Dual modes in Kerr spacetimes and the Whiting transform: Mode stability revisited
Abstract
The purpose of the paper is to place Whiting's classical growing mode stability argument, extended to real frequencies by Shlapentokh-Rothman for the scalar wave equation and by Andersson, Ma, Paganini and Whiting in general, in the framework of classical PDE theory. The key steps are: a description of the dual or adjoint modes, a singular phase space pairing argument which is technically executed via the Fourier transform, followed by a standard unique continuation result. One part of our description of the dual modes connects them directly to the standard mode solutions which are smooth over the event horizon. In the zero spin case, there is a geometric interpretation of this: quasinormal modes (for non-zero real frequencies) which are smooth across the future event horizon can be extended as distributional QNM solutions by 0 through the past event horizon. We also describe the behavior of mode solutions at the bifurcate sphere.
Related
- Generalized Whiting transform and Flux Estimates for Wave Equations in Subextremal Kerr
- Quasinormal modes on Kerr spacetimes
- Stability and instability of torus-symmetric Einstein spacetimes with square-integrable connection
- Mode stability for the Klein-Gordon equation on subextremal Kerr-de Sitter spacetimes
- Finite-energy spacetimes with torus symmetry. Cauchy stability of Einstein-Euler and Einstein-Navier-Stokes areal flows
BibTeX
@misc{petersen2026dual,
title = {{Dual modes in Kerr spacetimes and the Whiting transform: Mode stability revisited}},
author = {Oliver Petersen and András Vasy},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2608.26034},
eprint = {2608.26034},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.26034},
}
RIS
TY - UNPB
TI - Dual modes in Kerr spacetimes and the Whiting transform: Mode stability revisited
AU - Oliver Petersen
AU - András Vasy
PY - 2026
DA - 2026/09/21
PB - arXiv
DO - 10.48550/arxiv.2608.26034
UR - https://arxiv.org/abs/2608.26034
AB - The purpose of the paper is to place Whiting's classical growing mode stability argument, extended to real frequencies by Shlapentokh-Rothman for the scalar wave equation and by Andersson, Ma, Paganini and Whiting in general, in the framework of classical PDE theory. The key steps are: a description of the dual or adjoint modes, a singular phase space pairing argument which is technically executed via the Fourier transform, followed by a standard unique continuation result. One part of our description of the dual modes connects them directly to the standard mode solutions which are smooth over the event horizon. In the zero spin case, there is a geometric interpretation of this: quasinormal modes (for non-zero real frequencies) which are smooth across the future event horizon can be extended as distributional QNM solutions by 0 through the past event horizon. We also describe the behavior of mode solutions at the bifurcate sphere.
ER -