Dynamical Horizon Segments and Spacetime Isometries

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