Comment on the chaotic singularity in some magnetic Bianchi VI0_0 cosmologies

Description of the magnetic Bianchi VI$_0$ cosmologies of LeBlanc, Kerr, and Wainwright in the formalisms both of Belinskii, Khalatnikov, and Lifshitz, and of Misner allows qualitative understanding of the Mixmaster-like singularity in those models.Comment: 8 pages, Revtex, no figure

Report on A5. Computer Methods

Session A5 on numerical methods contained talks on colliding black holes, critical phenomena, investigation of singularities, and computer algebra.Comment: 9 pages, for GR16 proceeding

Influence of scalar fields on the approach to a cosmological singularity

The method of consistent potentials is used to explain how a minimally coupled (classical) scalar field can suppress Mixmaster oscillations in the approach to the singularity of generic cosmological spacetimes.Comment: 8 pages includes 5 figures. Uses revtex, psfi

Approach to the Singularity in Spatially Inhomogeneous Cosmologies

A combination of analytic and numerical methods has yielded a clear understanding of the approach to the singularity in spatially inhomogeneous cosmologies. Strong support is found for the longstanding claim by Belinskii, Khalatnikov, and Lifshitz that the collapse is dominated by local Kasner or Mixmaster behavior. The Method of Consistent Potentials is used to establish the consistency of asymptotic velocity term dominance (AVTD) (local Kasner behavior) in that no terms in Einstein's equations will grow exponentially when the VTD solution, obtained by neglecting all terms containing spatial derivatives, is substituted into the full equations. When the VTD solution is inconsistent, the exponential terms act dynamically as potentials either to drive the system into a consistent AVTD regime or to maintain an oscillatory approach to the singularity.Comment: 17 pages, in Differential Equations and Mathematical Physics : Proceedings of an International Conference Held at the University of Alabama in Birmingham, ed. by R. Weikard, G. Weinstein (American Mathematical Society, 2000

Numerical Investigation of Cosmological Singularities

We describe a numerical approach to address the BKL conjecture that the generic cosmological singularity is locally Mixmaster-like. We consider application of a symplectic PDE solver to three models of increasing complexity--spatially homogeneous (vacuum) Mixmaster cosmologies where we compare the symplectic ODE solver to a Runge-Kutta one, the (plane symmetric, vacuum) Gowdy universe on $T^3 \times R$ whose dynamical degrees of freedom satisfy nonlinearly coupled PDE's in one spatial dimension and time, and U(1) symmetric, vacuum cosmologies on $T^3 \times R$ which are the simplest spatially inhomogeneous universes in which local Mixmaster dynamics is allowed.Comment: Based on lectures given at WE-Heraeus-Seminar on Relativity and Scientific Computing. 24 pages, Latex, 9 figures in separate file BHfigs.uu, uses psfig.te

Numerical Investigation of Singularities

Numerical exploration of the properties of singularities could, in principle, yield detailed understanding of their nature in physically realistic cases. Examples of numerical investigations into the formation of naked singularities, critical behavior in collapse, passage through the Cauchy horizon, chaos of the Mixmaster singularity, and singularities in spatially inhomogeneous cosmololgies are discussed.Comment: Based on GR14 talk. 21 pages, Latex, 5 figures in separate file gr14.uu. Uses sprocl.sty, psfig.st

On the Nature of the Generic Big Bang

Spatially homogeneous but possibly anisotropic cosmologies have two main types of singularities: (1) asymptotically velocity term dominated (AVTD) - (reversing the time direction) the universe evolves to the singularity with fixed anisotropic collapse rates ; (2) Mixmaster-the anisotropic collapse rates change in a deterministicaly chaotic way. Much less is known about spatially inhomogeneous universes. It has been claimed that a generic universe would evolve toward the singularity as a different Mixmaster universe at each spatial point. I shall discuss how to predict whether a cosmology has an AVTD or Mixmaster singularity and whether or not our numerical simulations agree with these predictions.Comment: 7 pages, 5 figures, uses Revtex, epsf. Based on talk given at Symposium on Frontiers of Fundamental Physics, Hyderabad, India, 11-12 Dec 199

Signature for local Mixmaster dynamics in U(1) symmetric cosmologies

Previous studies \cite{berger98a} have provided strong support for a local, oscillatory approach to the singularity in U(1) symmetric, spatially inhomogeneous vacuum cosmologies on $T^3 \times R$. The description of a vacuum Bianchi type IX, spatially homogeneous Mixmaster cosmology (on $S^3 \times R$) in terms of the variables used to describe the U(1) symmetric cosmologies indicates that the oscillations in the latter are in fact those of local Mixmaster dynamics. One of the variables of the U(1) symmetric models increases only at the end of a Mixmaster era. Such an increase therefore yields a qualitative signature for local Mixmaster dynamics in spatially inhomogeneous cosmologies.Comment: 13 pages, uses RevTex, psfi

Exact U(1) symmetric cosmologies with local Mixmaster dynamics

By applying a standard solution generating technique, we transform an arbitrary vacuum Mixmaster solution on $S^3 \times {\bf R}$ to a new solution which is spatially inhomogeneous. We thereby obtain a family of exact, spatially inhomogeneous, vacuum spacetimes which exhibit Belinskii, Khalatnikov, and Lifshitz (BKL) oscillatory behavior. The solutions are constructed explicitly by performing the transformations on numerically generated, homogeneous Mixmaster solutions. Their behavior is found to be qualitatively like that seen in previous numerical simulations of generic U(1) symmetric cosmological spacetimes on $T^3 \times {\bf R}$.Comment: 13 pages, 5 figures, uses RevTeX and psfi

Stability Within T2T^2-Symmetric Expanding Spacetimes

We prove a nonpolarised analogue of the asymptotic characterization of $T^2$-symmetric Einstein Flow solutions completed recently by LeFloch and Smulevici. In this work, we impose a condition weaker than polarisation and so our result applies to a larger class. We obtain similar rates of decay for the normalized energy and associated quantities for this class. We describe numerical simulations which indicate that there is a locally attractive set for $T^2$-symmetric solutions not covered by our main theorem. This local attractor is distinct from the local attractor in our main theorem, thereby indicating that the polarised asymptotics are unstable.Comment: 19 pages, 5 figure