Singularities in General Relativity

Introduction

One of the outstanding open mathematical problems in General Relativity is to understand how singularities form in solutions to the Einstein equations. Singularities are typically found in the interior of black holes or at the Big Bang. The main objective of this proposal is to start a research group at the University of Crete that will develop novel mathematical techniques, combining ingredients from analysis, partial differential equations, and differential geometry, which will enable us to prove the formation of singularities in various new settings and test long-standing conjectures in the field.

Dynamics of Big Bang Singularities

In vacuum, the generic cosmological singularity is conjecturally spacelike, local, and oscillatory. The only rigorous evidence in favor of the latter scenario is restricted to homogeneous solutions.

We intend to:

1. Construct Big Bang singularities containing spikes or oscillations, without symmetries and analyticity, that test the conjectural picture.
2. Utilize previous work of the PI with J. Luk constructing Kasner-like singularities, as they are the building blocks of more complicated scenarios.
3. Dynamically study the submanifold of Big Bang singularities having no oscillations by advancing techniques recently developed in work of the PI and collaborators to prove stable Big Bang formation in the sub-critical regime.

The Black Hole Interior Problem

The stable phenomenon that has been observed so far in black hole regions is Cauchy horizon formation. Our goal is to show that in certain regimes, a similar situation to that of Big Bangs occurs, where part of the inner black hole boundary is spacelike and singular.

We consider:

• The Einstein-massless-scalar field system.
• The classical Oppenheimer-Snyder dust model of gravitational collapse for a class of initial data outside of spherical symmetry, where singularity formation has not yet been understood.