Mathematics – Logic
Scientific paper
Oct 1999
adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=1999phdt.........4w&link_type=abstract
Thesis (PhD). UNIVERSITY OF OREGON, Source DAI-B 60/04, p. 1653, Oct 1999, 152 pages.
Mathematics
Logic
11
Singularity, Mixmaster, Gowdy
Scientific paper
This dissertation is a report of research on the properties of cosmological solutions to Einstein's equation near the singularity. In the first part of this work methods from dynamical systems analysis are used to prove that the presence of a magnetic field perpendicular to the two commuting Killing vector fields in spatially homogeneous solutions of Bianchi type VI0 changes the dynamics from asymptotically velocity term dominated to oscillatory. In particular, it is shown that the α-limit set (for the time direction that puts the singularity in the past) of any of these magnetic solutions contains at least two Kasner points. Previous work by others provides evidence that these spacetimes are not only oscillatory but indeed mixmaster. However, this has never been proven. In the second part of this work evidence for mixmaster behavior in a class of solutions which is a spatially inhomogeneous generalization of locally homogeneous magnetic Bianchi VI0 is presented. In this generalization spatial variation is allowed in the direction of the magnetic field. These spacetimes are much like the Gowdy models, except for the presence of the magnetic field and a different spatial topology. Numerical results combined with qualitative analysis show that the presence of the magnetic field changes the dynamics from asymptotically velocity term dominated to mixmaster. These are the first results that provide clear evidence of the occurrence of mixmaster evolution in spatially inhomogeneous solutions to Einstein's equation. The evolution toward the singularity at almost every spatial point asymptotically approaches the evolution of some magnetic Bianchi VI0 spacetime. The evolution at different spatial points approaches the evolution of different magnetic Bianchi VI0 spacetimes. This behavior is in agreement with the description by Belinskii, Khalatnikov and Lifshitz of the oscillatory approach to the singularity. Using the same methods to study the Gowdy models having T 3 spatial topology leads to the conjecture that the presence of a magnetic field perpendicular to the symmetry directions changes the dynamics from asymptotically velocity term dominated to mixmaster. Finally, indications of oscillatory behavior in the vacuum (no magnetic field) T 2 symmetric spacetimes with nonvanishing twist are obtained by these methods.
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