Summary
A primary unresolved issue for cosmological singularities is whether or not their behavior is locally of the Mixmaster type [as conjectured by Belinskii, Khalatnikov, and Lifshitz (BKL)]. The Mixmaster dynamics first appears in spatially homogeneous cosmologies of Bianchi types VIII and IX. A multiple of the spatial scalar curvature acts as a closed potential leading, in the evolution toward the singularity (say τ → ∞), to an (almost certainly) infinite sequence of bounces whose parameters exhibit the sensitivity to the initial conditions usually associated with chaos. Other homogeneous cosmologies are characterized by open (or no) potentials leading to a last bounce as τ → ∞ . Such models are called asymptotically velocity term dominated (AVTD). Here we shall describe a numerical approach to address the BKL conjecture. Starting with a symplectic numerical method ideally suited to this problem, we shall consider application of the method to three models of increasing complexity. The first application is to spatially homogeneous (vacuum) Mixmaster cosmologies where we compare the symplectic ODE solver to a Runge-Kutta one. The second application is to the (plane symmetric, vacuum) Gowdy universe on T3 × R. The dynamical degrees of freedom satisfy nonlinearly coupled PDEs in one spatial dimension and time. We demonstrate support for conjectured AVTD behavior for this model and explain its observed nonlinear small-scale spatial structure. Finally, we study U(1) symmetric, vacuum cosmologies on T3 × R. These are the simplest spatially inhomogeneous universes in which local Mixmaster dynamics is allowed. The Gowdy code is easily generalized to this model, although the spatial differencing needed in the symplectic method is not trivial. For AVTD models, we expect the potential-like term in the Hamiltonian constraint to vanish as τ → ∞ while in local Mixmaster it becomes (locally) large from time to time. We show how the potential behaves for a variety of generic U(l) models.
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References
Fleck, J.A., Morris, J.R., Feit, M.D. (1976): Time-dependent propagation of high-energy laser-beams through atmosphere. Appl. Phys. 10, 129–160
Moncrief, V. (1983): Finite-difference approach to solving operator equations of motion in quantum theory. Phys. Rev. D 28, 2485–2490
Berger, B.K., Moncrief, V. (1993): Numerical investigation of cosmological singularities. Phys. Rev. D 48, 4676–4687
Blanco, S., Costa, A., Rosso, O.A. (195): Chaos in classical cosmology. II. Preprint University of Buenos Aires
Misner, C.W., Thorne, K.S., Wheeler, J.A. (1973): Gravitation. Freeman, San Francisco
Belinskii, V.A., Lifshitz, E.M., Khalatnikov, I.M. (1971): Oscillatory approach to the singular point in relativistic cosmology. Sov. Phys. Usp. 13, 745–765
Misner, C. W. (1969): Mixmaster universe. Phys. Rev. Lett. 22, 1071–1074
Ryan Jr., M.P., Shepley, L.C. (1975): Homogeneous relativistic cosmologies. Princeton University Press, Princeton
Chernoff, D.F., Barrow J.D. (1983): Chaos in the Mixmaster universe. Phys. Rev. Lett. 50, 134–137
Berger, B.K. (1994): How to determine approximate Mixmaster parameters from numerical evolution of Einstein’s equations. Phys. Rev. D 49, 1120–1123
Moser, A.R., Matzner, R.A., Ryan Jr., M.P. (1973): Numerical solutions for symmetric Bianchi IX universes. Ann. Phys. (N.Y.) 79, 558–579
Hobill, D., Burd, A., Coley, A. (eds.) (1994): Deterministic chaos in general relativity. Plenum, New York
Kasner, E. (1921): Geometrical theorems on Einstein’s cosmological equations. Am. J. Math. 43, 130
Gowdy, R.H. (1971): Gravitation waves in closed universes. Phys. Rev. Lett. 27, 826–829
Berger, B.K. (1974): Quantum graviton creation in a model universe. Ann. Phys. (N.Y.) 83, 458–490
Isenberg, J., Moncrief, V. (1990): Asymptotic behavior of the gravitational field and the nature of singularities in Gowdy spacetimes. Ann. Phys. (N.Y.) 199, 84–122
Grubišić, B., Moncrief, V. (1993): Asymptotic behavior of the T3 × R Gowdy space-times. Phys. Rev. D 47, 2371–2382
Moncrief, V. (1986): Reduction of Einstein’s equations for vacuum space-times with space-like U(1) isometry groups. Ann. Phys. (N.Y.) 167, 118–142
Suzuki, M. (1990): Fractal decomposition of exponential operators with applications to many-body theories and Monte Carlo simulations. Phys. Lett. A 146, 319–323
Berger, B.K. (1991): Comments on the computation of Liapunov exponents for the Mixmaster universe. Gen. Rel. Grav. 23, 1385–1402
Berger, B.K. (1990): Numerical study of initially expanding Mixmaster universes. Class. Quantum Grav. 7, 203–216
Press, W.H., Flannery, B.P., Teukolsky, S.A., Vetterling, W.T. (1992): Numerical recipes (second edition). Cambridge University Press, Cambridge
Garcia, A.L. (1994): Numerical methods for physicists. Prentice-Hall, Englewood Cliffs, NJ
Berger, B.K., Garfinkle, D., Strasser, E. (1995): A numerical study of Mixmaster dynamics in magnetic Bianchi V I. cosmologies. In preparation
Berger, B.K., Garfinkle, D.,Grubišić, B., Moncrief, V. (1995): Phenomenology of the Gowdy cosmology on T3 × R. In preparation
Grubišić, B., Moncrief, V. (1994): Mixmaster spacetime, Geroch’s transformation, and constants of motion. Phys. Rev. D 49, 2792–2800
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© 1996 Springer-Verlag Berlin Heidelberg
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Berger, B.K. (1996). Numerical Investigation of Cosmological Singularities. In: Hehl, F.W., Puntigam, R.A., Ruder, H. (eds) Relativity and Scientific Computing. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-95732-1_8
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DOI: https://doi.org/10.1007/978-3-642-95732-1_8
Publisher Name: Springer, Berlin, Heidelberg
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