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For an axial Killing vector $\xi$ and zero cosmological constant, set $F^{ab}={\nabla }^b{\xi }^a$. The Killing equation and Ricci identity give ${\nabla }_b{F}^{ab}=-R^a{}_b{\xi }^b$. In a vacuum region this vanishes, so Stokes theorem makes the Komar angular momentum equal on homologous enclosing surfaces. The region may contain gravitational waves with nonzero Weyl curvature: Ricci flatness, rather than complete flatness, is what removes the flux. This proves absence of axial angular-momentum transport by exactly axisymmetric gravitational radiation, without assuming that a radiating spacetime is stationary.

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.

Let $C^a{}_{bcd}={\partial }_d{\Gamma }_{bc}^a-{\partial }_c{\Gamma }_{bd}^a-{\Gamma }_{ec}^a{\Gamma }_{bd}^e+{\Gamma }_{ed}^a{\Gamma }_{bc}^e$, with all other index positions obtained by the metric tensor. This convention is the negative of the convention $[{\nabla }_c,{\nabla }_d]V^a=R^a{}_{bcd}{V}^b$. The Ricci identity is $k_{a;bc}-k_{a;cb}=C_a{}^d{}_{bc}{k}_d$, with $k_{a;bc}={\nabla }_c{\nabla }_b{k}_a$. For a Killing vector, the Killing equation makes this second derivative antisymmetric in $a,b$. Applying the commutator three times gives $2k_{a;bc}=(C_a{}^d{}_{bc}+C_c{}^d{}_{ba}-C_b{}^d{}_{ac})k_d=-2C_{abc}{}^d{k}_d$, where the last step uses the first Bianchi identity. Switching the sign of the Riemann curvature tensor switches the sign in this identity; moving a raised index by ordinary metric contraction does not switch it.

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.

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 a Killing vector field, set $F_{ab}={\nabla }_a{K}_b$ and $a^s=K^t{\nabla }_t{K}^s$. The Killing equation makes $F$ antisymmetric, and expansion of normalized three-index antisymmetrization gives the displayed identity. On a Killing horizon, hypersurface orthogonality removes its first right-hand term and relates $F^{ab}{F}_{ab}$ to surface gravity.

The Killing equation tests whether a candidate vector field generates a local isometry, without needing to solve for its flow of a vector field. For the Levi-Civita connection of a metric tensor $g$, it isParentheses mean averaging over the two index orders. The equation says the symmetric part of the covariant derivative vanishes, not that the entire covariant derivative vanishes. The Lie derivative of the metric tensor gives the coordinate formThe last two terms account for how tangent vectors move with the field; checking only $K^{\rho }{\partial }_{\rho }{g}_{\mu \nu }$ misses this. See Carroll, section 5, equations (5.33) and (5.41)-(5.42).

For the Euclidean Killing vector field $K=-y{\partial }_x+x{\partial }_y$, the Euclidean metric in Cartesian coordinates makes covariant derivatives ordinary partial derivatives. Its diagonal equations give ${\partial }_x{K}_x={\partial }_y{K}_y=0$, and its mixed equation gives ${\partial }_x{K}_y+{\partial }_y{K}_x=1-1=0$. Thus a varying vector field can preserve a constant metric tensor. In contrast, $K=x{\partial }_x+y{\partial }_y$ gives $({\mathcal{L}}_{K}g{)}_{xx}=2$ and fails the Killing equation, despite the constant metric components.

If $K={\partial }_z$ is a coordinate basis vector, its components are constant and the coordinate form reduces to ${\partial }_z{g}_{\mu \nu }=0$. Hence independence of the metric components from $z$ supplies a Killing vector field. This shortcut applies to that coordinate generator, not to an arbitrary varying vector field.

On a geodesically complete Riemannian manifold, every Killing vector field is complete and integrates to a global one-parameter subgroup of its isometry group. On the incomplete interval $(0,1)$ with its Euclidean metric, ${\partial }_x$ satisfies the Killing equation, but translations leave the interval in finite time. The full isometry group is only the identity and reflection. Local solutions of the Killing equation therefore need not equal the Lie algebra of global isometries without completeness.

With metric signature $(+---)$, normalize a timelike Killing vector field by $U^a=K^a/N$, $N^2=K^a{K}_a$. The Killing equation gives $K^b{\nabla }_b{K}_a=-\frac{1}{2}{\nabla }_a{N}^2$, while $K^b{\nabla }_{b}N=0$. Thus the four-acceleration of its normalized flow is the displayed gradient. With metric signature $(-+++)$ the corresponding lower-index formula has the opposite sign.

The Killing equation and the second covariant derivative of a Killing vector transport a Killing field and its antisymmetric derivative along any curve. Here $L_{ab}={\nabla }_b{\xi }_a$ and the curvature convention is the one in curvature-sign convention in Killing derivative identities. In a parallel frame the equations form a homogeneous linear system of ordinary differential equations. Initial value zero has unique zero continuation. Thus on a connected pseudo-Riemannian manifold, not just on a Riemannian manifold, the field is determined by its value and first derivative at one point.

A Killing vector field describes a direction in which geometry has a continuous symmetry: moving points together along its flow of a vector field preserves the metric tensor, including its pairing of tangent vectors. Such a symmetry gives a geodesic conserved quantity from a Killing vector, often a component of momentum or angular momentum. The direction may vary with position: rotation provides the basic example.

On a Riemannian manifold or Lorentzian manifold with metric tensor $g$, a smooth vector field $K$ is a Killing vector field when its Lie derivative of $g$ vanishes. If ${\Phi }_s$ is its local flow of a vector field, the equivalent statements areThe middle equality uses the pullback of a covariant tensor: the local flow maps are isometries on their domains. The last equality is the Killing equation, using the Levi-Civita connection. The infinitesimal condition holds throughout an open domain and integrates to exact local isometries; it is stronger than making the first derivatives of $g$ vanish at a single point. See Tong, section 4.3.1, equation (4.29).

For example, on the interval $0<x<1$ with Euclidean metric $d{x}^2$, $K={\partial }_x$ translates points by $x\mapsto x+s$. This is a local isometry, but a point reaches the end of the interval in finite flow parameter. A local flow of a vector field can be incomplete; see Killing fields on incomplete manifolds need not generate global isometries. The full-plane translation and rotation examples under Euclidean Killing vector field instead have flows for every real $s$.

With zero cosmological constant and the component Hodge star operator convention $(\star j{)}_{abc}={ϵ}_{abcd}{j}^d$, the Einstein field equations and Killing equation give $d\star dm=-16\pi \star (T_{ab}{m}^b-\frac{1}{2}T{m}_a)$. If $m$ is tangent to a spacelike hypersurface, the trace term pulls back to zero. For $\partial \Sigma =S_{\infty }-{\bigcup }_i{S}_i$, Stokes theorem yieldsThe inner Komar charges cannot be dropped for a vacuum Kerr black hole, whose angular momentum is nonzero although its stress-energy tensor vanishes. A matter-only volume formula requires a regular filling with no inner boundaries.

A Komar charge is an oriented surface integral constructed from a Killing vector field and the metric tensor. Let $k$ be the metric-dual differential one-form of a stationary Killing vector field. In four dimensions, with geometrized units $G=c=1$ and the standard future/outward orientation, its mass normalization iswhere $\star$ is the Hodge star operator and $S$ is a closed spacelike two-surface. The normalization of the Killing vector field fixes the time scale: for Schwarzschild spacetime, unit time translation at infinity gives the black-hole mass parameter.

The Killing equation relates the divergence of $dk$ to a contraction of the Ricci tensor. Consequently, homologous enclosing surfaces give the same mass in a vacuum region with zero cosmological constant. Matter or a nonzero cosmological constant introduces a volume contribution; an arbitrary finite-radius surface integral need not equal the asymptotic mass. Axial symmetry instead gives Komar angular momentum with a different normalization.

For a torsion-free connection, the Lie derivative of a metric tensor obeysThus the abbreviated formula ${\mathcal{L}}_{K}g=2{\nabla }_{(a}{K}_{b)}$ requires metric compatibility. The familiar Killing equation and second covariant derivative of a Killing vector use the Levi-Civita connection, which has both vanishing torsion tensor and metric compatibility. On Minkowski spacetime with metric signature $(-+++)$, a torsion-free connection with only ${\Gamma }^0{}_{00}=1$ supplies a counterexample: $K={\partial }_0$ is a Killing vector field for the constant Minkowski metric, but ${\nabla }_0{K}_0=1$.

For a non-null Killing vector field $V$ of a Levi-Civita connection, hypersurface orthogonality implies that $\eta =V^{♭}/F$, with $F=g(V,V)$, is a closed differential form. The Killing equation and $V^{♭}\wedge d{V}^{♭}=0$ give $2F{\nabla }_{\mu }{V}_{\nu }=V_{\nu }{\partial }_{\mu }F-V_{\mu }{\partial }_{\nu }F$, whence $d\eta =0$. The Poincaré lemma gives $\eta =d\varphi$ locally. For a timelike $V$, $\varphi$ supplies local static time coordinates. This does not assert global exactness.

The null expansion, null shear and null twist of a congruence tangent to the generators of a Killing horizon all vanish there. If $k$ is its Killing generator and $\ell =fk$ is an affine generator, the screen-space projector kills every term proportional to $k$. The symmetric part of the projected derivative of $\ell$ therefore vanishes by the Killing equation. The antisymmetric part vanishes by Frobenius theorem, since $\ell$ is hypersurface-orthogonal. These conclusions extend through a regular bifurcation surface by continuity.

A Killing vector field $K$ generates local isometries: its flow of a vector field ${\Phi }_s$ satisfies ${\Phi }_s^{\ast }g=g$. Equivalently, the Lie derivative of the metric tensor vanishes. For the Levi-Civita connection, metric compatibility converts the coordinate expression into the Killing equation:The Levi-Civita connection is an implicit extra assumption in this question. The printed assumption of vanishing torsion tensor alone does not imply metric compatibility. For example, on Minkowski spacetime, take the affine connection whose only nonzero Christoffel symbol is ${\Gamma }^0{}_{00}=1$. The vector field $K={\partial }_0$ preserves the Minkowski metric, but $K_0=-1$ gives ${\nabla }_0{K}_0=1$, contradicting the Killing equation. The following calculation, and the covariant formulas in part (b), use the Levi-Civita connection; this is the role of metric compatibility in Killing identities.

For the stated curvature sign convention, the Ricci identity on a covector has a minus sign:Write $A_{\mu \nu \rho }={\nabla }_{\mu }{\nabla }_{\nu }{K}_{\rho }$. Differentiating the Killing equation gives $A_{\mu \nu \rho }=-A_{\mu \rho \nu }$. Alternately using this relation and the Ricci identity givesThe first Bianchi identity and antisymmetry of the last two indices of the Riemann curvature tensor reduce the three curvature terms to $2R^{\sigma }{}_{\mu \nu \rho }{K}_{\sigma }$. Consequently the second covariant derivative of a Killing vector is

Write the affine function obtained in part (iii) as $\lambda =c+2b_{\mu }{x}^{\mu }$, with constant $c$ and $b_{\mu }$. A Conformal Killing vector field with precisely this conformal Killing factor isDirect partial differentiation verifies its Conformal Killing equation. Subtracting it from any other solution leaves a Killing vector field $Y$. The formula in part (iii), now with $\lambda =0$, says ${\partial }_{\mu }{\partial }_{\nu }{Y}_{\rho }=0$. Therefore $Y$ is affine, and the Killing equation makes its linear coefficient antisymmetric after lowering an index. Every solution is consequentlyThe four constant $a^{\mu }$ generate translations, the six independent ${\omega }_{\mu \nu }$ generate Lorentz transformations, $c$ generates a dilation, and the four $b^{\mu }$ generate special conformal transformations. Their constant, linear and quadratic terms are independent; within the linear terms, the antisymmetric coefficient cannot equal a nonzero multiple of the identity. Hence the conformal Killing algebra of Minkowski spacetime hasFor $\lambda =1$, the canonical choice is the dilation generator $D=x^{\mu }{\partial }_{\mu }$. Its flow of a vector field solves $d{x}^{\mu }/ds=x^{\mu }$, givingOne may add any Killing vector field without changing $\lambda =1$, so this choice is not unique. The displayed dilation gives a global one-parameter group of diffeomorphisms. The count of fifteen concerns independent Conformal Killing vector fields; some special conformal transformations have only local flows on all of affine Minkowski spacetime.

The product rule for the Lie derivative and the Killing equation giveConsequently $K$ is a Conformal Killing vector field for the rescaled metric tensor, with smooth conformal Killing factorIt remains an actual Killing vector field precisely when $K(\Lambda )=0$: the conformal factor must be constant along its integral curves of a vector field.