Boundedness and decay for the Teukolsky system of spin $\pm 2$ on Reissner-Nordström spacetime: the case $|Q|\ll M$

Abstract

We prove boundedness and polynomial decay statements for solutions to the spin $\pm 2$ generalized Teukolsky system on a Reissner-Nordström background with small charge. The first equation of the system is the generalization of the standard Teukolsky equation in Schwarzschild for the extreme component of the curvature $\alpha$. The second equation, coupled with the first one, is a new equation for a new gauge-invariant quantity involving the electromagnetic curvature components. The proof is based on the use of derived quantities, introduced in previous works on linear stability of Schwarzschild. These quantities verify a generalized coupled Regge-Wheeler system. These equations are the ones verified by the extreme null curvature and electromagnetic components of a gravitational and electromagnetic perturbation of the Reissner-Nordström spacetime. Consequently, as in the Schwarzschild case, these bounds provide the first step in proving the full linear stability of Reissner-Nordström metric for small charge to coupled gravitational and electromagnetic perturbations.

95 pages, 1 figure, version accepted for publication

References in corpus (5)

• The global non-linear stability of the Kerr-de Sitter family of black holes
• Non-linear stability of the Kerr-Newman-de Sitter family of charged black holes
• The linear stability of the Schwarzschild solution to gravitational perturbations in the generalised wave gauge
• The linear stability of Reissner-Nordström spacetime for small charge
• Boundedness and decay for the Teukolsky equation of spin $\pm 1$ on Reissner-Nordström spacetime: the $\ell =1$ spherical mode

Cited by in corpus (2)

• The linear stability of Reissner-Nordström spacetime: the full subextremal range
• Physical-space estimates for axisymmetric waves on extremal Kerr spacetime