Given a spacetime $(ℳ,g_{ab})$ admitting a dynamical horizon segment $\mathscr{H}$, we show that there are stringent constraints on the Killing fields ${\xi }^a$ that $g_{ab}$ can admit in a neighborhood of $\mathscr{H}$: Generically, ${\xi }^a$ can only be a rotational Killing field which, furthermore, leaves each marginally trapped 2-sphere cross section $\mathcal{S}$ of $\mathscr{H}$ invariant. Finally, if ${\xi }^a$ happens to be hypersurface orthogonal near $\mathscr{H}$—as, for example, in a spacetime representing a head-on collision of nonspinning black holes—then, not only the angular momentum but also all spin multipoles vanish on every $\mathcal{S}$; the entire spin structure of these dynamical horizon segments (DHSs) is indistinguishable from that of spherically symmetric DHSs!.

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• General relativity
• Gravitation
• Gravitational waves
• Classical black holes