Abstract
We prove boundedness and polynomial decay statements for solutions of the spin pm ,2 Teukolsky equation on a Kerr exterior background with parameters satisfying |a|ll M. The bounds are obtained by introducing generalisations of the higher order quantities P and {underline{P}} used in our previous work on the linear stability of Schwarzschild. The existence of these quantities in the Schwarzschild case is related to the transformation theory of Chandrasekhar. In a followup paper, we shall extend this result to the general sub-extremal range of parameters |a|<M. As in the Schwarzschild case, these bounds provide the first step in proving the full linear stability of the Kerr metric to gravitational perturbations.
Highlights
The stability of the celebrated Schwarzschild [100] and Kerr metrics [72] remains one of the most important open problems of classical general relativity and has generated a large number of studies over the years since the pioneering paper of Regge–Wheeler [98]
The ultimate question is that of nonlinear stability, that is to say, the dynamic stability of the Kerr family (M, ga,M ), without symmetry assumptions, as solutions to the Einstein vacuum equations
The original approach to linear stability in the Schwarzschild case centred on socalled metric perturbations, leading to the decoupled equations of Regge–Wheeler [98] and Zerilli [113]
Summary
The stability of the celebrated Schwarzschild [100] and Kerr metrics [72] remains one of the most important open problems of classical general relativity and has generated a large number of studies over the years since the pioneering paper of Regge–Wheeler [98]. A necessary step to understand nonlinear stability is proving suitable versions of linear stability, i.e. boundedness and decay statements for the linearisation of (1) around the Schwarzschild and Kerr solutions. This requires first imposing a gauge in which the equations (1) become well-posed. In a separate paper, following our previous work on Schwarzschild [31], we will use the above result to show the full linear stability of the Kerr solution in an appropriate gauge
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