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 crucial new ingredient is the control of the boundary terms generated by the Morawetz estimates. We use as input the results of our companion paper \cite{He-K2}, which provides a frequency independent estimate for the flux based on a purely physical space version of the seminal Whiting transform \cite{W}. 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. In this new version we have added a new, much simpler, proof which covers the full subextremal case. The results proved here are restricted to scalar wave equations, corresponding to spin $0$, in the range |a|/m≤ 0.75. We expect this restriction to be technical, and the methods developed in this paper to extend to the Teukolsky equation.
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BibTeX
@misc{he2026physical,
title = {{A Physical space derivation of Morawetz-Energy estimates in Kerr spacetimes with large angular momentum}},
author = {Lili He and Sergiu Klainerman},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2607.08958},
eprint = {2607.08958},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.08958},
}
RIS
TY - UNPB
TI - A Physical space derivation of Morawetz-Energy estimates in Kerr spacetimes with large angular momentum
AU - Lili He
AU - Sergiu Klainerman
PY - 2026
DA - 2026/09/29
PB - arXiv
DO - 10.48550/arxiv.2607.08958
UR - https://arxiv.org/abs/2607.08958
AB - 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 crucial new ingredient is the control of the boundary terms generated by the Morawetz estimates. We use as input the results of our companion paper \cite{He-K2}, which provides a frequency independent estimate for the flux based on a purely physical space version of the seminal Whiting transform \cite{W}. 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. In this new version we have added a new, much simpler, proof which covers the full subextremal case. The results proved here are restricted to scalar wave equations, corresponding to spin $0$, in the range |a|/m≤ 0.75. We expect this restriction to be technical, and the methods developed in this paper to extend to the Teukolsky equation.
ER -