Possible PhD projcts

The question: what can Einstein's general theory of relativity do when pushed to extremes? In particular:

• Can it generate naked singularities in collapse from smooth initial data?
• Can it generate extremal black holes?

The tools:

• Simulating extreme spacetimes through numerical time evolutions to discover what happens.
• Then building mathematical models for it (which may also require numerical work, but of a different kind).

With Hilditch, Baumgarte and others, I am currently trying to verify strong indications from numerical relativity simulations that naked singularities can form from the collapse of very strong gravitational waves in vacuum, at the threshold of collapse. This is interesting because it tests general relativity on its own, without the complications of matter.

A long-term research project is to investigate the usefulness of null coordinates in numerical relativity for collapse simulations, and in particular critical collapse. Null coordinates are optimal for many problems in spherical symmetry, but have not been used much for strong-field situations beyond spherical symmetry. In an onging series of papers, I have revisited the foundations, and applied the method to spherical charged scalar field collapse and axisymmetric scalar field collapse.

I am also interested in gravity in 2+1 spacetime dimensions. In 2+1 dimensions, general relativity is simple (there are no gravitational waves, and the vacuum solution is locally unique), but not too simple (vaccuum black holes exist, and black holes can be formed in the gravitational collapse of matter, if we assume a negative cosmological constant).

Here are some possibilities for PhD projects:

• Explore critical collapse with angular momentum (in 3+1 spacetime dimensions), using an axisymmetric numerical code based on null coordinates.
• Explore critical collapse of electromagnetic waves (in 3+1 spacetime dimensions), using an axisymmetric numerical code based on null coordinates.
• Improve results from a first paper on axisymmetric scalar field critical collapse using recently improved methods for null coordinates.
• Use an existing public numerical relativity code, such as the Einstein Toolkit or GRChombo, to investigate critical collapse, using our improved understanding of gauges required.
• In spherical symmetry, using null coordinates, revisit older results on scalar field critical collapse in higher dimensions.