Abstract
Talk
Goal: Understand the classical stability of of spacetimes relevant in theory and supergravity.
Definition
A Spacetime is a space time is a manifold with dimension in space and one in dime whose metric fulfills the Lorentzian metric fulfils some curvature condition.
We can describe spacetime in Minkowski (globally useful) or Schwarzschild (useful around black holes and big objects) metric.
We study compact Ricci-flat Riemannsche Mannigfaltigkeit (z.B. flat torus, Calabi-Yau manifold) The idea of String theory and supersymmetry is that there might be some very small dimensions hidden from view.
Definition (Cauchy problem in general relativity)
Study dynamical ecolution of geometric data under the Einstein equations.
A small bump of gravitational ‘energy’ can have three general ways to evolve. It can contract into a point, it can dissipate into gravitational waves or maybe even stay constant. Are thse solutions stable? Asymptotically stable? Periodic?
We assume that the geometry is flat, then we can simplify the PDE and simplify it symbolically, by using the box operator.
Minkowski stability
We mention rapid decay. We first studied the linear problem. When we described the non-linear problem, we got worried. Some of the terms tend to blow up as time goes on. But luckily those terms cancel and in the end we get a quite nice problem.
To make the study much easier, we will only study supersymmetric product spacetimes calle Kaluza-Klein spacetime.
Theorem (Huneau-Stingo-ZW)
The Kaluza-Klein-spacetime is asymptotically stable to small initial data perturbations under the vacuum Einstein equations.