Rasmus Jouttijärvi: Towards Stability of Einstein Manifolds with Boundary

In the case of a compact manifold without boundary, the space of symmetric tensors preserving the volume form (point-wise) and the scalar curvature are used to define the concept of stability for the linearized Einstein equations. These tensors are known as transverse-traceless (TT) tensors.

Upon the introduction of a boundary, the space of TT tensors no longer arise naturally. Thus we must settle for the larger subspace of tensors preserving the scalar curvature and the total volume (TV tensors). In this talk we will discuss a set of geometric boundary conditions that is compatible with a notion of stability as defined on TV tensors.

As a test-case, we shall discuss stability of the Riemannian Schwarzschild-Anti-de-Sitter family of metrics; a problem commonly described as "a Black hole in a box".