= Angular momentum conservation in an axisymmetric vacuum region
{title2=$J_{S_2}=J_{S_1}$}
For an axial <Killing vector> $\xi$ and zero <cosmological constant>, set $F^{ab}=\nabla^b\xi^a$. The <Killing equation> and <Ricci identity> give $\nabla_bF^{ab}=-R^a{}_b\xi^b$. 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.
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