Scattering theories are of large importance for many problems in General Relativity, and scattering in the interior of a black hole is particularly relevant in the context of the cosmic censorship conjecture and the related Cauchy horizon instability problem. The Cauchy horizon instability is thought to be directly linked to a notion of gravitational blue-shift at the horizon, which manifests itself as a blow-up in some observed quantity. By constructing the scattering channels, one aims to recover information about the behavior of the field near the horizons, where the instabilities might be seen as the unboundedness of the scattering operators (energy blow-up) or from the lack of regularity at the horizon of the propagating field (C^1-blow-up). In this talk I will present the recent development of the scattering theories in the interior black holes between the Cauchy horizon and the event horizon, and the various blow-up results. From a mathematical point of view, different fields exhibit contrasting scattering phenomena. For example, the scattering of linear waves is delicate and surprising breakdowns of scattering happens in generic situations.

We discuss the problem of global uniqueness for the Einstein equations with a positive cosmological constant. After reviewing the modern formulations of strong cosmic censorship (SCCC), we prove a stability result suggesting a potential failure of SCCC in the spherically symmetric framework. In particular, we study the characteristic initial value problem (IVP) for the spherically symmetric Einstein-Maxwell-charged-Klein-Gordon system, with initial data as expected to arise from charged gravitational collapse. The solution to the IVP describes the interior of a black hole asymptotically approaching a sub-extremal Reissner-Nordström-de Sitter spacetime. By using bootstrap methods in the black hole interior, we establish the $H^1$ non-linear stability of the Cauchy horizon for a large set of initial data.

This is an introductory course on the wave equation, to be delivered in weekly two-hour sessions. Topics to be covered include Fourier and physical space methods, existence and uniqueness of solutions to linear equations with constant or variable coefficients, energy estimates, vector fields, Strichartz estimates and local/global well posedness for nonlinear wave equations. Time permitting, singularity formation and geometric optics may also be addressed.

References

This is an introductory course on the wave equation, to be delivered in weekly two-hour sessions. Topics to be covered include Fourier and physical space methods, existence and uniqueness of solutions to linear equations with constant or variable coefficients, energy estimates, vector fields, Strichartz estimates and local/global well posedness for nonlinear wave equations. Time permitting, singularity formation and geometric optics may also be addressed.

References

This is an introductory course on the wave equation, to be delivered in weekly two-hour sessions. Topics to be covered include Fourier and physical space methods, existence and uniqueness of solutions to linear equations with constant or variable coefficients, energy estimates, vector fields, Strichartz estimates and local/global well posedness for nonlinear wave equations. Time permitting, singularity formation and geometric optics may also be addressed.

References