Critical phenomena in gravitational collapse

Bourg, Patrick (2022) Critical phenomena in gravitational collapse. University of Southampton, Doctoral Thesis, 244pp.

Record type: Thesis (Doctoral)

Abstract

This thesis’ main aim is the study of the critical collapse of an ultrarelativistic perfect fluid in axisymmetry in 2+ 1 dimensions. The study of critical collapse in this dimension is easier than in higher dimensions, notably because, even in axisymmetry, all variables only depend on time and radius. The main downside, however, is that studying critical collapse in 2 + 1 dimensions necessitates the presence of a negative cosmological constant Λ. To set the stage, we review the relevant literature concerning critical collapse. We also thoroughly study the above system in 3 + 1 dimensions, restricted to spherical symmetry, both numerically and semi-analytically. Finally, we give a brief overview of anti-de Sitter space in any dimensions, as well as the rotating black-hole solution in 2 + 1 dimensions. We proceed by closely analysing all rigidly rotating perfect fluid solutions in 2 + 1 dimensions, with negative Λ and with general barotropic equation of state P = P(ρ). These solutions turn out to be of crucial importance to the critical collapse of the perfect fluid. With the restriction of a regular center and finite total mass M and angular momentum J (rotating stars), they form a two-parameter family of solutions. Not all pairs (M, J) can be realized as stars, but when they do, spacetime is asymptotically anti-de Sitter, without horizons or closed timelike curves. In order to perform our study, we develop a high-resolution shock-capturing formulation of the Einstein-perfect fluid system. In axisymmetry, in 2+1 dimensions, there are two conserved quantities. They correspond to the angular momentum J, and a generalization of the Kodama conserved mass M. The associated two conserved matter currents are used in the stress-energy conservation, so that M and J are conserved during the evolution. We demonstrate pointwise and L ² -norm convergence of the code in the test cases of generic dispersion and collapse, and stable and unstable rotating stars. We then carry out the study of the critical collapse of a perfect fluid, with linear equation of state P = κρ, first restricting to spherical symmetry. We find that the associated critical phenomena are quite different from their 3 + 1 counterparts: the critical solution is type I or II, depending on the value of κ. Furthermore, the type I critical solution is shown to be static, as expected, while the type II critical solution is surprisingly not self-similar. Instead, it is quasistatic, meaning that the solution shrinks adiabatically to zero size, going through the sequence of static solutions. Finally, we generalize the above study to rotating initial data. We still find the same demarcation separating type I and II phenomena. In the type I case, the critical solution is now stationary. In the type II case, the picture is more subtle: for small angular momenta, we find type II phenomena and the critical solution is quasi stationary. As the black-hole threshold is approached, the spin-to-mass ratio of the critical solution increases as it contracts, and hence so does that of the black hole created at the end. As extremality is approached, the contraction of the critical solution smoothly ends, so that the formation of extremal black holes is avoided.