Global solutions of the vacuum Einstein equations with initial data on a light-cone

At the heart of general relativity lie Einstein’s field equations which describe in which way the presence of matter curves spacetime. They form a complicated system of partial differential equations for the gravitational field. A key direction of research is the construction of solutions to these equations which represent physically relevant configurations. In particular, one is interested in so-called asymptotically flat spacetimes where the gravitational field shows a specific fall-off behavior at infinity, which then provide models for isolated gravitational systems.
The main aim of this project was to construct such space-times. For this, a new method to construct such spacetimes has been developed, based on a new approach to the characteristic initial value problem, using a new reformulation of the Einstein equations. Detailed mathematical results concerning these equations has been proved, as needed to rigorously prove existence of the associated solutions. We have thus successfully attained our goal, to construct new large classes of asymptotically flat solutions to Einstein’s field equations. The large-distance properties of the solutions have been described in detail.
Because solutions of symmetries play a distinguished role in physics, we have extended the initial goals and given an exhaustive description of characteristic initial data sets leading to solutions to Einstein’s equations with symmetries. Combining this with our previous analysis, one obtains new families of spacetimes which are both asymptotically flat and admit symmetries.