Linear wave equations with initial data on compact Cauchy horizons

13.07.2017, 16:15 Uhr  –  Haus 9, Raum 0.14
Forschungsseminar Differentialgeometrie

Oliver Lindblad Petersen

When studying wave equations in a region of a curved spacetime, one usually assumes that the region satisfies certain causality conditions. Expressed in mathematical terms, one assumes that the region is globally hyperbolic. The past (resp. future) boundary of the globally hyperbolic region is called past (resp. future) Cauchy horizon. The central question of this talk is: Given initial data on a compact Cauchy horizon, does there exist a unique solution to the wave equation?
Our main result concerns spacetimes with a so called non-degenerate compact Cauchy horizon. Examples include the Taub-NUT spacetime and the Misner spacetime. We show that given initial data on the Cauchy horizon,  there exists a unique solution on the globally hyperbolic region that is smooth up to the boundary. The proof requires precise energy estimates for the wave equation close to the Cauchy horizon. If one drops the assumption that the Cauchy horizon is non-degenerate, simple counter examples show that uniqueness does not hold in general.