Wave decay and horizon instability on strongly charged extremal Kerr-Newman black holes

Abstract

We prove the first boundedness and pointwise decay result for the scalar wave equation on rotating extremal black holes without any symmetry assumptions. The result applies to slowly rotating (equivalently, strongly charged) extremal Kerr-Newman spacetimes. We establish uniform energy boundedness, integrated local energy decay, and a hierarchy of boundary-weighted estimates at the extremal horizon and at null infinity, from which inverse-polynomial pointwise decay follows in the entire exterior region. As a consequence, we also prove the expected Aretakis instability: for generic initial data, suitable transversal derivatives fail to decay along the event horizon, and higher transversal derivatives blow up asymptotically. The proof uses the $b$-structure of the wave operator near the two boundary hypersurfaces, together with a treatment of normally hyperbolic trapping on extremal Kerr--Newman.