Mode analysis for the linearized Einstein equations on the Kerr metric: the large $\mathfrak{a}$ case

Abstract

We give a complete analysis of mode solutions for the linearized Einstein equations and the $1$-form wave operator on the Kerr metric in the large $\mathfrak{a}$ case. By mode solutions we mean solutions of the form $e^{-i{t}_{\ast }\sigma }\tilde{h}(r,\theta ,\phi )$ where $t_{\ast }$ is a suitable time variable. The corresponding Fourier-transformed $1$-form wave operator and linearized Einstein operator are shown to be Fredholm between suitable function spaces and $\tilde{h}$ has to lie in the domain of these operators. These spaces are constructed following the general framework of Vasy (2013, 2021) along the lines of Häfner et al. (2021). No mode solutions exist for $\Im \sigma \ge 0,\sigma \ne 0$. For $\sigma =0$ mode solutions are Coulomb solutions for the $1$-form wave operator and linearized Kerr solutions plus pure gauge terms in the case of the linearized Einstein equations. If we fix a De Turck/wave map gauge, then the zero mode solutions for the linearized Einstein equations lie in a fixed seven-dimensional space. The proof relies on the absence of modes for the Teukolsky equation shown by Whiting (1989) and Anderson et al. (2019) and a complete classification of the gauge invariants of linearized gravity on the Kerr spacetime by Aksteiner and Bäckdahl (2018) and Aksteiner et al. (2021).