A Physical space derivation of Morawetz-Energy estimates in Kerr spacetimes with large angular momentum

Abstract

We revisit the derivation of Morawetz--energy estimates for scalar wave equations in the domain of outer communication of a Kerr spacetime \(\KK(a,m)\). Our goal is to develop robust physical-space methods which are well suited for extension to realistic perturbations of Kerr. The proof rests on several ingredients. First, we derive conditional Morawetz estimates which extend the physical-space techniques initiated by Andersson and Blue \cite{AB}, and later adapted in \cite{GKS} to perturbations of slowly rotating Kerr, by exploiting a physical-space characterization of the full \(r\)-range of trapped null geodesics. Second, we use an idea introduced by Stogin \cite{St} in the axially symmetric case to handle the low-frequency difficulties in the Morawetz estimates. In the general case, the control of the lower-order terms also requires making full use of the principal trapping term in the Morawetz bulk norm, together with a new use of Hardy-type inequalities. A further new ingredient is the control of the boundary terms generated by the Morawetz estimates. This is based on two additional ideas: physical-space versions of Whiting's transform \cite{W}, developed in a forthcoming paper \cite{H-K2}, which yield a flux-independent energy estimate; and an adaptation of the Andersson--Blue invariant-operator method, which turns that estimate into a bound for the horizon flux. Finally, a continuity argument yields an unconditional global-in-time Morawetz estimate, while a new energy estimate is obtained from the construction of a causal vectorfield which is Killing on the trapping set. The results proved here are restricted to scalar wave equations, corresponding to spin \(0\), in the range \(|a|/m\leq 0.75\). We expect this restriction to be technical, and the methods developed in this paper to extend to the Teukolsky equation.