Deterministic Chaos in General Relativity

1. Front Matter

   Pages i-xi

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2. A Brief Review of “Deterministic Chaos in General Relativity”

   1. A Brief Review of “Deterministic Chaos in General Relativity”

      • David Hobill

      Pages 1-16

3. Mathematical Preliminaries

   1. Front Matter

      Pages 17-17

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   2. Introduction to Dynamical Systems

      • John Wainwright

      Pages 19-61

   3. A Short Course on Chaotic Hamiltonian Systems

      • Richard C. Churchill

      Pages 63-88

   4. Geometry of Perturbation Theory

      • Richard Cushman

      Pages 89-101

   5. Between Integrability and Chaos

      • David L. Rod, Richard C. Churchill

      Pages 103-105

   6. On Defining Chaos in the Absence of Time

      • Richard C. Churchill

      Pages 107-112

   7. On the Dynamics of Generators of Cauchy Horizons

      • P. T. Chruściel, J. Isenberg

      Pages 113-125

4. Compact Relativistic Systems

   1. Front Matter

      Pages 127-127

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   2. Chaos in the Case of Two Fixed Black Holes

      • G. Contopoulos

      Pages 129-144

   3. Particle Motion Around Perturbed Black Holes: The Onset of Chaos

      • Luca Bombelli

      Pages 145-154

   4. Critical Behaviour in Scalar Field Collapse

      • M. W. Choptuik

      Pages 155-175

5. Cosmological Systems

   1. Front Matter

      Pages 177-177

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   2. Relativistic Cosmologies

      • M. A. H. MacCallum

      Pages 179-201

   3. Homoclinic Chaos in Relativistic Cosmology

      • Esteban Calzetta

      Pages 203-235

   4. Mixing Properties of Compact K = −1 FLRW Models

      • George Ellis, Reza Tavakol

      Pages 237-250

   5. Classical and Quantum Chaos in Robertson-Walker Cosmologies

      • Roman Tomaschitz

      Pages 251-268

   6. Relativistic Fractal Cosmologies

      • Marcelo B. Ribeiro

      Pages 269-296

   7. Self-Similar Asymptotic Solutions of Einstein’s Equations

      • A. A. Coley, R. J. van den Hoogen

      Pages 297-306

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.

• chaos
• dynamical systems
• dynamische Systeme
• geometry
• numerical methods

• University of Calgary, Calgary, Canada

  David Hobill

• Dalhousie University, Halifax, Canada

  Adrian Burd, Alan Coley

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