Critical phenomena in gravitational collapse

Critical phenomena in gravitational collapse

Overview

One key prediction of general relativity is that when matter is sufficiently dense it must keep collapsing until it dissappears from sight inside a black hole. A process such as the collapse of a massive star can therefore have two qualitatively different outcomes: either a black hole forms or it does not.

Consider fine-tuning any one parameter in the initial data, for example the mass of the collapsing star, right to the threshold of black hole formation. At the black hole threshold, interesting things happen:

Mathematically, these phenomena (universality and scaling) are closely related to critical phase transitions in thermodynamics. The critical exponents can be calculated renormalisation methods, but uniquely in general relativity the action of the renormalisation group is also a time evolution.

Fine-tuning generic initial data, for generic systems, to the black hole threshold thus allows one to make arbitrarily small black holes, and this means arbitrarily large spacetime curvature outside the black hole. In the limit of perfect fine-tuning, a naked singularity is created from regular initial data: classical relativity then breaks down (and presumably quantum gravity takes over).

These numerical results have forced us to refine the cosmic censorship conjecture, which states that naked singularities cannot be created in collapse from regular initial data for reasonable systems: we now have to emphasise that this cannot happen for generic initial data, because we now know that it does happen for a codimension-1 set of smooth initial data, namely those which have been fine-tuned to the black hole threshold. (See the bet between John Preskill, Stephen Hawking and Kip Thorne.)

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Current research

Beyond spherical symmetry

Black holes have angular momentum as well as mass (but no other properties), and we would like to know how that scales at the black hole threshold. On the other hand, angular momentum resists collapse. This suggests that the mathematical similarity with critical phase transitions can be extended much further and gives rise to new predictions: C. Gundlach, Critical collapse with angular momentum: from critical exponents to scaling functions, Phys. Rev. D 65, 64019 (2002).

This has motivated my work with Thomas Baumgarte on axisymmetric fluid collapse with angular momentum, starting with Phys. Rev. Lett. 116, 22103 (2016), and continuing.

Next to scalar fields and perfect fluids, electromagnetic fields are also well-motivated and clean sources of curvature. But electromagnetic waves cannot be spherically symmetric, so, as with angular momentum, one must relax the symmetry at least to axisymmetry. I am working on this with Thomas Baumgarte and David Hilditch, starting with Phys. Rev. Lett. 123, 171103 (2019), and continuing.

Vacuum critical collapse

However, the ultimate goal is the formation of naked singularities in vacuum gravitational collapse, that is the implosion of ultr-strong gravitational waves under their own weight. This is unlikely to happen in nature, but puts the question really sharply: does classical general relativity break down for a codimension-one set of initial data, because a naked singularity develops? In vacuum one cannot blame the matter for this, so it is the cleanest system in which to ask this question.

It turns out that vacuum gravity is also the most difficult system in which to investigate the threshold of collapse, and hence potential naked singularities. Again, gravitational waves cannot be spherically symmetric, and one needs to relax at least to axisymmetry. But it turns out that numerical simulations are much harder in axisymmetric vacuum collapse than in axisymmetric collapse with matter, and we are still at an early stage here.

However, the phenoma we see are already quite complex. There are many indications of curvature scaling (and hence at the creation of naked singularities), but there indications both for and against this happening through (discrete) self-similarity, and universality is also currently unclear.

My own current approach is to use time evolutions in null coordinates. These have been used a lot in spherical symmmetry, but not so much beyond. They have a number of advantages:

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