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

#### Autoren: Lars Andersson, Dietrich Häfner, Bernard F. Whiting (2022)

We give a complete analysis of mode solutions for the linearized Einstein equations and the 1form wave operator on the Kerr metric in the large a case. By mode solutions we mean solutions of the form $$e^{-i t_\ast \sigma} \tilde h (r,\theta,\varphi)$$ where $$t_\ast$$ is a suitable time variable. The corresponding Fourier transformed 1form 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. No mode solutions exist for $$\mathfrak{J} \sigma \geq 0$$, $$\sigma \neq 0$$.  For $$\sigma=0$$  mode solutions are Coulomb solutions for the 1form 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 7dimensional space. The proof relies on the absence of modes for the Teukolsky equation shown by the third author and a complete classification of the gauge invariants of linearized gravity on the Kerr spacetime due to Aksteiner et al.

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