K={\partial }_t$ and $\Phi ={\partial }_{\varphi }$, these are $E=-p_t$ and $L_z=p_{\varphi }$. Both are conserved quantities along the geodesic, even if the metric has a time-angle cross term. A future timelike $K$ makes $E$ positive for a future massive particle; the scale of $E$ depends on the normalization of $K$. Not every spacetime has a global timelike Killing vector field, and a stationary field can become spacelike in an ergoregion of a stationary spacetime. Conservation on the field's domain does not imply a universal global positive energy; see Tong, sections 4.3.1-4.3.2 on the timelike-domain qualification and conserved charges.

For metric signature $(+---)$, a stationary perfect fluid follows a timelike Killing vector field $K$ with four-velocity $U=K/N$. Its Killing-flow four-acceleration is $U^b{\nabla }_b{U}_a=-{\nabla }_a\log N$. The relativistic Euler equation and stationarity of the pressure give the displayed hydrostatic equilibrium equation. This uses stationarity along $K$ and does not require hypersurface orthogonality.

Ingoing Kerr coordinates cancel the explicit inverse-$\Delta$ singularities of the Boyer-Lindquist coordinates metric. They give a smooth, nondegenerate Kerr metric at the future outer horizon when $M>|a|$. Its stationary and axial Killing vector fields are ${\partial }_v$ and ${\partial }_{\chi }$. The advanced chart need not cover every horizon branch of the maximal extension.

On a compact flat Riemannian manifold, the integrated Bochner identity for Killing vector fields makes every Killing vector field parallel. Parallel vector fields are exactly the fixed vectors of the linear holonomy. Thus the six orientable closed flat three-manifold types have identity-component isometry dimensions three for the torus, one for each cyclic nontrivial holonomy, and zero for the Hantzsche-Wendt manifold.