= Solution
A <gravitational wave> can propagate through a region with vanishing <stress-energy tensor>; its gravitational field is not a matter current $T_{ab}m^b$. Nevertheless, exact <axisymmetry> gives an axial <Killing vector field>, and the vacuum identity $d\star dm=0$ makes the <Komar angular momentum> equal on enclosing <spacelike submanifolds> in the same <homology class>. Applying the same identity to the <spacetime> tube between surrounding <spacelike submanifolds> at different times gives zero net flux of this charge through the tube.
Consequently \b[<gravitational waves> in exact <axisymmetry> carry no net <angular momentum> about the symmetry axis]. They can still carry <energy> and decrease the total <mass>. In a mode description, rotationally invariant radiative data have zero azimuthal mode number; exact <axisymmetry> therefore excludes the axial <angular momentum> flux present in nonaxisymmetric merger radiation. This does not mean there are no <gravitational waves>, or that radiation cannot transport <angular momentum> when the exact symmetry is absent. The relevant conserved charge includes any horizon contributions identified in the preceding <Komar angular momentum with inner boundaries> identity.
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