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Scientific paper
Dec 2002
adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=2002phdt........50v&link_type=abstract
ProQuest Dissertations And Theses; Thesis (Ph.D.)--The University of Texas at Austin, 2002.; Publication Number: AAT3110697; ISB
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1
Scientific paper
This dissertation presents the numerical solution and investigation of spin-½ fields coupled to gravity (Einstein-Dirac system). The primary focus is on the behavior at the threshold of black hole formation. A spherically symmetric system of massive spin-½ fields in a singlet spinor state is studied and shown to exhibit both unstable and stable solutions. The unstable solutions correspond to the threshold between black hole collapse and dispersal. There are a continuum of stable solutions that are solitonic in nature. These tended to oscillate and approach the stable, static solutions that we found through independent techniques. A spherically symmetric system is constructed from massless spin-½ fields by using spinor harmonics for their angular part and taking an incoherent sum of their individual stress tensors. The result is a spherically symmetric system with no net spin-angular momentum. Instead the system feels the effect of a "spin-angular momentum barrier." The strength of the barrier is controlled by the spin-angular momentum quantum number, l. The lowest value of l = ½ corresponds to two counter-rotating shells. In this case, black hole formation occurs at infinitesimal mass (Type II). This new, continuously self-similar solution is found by solving the Einstein-massless-Dirac system of nonlinear partial differential equations. A self-similar ansatz is then taken which reduces the partial differential equations to a set of ordinary differential equations. These new equations are solved and the solution of the PDEs are shown to agree with the solution of the ODES. The Einstein-massless-Dirac system of PDEs is then solved for other values of l. As l is increased, the scaling exponent, lambda of the Type II solutions is shown to decrease. The final chapter describes a new two-dimensional, axisymmetric code which uses a combination of harmonic coordinates and Chebyshev pseudospectral collocation methods to solve Einstein's equations. This evolution code is a hybrid of finite-difference and spectral techniques---the temporal derivatives are approximated by finite-difference operators while the spatial derivatives are found using spectral methods. The constraint equations are solved using a purely spectral nonlinear elliptic solver which uses the Newton-Kantorovich method.
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