The Aretakis constants and instability in general spherically symmetric extremal black hole spacetimes: higher multipole modes, late-time tails, and geometrical meanings

We study late-time behaviors of massive scalar fields in general static and spherically symmetric extremal black hole spacetimes in arbitrary dimensions. We show the existence of conserved quantities on the extremal black hole horizons for specific mass squared and multipole modes of the scalar fields. Those quantities on the horizon are called the Aretakis constants and are constructed from the higher-order derivatives of the fields. Focusing on the region near the horizon at late times, where is well approximated by the near-horizon geometry, we show that the leading behaviors of the fields are described by power-law tails. The late-time power-law tails lead to the Atetakis instability: blowups of the transverse derivatives of the fields on the horizon. We further argue that the Aretakis constants and instability correspond to respectively constants and blowups of components of covariant derivatives of the fields at the late time in the parallelly propagated null geodesic frame along the horizon. We finally discuss the relation between the Aretakis constants and ladder operators constructed from the approximate spacetime conformal symmetry near the extremal black hole horizons.