Killing-Vector Reduction and Twist Potentials

A nonnull Killing field removes one coordinate dependence, but the removed direction leaves two invariant traces: its norm and its twist. Together with a metric on the three-dimensional orbit space, these fields reconstruct the original four-metric locally. The twist distinguishes a merely stationary geometry from a hypersurface-orthogonal, static one and becomes a scalar potential only when the Ricci tensor supplies the required closure condition.

Required background. Use Killing fields and isometry groups for Killing identities and the Einstein equation for the Ricci source of twist.

Helpful background. Hypersurface normals and tangential projectors supplies the projection and hypersurface-orthogonality criteria.

Norm, quotient metric, and twist

Let $\xi^\mu$ be a Killing field with nonzero norm

$$\lambda=g_{\mu\nu}\xi^\mu\xi^\nu.$$

This page uses the site’s orientation $\epsilon_{0123}=+\sqrt{-g}$ and defines the twist one-form by

$$\omega_\mu=\epsilon_{\mu\nu\rho\sigma} \xi^\nu\nabla^\rho\xi^\sigma.$$

Both $\lambda$ and $\omega_\mu$ are invariant along the Killing flow. On a region where the flow has a regular orbit space $S=M/\mathbb R$ or $M/U(1)$, define

$$h_{\mu\nu}=g_{\mu\nu}-\lambda^{-1}\xi_\mu\xi_\nu.$$

It annihilates $\xi^\nu$ and descends to a three-metric on $S$. If $\xi$ is timelike, $\lambda<0$ and $h$ is positive definite; if $\xi$ is spacelike, $h$ is Lorentzian. Geroch develops this quotient construction and the norm–twist variables in Geroch 1971, pp. 919–921.

The derivative of a Killing field can be recovered from its norm and twist:

$$\nabla_\mu\xi_\nu =\frac{1}{2\lambda} \left( \xi_\nu\nabla_\mu\lambda -\xi_\mu\nabla_\nu\lambda +\epsilon_{\mu\nu\rho\sigma}\xi^\rho\omega^\sigma \right).$$

This identity makes the information content transparent. The gradient of $\lambda$ measures the changing size of the symmetry direction. The twist measures the failure of its orthogonal three-planes to fit together into hypersurfaces.

Closure and the twist potential

The Killing identity $\nabla_\mu\nabla_\nu\xi_\rho=R_{\rho\nu\mu\sigma}\xi^\sigma$ gives

$$\nabla_{[\mu}\omega_{\nu]} =\epsilon_{\mu\nu\rho\sigma}\xi^\rho R^\sigma{}_{\tau}\xi^\tau.$$

Thus $d\omega=0$ in vacuum, in an Einstein space $R_{\mu\nu}=\Lambda g_{\mu\nu}$, or more generally whenever $R^\mu{}_{\nu}\xi^\nu$ is proportional to $\xi^\mu$. On a simply connected orbit-space patch there is then a twist potential $\chi$ with

$$\omega_\mu=\nabla_\mu\chi.$$

Closure is local; exactness is global. A closed twist one-form can have nonzero periods when the quotient has nontrivial first cohomology. Matter currents can also source $d\omega$, so introducing $\chi$ without checking the Ricci condition can delete physical configurations.

Frobenius’ theorem supplies the static limit:

$$\xi_{[\mu}\nabla_\nu\xi_{\rho]}=0 \quad\Longleftrightarrow\quad \omega_\mu=0.$$

For timelike $\xi$, vanishing twist means that the stationary flow is locally orthogonal to spacelike hypersurfaces. Stationarity plus $\omega=0$ is the local geometric content of staticity; a coordinate cross term is only indirect evidence.

Reduced vacuum equations

It is convenient to conformally rescale the orbit metric to

$$\gamma_{ab}=|\lambda|\,h_{ab}.$$

With $D_a$ and $R_{ab}[\gamma]$ built from $\gamma_{ab}$, four-dimensional vacuum Einstein equations reduce to the three-dimensional gravity–sigma-model system

$$R_{ab}[\gamma] =\frac{1}{2\lambda^2} \left(D_a\lambda D_b\lambda+D_a\chi D_b\chi\right),$$ $$D^aD_a\lambda =\lambda^{-1}\left[(D\lambda)^2-(D\chi)^2\right], \qquad D^aD_a\chi =2\lambda^{-1}D^a\lambda D_a\chi.$$

Changing the conformal convention changes the displayed coefficients, not the four-dimensional content. A reliable translation therefore records whether the source uses $h$, $|\lambda|h$, or another orbit metric.

For Minkowski spacetime with $\xi=\partial_t$, $\lambda=-1$ and $\chi$ is constant, so the reduced source vanishes and $\gamma$ is Euclidean. For a rotating stationary spacetime, the timelike Killing twist is generally nonzero. At an ergosurface $\lambda=0$, this one-Killing reduction becomes singular even when the spacetime is smooth; one must change the reduction or work with the full two-Killing system.

Exercise

In cylindrical Minkowski spacetime, reduce along the axial Killing field $\xi=\partial_\phi$. Find $\lambda$, $\omega$, $h$, and explain the behavior at the axis.

Solution

From

$$ds^2=-dt^2+dr^2+r^2d\phi^2+dz^2$$

one gets $\lambda=r^2$. The Killing field is hypersurface orthogonal, so $\omega=0$. Removing its vertical component gives

$$h=-dt^2+dr^2+dz^2.$$

The regular-orbit variables are valid for $r>0$. At $r=0$, $\xi$ vanishes and $\lambda^{-1}$ in the projector diverges, although the parent metric is flat. This is an exceptional orbit of the rotation action, not a curvature singularity. Smooth reconstruction requires the axial regularity condition that the circumference-to-radius ratio approach $2\pi$.

References

• Geroch, Robert. “A Method for Generating Solutions of Einstein’s Equations.” Journal of Mathematical Physics 12, no. 6 (1971): 918–924. DOI.