# Quasilinear wave equations on asymptotically flat spacetimes with applications to Kerr black holes

**Authors:** Mihalis Dafermos, Gustav Holzegel, Igor Rodnianski, Martin Taylor  
**arXiv:** [2212.14093](https://arxiv.org/abs/2212.14093)  
**Date:** 2022-12-28  
**Summary page:** https://www.summarizepaper.com/en/paper/2212.14093v1/

## Key points

- The authors prove global existence and decay for small-data solutions to quasilinear wave equations on asymptotically flat spacetime backgrounds
- This includes black holes in general relativity with horizons, ergoregions, and trapped null geodesics
- The approach is dyadically localized and uses time-translation invariant rp-weighted estimates on dyadic time-slabs
- A "blackbox" linear inhomogeneous energy estimate is used on exactly stationary metrics along with a novel physical space top order identity that does not need to capture trapping structure
- The approach circumvents the need for producing a purely physical space identity or analyzing commutation properties of frequency projections with the wave operator of time-dependent metrics
- In the case of Kerr black holes, the required linear inhomogeneous estimate can be quoted directly from literature and an additional top order physical space identity can be shown easily in many cases
- Overall, this paper provides important insights into quasilinear wave equations on asymptotically flat spacetimes with applications to Kerr black holes.

## Summary

In this paper, the authors prove global existence and decay for small-data solutions to a class of quasilinear wave equations on a wide variety of asymptotically flat spacetime backgrounds. This includes the presence of horizons, ergoregions, and trapped null geodesics, as well as the Schwarzschild and very slowly rotating Kerr family of black holes in general relativity. The authors' approach has two distinguishing aspects. The first aspect is its dyadically localised nature; the nontrivial part of the analysis is reduced entirely to time-translation invariant rp-weighted estimates in the spirit of [DR09], to be applied on dyadic time-slabs which for large r are outgoing. Global existence and decay then both immediately follow by elementary iteration on consecutive such time-slabs without further global bootstrap. The second, more fundamental aspect is their direct use of a "blackbox" linear inhomogeneous energy estimate on exactly stationary metrics together with a novel but elementary physical space top order identity that need not capture the structure of trapping and is robust to perturbation. In particular, this approach circumvents the need either for producing a purely physical space identity capturing trapping or for a careful analysis of the commutation properties of frequency projections with the wave operator of time-dependent metrics. In the specific example of Kerr black holes, the required linear inhomogeneous estimate can be quoted directly from literature [DRSR16], while an additional top order physical space identity can be shown easily in many cases (the authors include in an appendix a proof for Kerr case |a| (cid:28) M). Overall, this paper provides important insights into quasilinear wave equations on asymptotically flat spacetimes with applications to Kerr black holes. The authors' innovative approach offers new ways to analyze these complex systems and could have significant implications for future research in this area.