Deterministic Chaos in General Relativity
David Hobill, Adrian Burd, A.A. Coley
Springer US, Sep 30, 1994 - Science - 472 pages
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.
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anisotropy approach approximation asymptotic attractor Belinskii Bianchi IX model Bianchi Type black hole bounce Cauchy horizons chaos chaotic behavior classical compact computations consider constant coordinate corresponding cosmological models cosmology curvature curve defined density described differential equations discussed dynamical system Einstein equations Einstein field equations equilibrium point evolution example field equations Figure finite fractal Friedmann function gauge geodesic flow given global gravitational collapse Hamiltonian system Hobill homoclinic hyperbolic infinite inhomogeneous initial conditions integrable invariant Kasner epochs Kasner ring Khalatnikov Lifshitz linear Lyapunov exponent manifold Math Misner Mixmaster Universe non-linear numerical observed oscillations parameter particle periodic orbits perturbation phase space Phys physical plane Poincare disc potential walls Quantum Grav relativistic Relativity S.E. Rugh scalar field scale self-similar separatrix sequence singularity solution spatially homogeneous structure symmetry Taub tensor Theorem theory topological trajectory Universe variables vector field zero