14 research outputs found
Cosmic String and Black Hole Limits of Toroidal Vlasov Bodies in General Relativity
We numerically investigate limits of a two-parameter family of stationary
solutions to the Einstein-Vlasov system. The solutions are toroidal and have
non-vanishing angular momentum. As one tunes to more relativistic solutions
(measured for example by an increasing redshift) there exists a sequence of
solutions which approaches the extreme Kerr black hole family. Solutions with
angular momentum larger than the square of the mass are also investigated, and
in the relativistic limit the near-field geometry of such solutions is observed
to become conical in the sense that there is a deficit angle. Such solutions
may provide self-consistent models for rotating circular cosmic strings.Comment: 13 page
Stability of AVTD Behavior within the Polarized -symmetric vacuum spacetimes
We prove stability of the family of Kasner solutions within the class of
polarized -symmetric solutions of the vacuum Einstein equations in the
contracting time direction with respect to an areal time foliation. All Kasner
solutions for which the asymptotic velocity parameter satisfies
are non-linearly stable, and all sufficiently small perturbations exhibit
asymptotically velocity term dominated (AVTD) behavior and blow-up of the
Kretschmann scalar.Comment: Update references and clarify discussion. Results and conclusions are
the sam
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Quasilinear hyperbolic Fuchsian systems and AVTD behavior in T2-symmetric vacuum spacetimes
We set up the singular initial value problem for quasilinear hyperbolic
Fuchsian systems of first order and establish an existence and uniqueness
theory for this problem with smooth data and smooth coefficients (and with even
lower regularity). We apply this theory in order to show the existence of
smooth (generally not analytic) T2-symmetric solutions to the vacuum Einstein
equations, which exhibit AVTD (asymptotically velocity term dominated) behavior
in the neighborhood of their singularities and are polarized or half-polarized.Comment: 78 page
Singular Symmetric Hyperbolic Systems and Cosmological Solutions to the Einstein Equations
Characterizing the long-time behavior of solutions to the Einstein field equations remains an active area of research today. In certain types of coordinates the Einstein equations form a coupled system of quasilinear wave equations. The investigation of the nature and properties of solutions to these equations lies in the field of geometric analysis. We make several contributions to the study of solution dynamics near singularities. While singularities are known to occur quite generally in solutions to the Einstein equations, the singular behavior of solutions is not well-understood. A valuable tool in this program has been to prove the existence of families of solutions which are so-called asymptotically velocity term dominated (AVTD). It turns out that a method, known as the Fuchsian method, is well-suited to proving the existence of families of such solutions. We formulate and prove a Fuchsian-type theorem for a class of quasilinear hyperbolic partial differential equations and show that the Einstein equations can be formulated as such a Fuchsian system in certain gauges, notably wave gauges. This formulation of Einstein equations provides a convenient general framework with which to study solutions within particular symmetry classes. The theorem mentioned above is applied to the class of solutions with two spatial symmetries -- both in the polarized and in the Gowdy cases -- in order to prove the existence of families of AVTD solutions. In the polarized case we find families of solutions in the smooth and Sobolev regularity classes in the areal gauge. In the Gowdy case we find a family of wave gauges, which contain the areal gauge, such that there exists a family of smooth AVTD solutions in each gauge
Computational Models of Galaxies in Kinetic Theory
In this research we model the distribution of mass in simulated galaxies by solving the Vlasov-Poisson system of equations. We\u27ve recently expanded our simulations to include multiple species of matter. This allows us to visualize the individual spatial density distributions of, for example, stars and dark matter as well as the joint gravitational potential. We have developed a library ofcomputational tools to allow us to investigate a number of the physical properties of these galaxies. In future work we will use these tools to compare the characteristics of our model galaxies to those of observed galaxies.https://digitalcommons.humboldt.edu/ideafest_posters/1266/thumbnail.jp
On axisymmetric and stationary solutions of the self-gravitating Vlasov system
Axisymmetric and stationary solutions are constructed to the Einstein-Vlasov and Vlasov-Poisson systems. These solutions are constructed numerically, using finite element methods and a fixed-point iteration in which the total mass is fixed at each step. A variety of axisymmetric stationary solutions are exhibited, including solutions with toroidal, disk-like, spindle-like, and composite spatial density configurations, as are solutions with non-vanishing net angular momentum. In the case of toroidal solutions, we show for the first time, solutions of the Einstein-Vlasov system which contain ergoregions