Beverly K. Berger

Global Foliations of Vacuum Spacetimes withT2Isometry

Abstract We prove a global existence theorem (with respect to a geometrically defined time) for globally hyperbolic solutions of the vacuum Einstein equations which admit a T 2 isometry group with two-dimensional spacelike orbits, acting on T 3 spacelike surfaces.

THE SINGULARITY IN GENERIC GRAVITATIONAL COLLAPSE IS SPACELIKE, LOCAL AND OSCILLATORY

A longstanding conjecture by Belinskii, Khalatnikov and Lifshitz that the singularity in generic gravitational collapse is spacelike, local and oscillatory is explored analytically and numerically in spatially inhomogeneous cosmological space–times. With a convenient choice of variables, it can be seen analytically how nonlinear terms in Einsteins equations control the approach to the singularity and cause oscillatory behavior. The analytic picture requires the drastic assumption that each spatial point evolves toward the singularity as an independent spatially homogeneous universe. In every case, detailed numerical simulations of the full Einstein evolution equations support this assumption.

MIXMASTER BEHAVIOR IN INHOMOGENEOUS COSMOLOGICAL SPACETIMES

Numerical investigation of a class of inhomogeneous cosmological spacetimes shows evidence that at a generic point in space the evolution toward the initial singularity is asymptotically that of a spatially homogeneous spacetime with Mixmaster behavior. This supports a long-standing conjecture due to Belinskii et al. on the nature of the generic singularity in Einsteins equations.

New algorithm for Mixmaster dynamics

We present a new numerical algorithm for evolving the Mixmaster spacetimes. By using symplectic integration techniques to take advantage of the exact Taub solution for the scattering between asymptotic Kasner regimes, we evolve these spacetimes with higher accuracy using much larger time steps than previously possible. The longer Mixmaster evolution thus allowed a detailed comparison with the Belinskii - Khalatnikov - Lifshitz (BKL) approximate Mixmaster dynamics. In particular, we show that errors between the BKL prediction and the measured parameters early in the simulation can be eliminated by relaxing the BKL assumptions to yield an improved map. The improved map has different predictions for vacuum Bianchi type IX and magnetic Bianchi type Mixmaster models which are clearly matched in the simulation.

Evidence for an oscillary singularity in generic U(1) symmetric cosmologies on T3 times R

A longstanding conjecture by Belinskii, Lifshitz, and Khalatnikov that the singularity in generic gravitational collapse is locally oscillatory is tested numerically in vacuum, U(1) symmetric cosmological spacetimes on

General Relativity and Gravitation: A Centennial Perspective

T^3 \times R

Exact U ( 1 ) symmetric cosmologies with local mixmaster dynamics

. If the velocity term dominated (VTD) solution to Einsteins equations is substituted into the Hamiltonian for the full Einstein evolution equations, one term is found to grow exponentially. This generates a prediction that oscillatory behavior involving this term and another (which the VTD solution causes to decay exponentially) should be observed in the approach to the singularity. Numerical simulations strongly support this prediction.

Numerical evidence that the singularity in polarized U(1) symmetric cosmologies on T3 times R is velocity dominated

Part I. Einsteins Triumph: 1. 100 years of general relativity George F. R. Ellis 2. Was Einstein right? Clifford M. Will 3. Cosmology David Wands, Misao Sasaki, Eiichiro Komatsu, Roy Maartens and Malcolm A. H. MacCallum 4. Relativistic astrophysics Peter Schneider, Ramesh Narayan, Jeffrey E. McClintock, Peter Meszaros and Martin J. Rees Part II. New Window on the Universe: 5. Receiving gravitational waves Beverly K. Berger, Karsten Danzmann, Gabriela Gonzalez, Andrea Lommen, Guido Mueller, Albrecht Rudiger and William Joseph Weber 6. Sources of gravitational waves. Theory and observations Alessandra Buonanno and B. S. Sathyaprakash Part III. Gravity is Geometry, After All: 7. Probing strong field gravity through numerical simulations Frans Pretorius, Matthew W. Choptuik and Luis Lehner 8. The initial value problem of general relativity and its implications Gregory J. Galloway, Pengzi Miao and Richard Schoen 9. Global behavior of solutions to Einsteins equations Stefanos Aretakis, James Isenberg, Vincent Moncrief and Igor Rodnianski Part IV. Beyond Einstein: 10. Quantum fields in curved space-times Stefan Hollands and Robert M. Wald 11. From general relativity to quantum gravity Abhay Ashtekar, Martin Reuter and Carlo Rovelli 12. Quantum gravity via unification Henriette Elvang and Gary T. Horowitz.

Numerical study of initially expanding mixmaster universes

By applying a standard solution generating technique, we transform an arbitrary vacuum Mixmaster solution on

Influence of scalar fields on the approach to a cosmological singularity

{S}^{3}\ifmmode\times\else\texttimes\fi{}\mathbf{R}