Conformal methods in General Relativity and applications: Perturbations of the asymptotic region of the Schwarzschild-de Sitter spacetime

Edgar Gasperín (Queen Mary)

Although the study of the Cauchy problem in General Relativity started in the decade of 1950 with the work of Foures-Bruhat, addressing the problem of global non-linear stability of solutions to the Einstein field equations is in general a hard problem. The first non-linear global non-linear stability result in General Relativity was obtained for the de-Sitter spacetime by H. Friedrich in the decade of 1980. In this talk the main tool used in the above result is introduced: a conformal (regular) representation of the Einstein field equations ---the so-called conformal Einstein field equations (CEFE). Then, the conformal structure of the Schwarzschild-de Sitter spacetime is analyzed using the extended conformal Einstein field equations (XCEFE). To this end, initial data for an asymptotic initial value problem for the Schwarzschild-de Sitter spacetime are obtained. This initial data allow to understand the singular behavior of the conformal structure at the asymptotic points where the horizons of the Schwarzschild-de Sitter spacetime meet the conformal boundary. Using the insights gained from the analysis of the Schwarzschild-de Sitter spacetime in a conformal Gaussian gauge, we consider nonlinear perturbations close to the Schwarzschild-de Sitter spacetime in the asymptotic region. Finally, we'll show that small enough perturbations of asymptotic initial data for the Schwarzschild-de Sitter spacetime give rise to a solution to the Einstein field equations which exists to the future and has an asymptotic structure similar to that of the Schwarzschild-de Sitter spacetime.