A scattering theory construction of dynamical vacuum black holes

Abstract

We construct a large class of dynamical vacuum black hole spacetimes whose exterior geometry asymptotically settles down to a fixed Schwarzschild or Kerr metric. The construction proceeds by solving a backwards scattering problem for the Einstein vacuum equations with characteristic data prescribed on the event horizon and (in the limit) at null infinity. The class admits the full "functional" degrees of freedom for the vacuum equations, and thus our solutions will in general possess no geometric or algebraic symmetries. It is essential, however, for the construction that the scattering data (and the resulting solution spacetime) converge to stationarity exponentially fast, in advanced and retarded time, their rate of decay intimately related to the surface gravity of the event horizon. This can be traced back to the celebrated redshift effect, which in the context of backwards evolution is seen as a blueshift.

88 pages, 14 figures, v2: minor changes, references added

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Cited by in corpus (21)

• The global non-linear stability of the Kerr-de Sitter family of black holes
• 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
• Global well-posedness of quasilinear wave equations on asymptotically de Sitter spaces
• Global analysis of quasilinear wave equations on asymptotically Kerr-de Sitter spaces
• The weak null condition and global existence using the p-weighted energy method
• Asymptotics for the wave equation on differential forms on Kerr-de Sitter space
• A Scattering Theory for Linearised Gravity on the Exterior of the Schwarzschild Black Hole I: The Teukolsky Equations
• The Stable Trapping Phenomenon for Black Strings and Black Rings and its Obstructions on the Decay of Linear Waves
• Mathematical General Relativity
• The $r^p$-weighted energy method of Dafermos and Rodnianski in general asymptotically flat spacetimes and applications
• The non-linear stability of the Schwarzschild family of black holes
• Asymptotic completeness for superradiant Klein-Gordon equations and applications to the De Sitter Kerr metric
• Formation of Trapped Surfaces from Past Null Infinity
• Trapped surfaces in vacuum arising dynamically from mild incoming radiation
• A Scale-Critical Trapped Surface Formation Criterion: A New Proof via Signature for Decay Rates
• Scattering theory for the Dirac equation on the Schwarzschild-Anti-de Sitter spacetime
• Instability of the extreme Kerr-Newman black-holes
• Modification to the Hawking temperature of a dynamical black hole by a flow-induced supertranslation
• Regularity of a double null coordinate system for Kerr-Newman-de Sitter spacetimes
• Timelike asymptotics for global solutions to a scalar quasilinear wave equation satisfying the weak null condition