Killing Fields and Isometry Groups

A vector field generates metric-preserving motion exactly when its symmetrized covariant derivative vanishes. This Killing equation converts a geometric statement about a local flow into a linear first-order differential equation, explains why Killing fields form a Lie algebra, and places a sharp finite-dimensional bound on the continuous symmetries of an $n$-dimensional metric.

Required background. Tensor Fields and Natural Operations supplies pointwise tensor products, contractions, and symmetry operations; Flows and Lie Derivatives supplies the flow-defined tensor pullback and Lie derivative. Affine Connections and Covariant Derivatives supplies the general connection framework, and The Levi-Civita Connection supplies the metric-compatible derivative used in Killing’s equation.

Helpful background. Calculus and Linear Algebra for Relativity supplies linear maps and component changes; the needed differential system is developed below. Affine and Variational Geodesics supplies the affine geodesic equation that motivates the conserved quantities developed next.

Metric-preserving flows

Let $X$ generate a local flow $\Phi_s$. The flow is a local isometry when $\Phi_s^*g=g$ wherever it is defined. Differentiating at $s=0$ gives

$$\mathcal L_Xg=0.$$

For the Levi-Civita connection,

$$(\mathcal L_Xg)_{ab} =X^c\nabla_cg_{ab}+g_{cb}\nabla_aX^c+g_{ac}\nabla_bX^c =2\nabla_{(a}X_{b)}.$$

Thus the infinitesimal condition is Killing’s equation

$$\nabla_{(a}X_{b)}=0,$$

where parentheses include the factor $1/2$. Conversely,

$$\frac{d}{ds}\Phi_s^*g=\Phi_s^*(\mathcal L_Xg)=0$$

implies $\Phi_s^*g=g$ from its value at $s=0$. The equivalence is local: a Killing field on an incomplete manifold need not have a flow defined for all parameter values. Translation $\partial_x$ on the open interval $(0,1)$ is the simplest counterexample.

The flow-to-equation argument and its coordinate form are given in Carroll 1997, §5, pp. 138–141, eqs. (5.34)–(5.46).

The Lie algebra of generators

If $X$ and $Y$ are Killing fields, then

$$\mathcal L_{[X,Y]}g =[\mathcal L_X,\mathcal L_Y]g=0.$$

They therefore close under the Lie bracket. This algebra is the infinitesimal symmetry algebra; only complete generators are guaranteed to exponentiate to globally defined one-parameter subgroups on the entire manifold. Discrete isometries, such as a reflection, have no nonzero infinitesimal generator and are not detected by solving Killing’s equation alone.

At a point $p$, Killing’s equation makes $(\nabla_aX_b)_p$ antisymmetric. A Killing field on a connected metric manifold is fixed by the data

$$X_a(p), \qquad \nabla_{[a}X_{b]}(p).$$

There are at most $n+n(n-1)/2=n(n+1)/2$ independent components. To see why these data determine the field, an isometry sends affine geodesics to affine geodesics. Its infinitesimal generator restricted to any geodesic is therefore a Jacobi field, determined by $X(p)$ and $\nabla_uX(p)$. Zero value and first derivative force it to vanish along every geodesic from $p$, hence on a normal neighborhood; overlapping neighborhoods propagate this conclusion through a connected manifold. The Killing algebra therefore has dimension at most $n(n+1)/2$.

Constant-curvature models attain the bound locally. Global identifications can remove generators: a flat two-torus has only two independent continuous translation symmetries, whereas the Euclidean plane has three local generators including rotation. A global rotation field generally fails to descend through the torus identifications. For a generic metric there are no nonzero Killing fields even locally.

This is not merely a count of equations versus unknowns. Killing’s equation is overdetermined, and curvature supplies integrability conditions. At a chosen point one may propose a value and an antisymmetric first derivative, but curvature can prevent those data from extending to a solution. The upper bound is universal, whereas attaining it is a strong local-geometric statement. In four dimensions the maximum is ten, realized locally by flat, de Sitter, and anti-de Sitter geometries.

If two Killing fields have the same value and first derivative at one point of a connected region, their difference has zero initial data and vanishes throughout that region. This uniqueness is useful computationally: comparing initial data can establish linear independence without sampling the entire manifold.

Solving and checking the equation

In coordinates, the equation can be evaluated without first computing $X_a$:

$$(\mathcal L_Xg)_{\mu\nu} =X^\rho\partial_\rho g_{\mu\nu} +g_{\rho\nu}\partial_\mu X^\rho +g_{\mu\rho}\partial_\nu X^\rho=0.$$

This form is often the quickest route when the metric coefficients are simple. In Minkowski coordinates, every translation $X=a^\mu\partial_\mu$ solves the equation because both $g_{\mu\nu}$ and $X^\rho$ have constant components. A spatial rotation $X=-y\partial_x+x\partial_y$ also solves it: its derivative matrix is antisymmetric in the Euclidean spatial block. By contrast, $X=x\partial_x$ gives $(\mathcal L_Xg)_{xx}=2$, so a dilation is not an isometry.

The bracket provides an independent check. With $P_x=\partial_x$, $P_y=\partial_y$, and $J=-y\partial_x+x\partial_y$, direct calculation gives $[J,P_x]=-P_y$ and $[J,P_y]=P_x$, and each result again satisfies Killing’s equation.

For a less trivial metric, one should check more than coordinate independence of the coefficients. A coordinate field $\partial_q$ is Killing if the metric components are independent of $q$ in that chart, but a Killing field need not coincide with a coordinate vector before adapted coordinates are constructed. Conversely, a component that vanishes at an axis need not signal a singular generator; the round-sphere exercise below shows how the invariant norm and group action resolve the issue.

Local symmetry versus physical charge

Killing’s equation states only that a metric is invariant under a geometric flow. It does not by itself define energy, angular momentum, or a gravitational charge. Such quantities require additional dynamics, boundary conditions, normalizations, and sometimes asymptotic rather than exact symmetries. Here the conclusion is deliberately narrower: an exact continuous metric symmetry supplies a Killing field and, as shown on the next page, a first integral of affinely parametrized geodesic motion.

Exercises

Show that the round unit two-sphere

$$ds^2=d\theta^2+\sin^2\theta\,d\phi^2$$

admits the Killing field $X=\partial_\phi$, and identify where that generator vanishes geometrically.

Solution

The metric components do not depend on $\phi$, and $X^\theta=0$, $X^\phi=1$ are constant. Every term in the coordinate expression for $\mathcal L_Xg$ therefore vanishes. The coordinate vector $\partial_\phi$ has squared norm $g_{\phi\phi}=\sin^2\theta$, so its geometric vector vanishes at $\theta=0$ and $\theta=\pi$. Those are the fixed points of the rotation, even though the angular coordinate itself is ill-defined there.

References

• Sean M. Carroll, Lecture Notes on General Relativity, arXiv:gr-qc/9712019 (1997), arXiv.