Boundedness and decay of scalar waves at the Cauchy horizon of the Kerr spacetime

Abstract

Adapting and extending the techniques developed in recent work with Vasy for the study of the Cauchy horizon of cosmological spacetimes, we obtain boundedness, regularity and decay of linear scalar waves on subextremal Reissner-Nordström and (slowly rotating) Kerr spacetimes, without any symmetry assumptions; in particular, we provide simple microlocal and scattering theoretic proofs of analogous results by Franzen. We show polynomial decay of linear waves relative to a Sobolev space of order slightly above $1/2$. This complements the generic $H_{\mathrm{loc}}^1$ blow-up result of Luk and Oh.

31 pages, 9 figures. v2 is the published version, with minor improvements

References in corpus (6)

• Analysis of linear waves near the Cauchy horizon of cosmological black holes
• Proof of linear instability of the Reissner-Nordström Cauchy horizon under scalar perturbations
• Time-Translation Invariance of Scattering Maps and Blue-Shift Instabilities on Kerr Black Hole Spacetimes
• Linear waves in the interior of extremal black holes I
• Boundedness of massless scalar waves on Kerr interior backgrounds
• Asymptotics for the wave equation on differential forms on Kerr-de Sitter space

Cited by in corpus (19)

• Analysis of linear waves near the Cauchy horizon of cosmological black holes
• Time-Translation Invariance of Scattering Maps and Blue-Shift Instabilities on Kerr Black Hole Spacetimes
• Late-time asymptotics for the wave equation on extremal Reissner-Nordström backgrounds
• Linear waves in the interior of extremal black holes I
• Rough initial data and the strength of the blue-shift instability on cosmological black holes with $\Lambda >0$
• A scattering theory for linear waves on the interior of Reissner-Nordström black holes
• Boundedness of massless scalar waves on Kerr interior backgrounds
• Generic blow-up results for the wave equation in the interior of a Schwarzschild black hole
• The interior of dynamical extremal black holes in spherical symmetry
• A non-degenerate scattering theory for the wave equation on extremal Reissner-Nordstrom
• Polynomial Blow-up Upper Bounds for the Einstein-scalar field System Under Spherical Symmetry
• Asymptotics for scalar perturbations from a neighborhood of the bifurcation sphere
• A scattering theory approach to Cauchy horizon instability and applications to mass inflation
• Curvature blow-up rates in spherically symmetric gravitational collapse to a Schwarzschild black hole
• Spectral instability of horizonless compact objects within astrophysical environments
• Overdamped quasibound states inside a Schwarzschild black hole
• Precise asymptotics of the spin $+2$ Teukolsky field in the Kerr black hole interior
• Investigations of strong cosmic censorship in 3-dimensional black strings
• Linear Instability of the Reissner-Nordström Cauchy Horizon