Gravitational Collapse with Distributed AMR
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Axisymmetric Gravitational Collapse
Assuming rotational symmetry about the z axis, we model the collapse of a scalar field and Brill vacuum collapse. With such a model, we can model the head-on collision of two black holes as well as black hole critical phenomena. We have implemented black hole excision and have begun adding adaptive mesh refinement (AMR). Such models are needed to predict gravitational wave forms for comparison to the highly anticipated results of LIGO scheduled to come online soon. We also consider the validity of the Hoop Conjecture and other cosmological issues related to Cosmic Censorship.
Black Hole Critical Phenomena
Fascinating phenomena found by Choptuik in the early 1990s occurs at the threshold of black hole formation. Imagine the ability to form black holes at will by squeezing together various amounts of energy. With lots of energy you form a big black hole. With smaller amounts, you form smaller black holes. Keep decreasing this amount to see how small you can make a black hole. This is the threshold for black hole formation, and in this region of parameter space phenomena such as power-law scaling, universality, and self-similarity occurs.
Computational Methods
One of the ultimate goals of numerical relativity is the ability to model the fully nonlinear Einstein equations with no assumed symmetry (i.e. in three dimensions). Attainment of that goal involves overcoming various obstacles, one of which is the computational power necessary.
Adaptive mesh refinement (AMR)
dynamically creates fine resolution sub-grids which overlay the coarse
grid and provide resolution as neeeded. I have begun development of
AMR in both 2 and 3 dimensions.
Many of these computations are run on expensive supercomputers. However, recently the idea of harnassing the power of a large number of cheap personal computers has proved productive. The challenge though is to be able to distribute your computational problem across these various computers in what is called a cluster. MPI is a tool which allows programmers to distribute their problem and achieve the scalable power. I have begun a model in 3 dimensions using MPI.
Critical Collapse without gravity.
Recent work has found a type of critical behavior manifest at the threshold of singular collapse in the nonlinear sigma model. Finding critical behavior without gravity suggests that many other models might have such interesting phenomena.
Global Monopoles and Topological Inflation
Inflation has been around a while and it solves a number of cosmological problems. However, it also introduces a few of its own, the primary one being that conditions in the Universe have to be tuned precisely to produce what we see today. However, inflation can occur inside a monopole with no tuning! This type of inflation is called topological inflation.
Textures and Defect Dynamics
Topological defects have relevance in cosmology as sources of structure formation, as well as condensed matter systems, particle physics, and the mathematics of partial differential equations. Computational methods provide an avenue for studying the dynamics of such defects. In particular, I have been able to demonstrate under what conditions the collapse of a texture (a particular type of defect) nucleates a pair of monopoles (another type of defect).
A monopole-antimonopole pair about to anhilate.