Deterministic Chaos in General Relativity

✍ Scribed by David Hobill (auth.), David Hobill, Adrian Burd, Alan Coley (eds.)

• Publisher: Springer US
• Year: 1994
• Tongue: English
• Leaves: 472
• Series: NATO ASI Series 332
• Edition: 1
• Category: Library

✦ Synopsis

Nonlinear dynamical systems play an important role in a number of disciplines. The physical, biological, economic and even sociological worlds are comprised of com­ plex nonlinear systems that cannot be broken down into the behavior of their con­ stituents and then reassembled to form the whole. The lack of a superposition principle in such systems has challenged researchers to use a variety of analytic and numerical methods in attempts to understand the interesting nonlinear interactions that occur in the World around us. General relativity is a nonlinear dynamical theory par excellence. Only recently has the nonlinear evolution of the gravitational field described by the theory been tackled through the use of methods used in other disciplines to study the importance of time dependent nonlinearities. The complexity of the equations of general relativity has been (and still remains) a major hurdle in the formulation of concrete mathematical concepts. In the past the imposition of a high degree of symmetry has allowed the construction of exact solutions to the Einstein equations. However, most of those solutions are nonphysical and of those that do have a physical significance, many are often highly idealized or time independent.

✦ Table of Contents

Front Matter....Pages i-xi
A Brief Review of “Deterministic Chaos in General Relativity”....Pages 1-16
Front Matter....Pages 17-17
Introduction to Dynamical Systems....Pages 19-61
A Short Course on Chaotic Hamiltonian Systems....Pages 63-88
Geometry of Perturbation Theory....Pages 89-101
Between Integrability and Chaos....Pages 103-105
On Defining Chaos in the Absence of Time....Pages 107-112
On the Dynamics of Generators of Cauchy Horizons....Pages 113-125
Front Matter....Pages 127-127
Chaos in the Case of Two Fixed Black Holes....Pages 129-144
Particle Motion Around Perturbed Black Holes: The Onset of Chaos....Pages 145-154
Critical Behaviour in Scalar Field Collapse....Pages 155-175
Front Matter....Pages 177-177
Relativistic Cosmologies....Pages 179-201
Homoclinic Chaos in Relativistic Cosmology....Pages 203-235
Mixing Properties of Compact K = −1 FLRW Models....Pages 237-250
Classical and Quantum Chaos in Robertson-Walker Cosmologies....Pages 251-268
Relativistic Fractal Cosmologies....Pages 269-296
Self-Similar Asymptotic Solutions of Einstein’s Equations....Pages 297-306
Nonlinearly Interacting Gravitational Waves in the Gowdy T 3 Cosmology....Pages 307-313
Front Matter....Pages 315-315
The Mixmaster Cosmological Metrics....Pages 317-328
The Belinskii-Khalatnikov-Lifshitz Discrete Evolution as an Approximation to Mixmaster Dynamics....Pages 329-344
How Can You Tell if the Bianchi IX Models Are Chaotic?....Pages 345-354
Front Matter....Pages 315-315
The Chaoticity of the Bianchi IX Cosmological Model....Pages 355-358
Chaos in the Einstein Equations — Characterization and Importance?....Pages 359-422
Integrability of the Mixmaster Universe....Pages 423-432
Continuous Time Dynamics and Iterative Maps of Ellis-MacCallum-Wainwright Variables....Pages 433-448
A Dynamical Systems Approach to the Oscillatory Singularity in Bianchi Cosmologies....Pages 449-462
Back Matter....Pages 463-472

✦ Subjects

Theoretical, Mathematical and Computational Physics