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Unformatted text preview: Chaotic Spacetime: Fractal Sets in the Mixmaster Universe Steve Flammia (Dated: May 20, 2004) I. INTRODUCTION When a physical system is described by nonlinear equations of motion, the complex phenomenon known as chaos often results. By chaos it is meant that the systems dynamics are so complicated and so dependent on small uncertainties that any attempt at predicting the future behavior of the system is futile. Einstein’s theory of General Relativity is governed by a set of equations that are nonlinear, especially when spacetime is highly curved. Thus, one might expect that near singularities the dynamics of spacetime is chaotic. In this paper, we focus on a specific model known as the mixmaster universe and study the behavior near the singularity at the big bang. If one is to prove the existence of chaos in a General Relativistic setting, then one must deal with the additional constraint that the proof remain valid for all observers. Standard indicators of chaos such as the Lyapunov exponent or the metric entropy have been shown to be observer dependent , and are thus unsuitable for use in GR. Fractals, however, are fundamentally observer independent because they are selfaffine (selfsimilar). The existence of fractal sets in the phase space of the mixmaster universe would then conclusively prove that the dynamics are chaotic. The paper is organized as follows: in section II we describe the mixmaster universe and the discretetime approximation of the dynamics; in section IIIA we prove that the set of fixed points for the discrete approximation is a repeller; in section IV we prove that the repeller is a fractal set, and thus the mixmaster universe is chaotic; in section V we discuss the emergence of a new fractal set in the discrete dynamics that (to the best of the author’s knowledge) has not yet been discussed in the literature. We conclude in section VI. All though no formal citations are given during the course of the paper, a brief discussion of the literature that was consulted can be found in section VII. In Appendix A, the Mathematica code used to calculate all of the figures and quantities presented in this paper is listed. The figures are placed at the end of the paper. II. THE MIXMASTER UNIVERSE The metric for the mixmaster universe is g = a 2 dx 2 + b 2 dy 2 + c 2 dz 2 dt 2 , (1) where the factors a,b,c are scaling factors which depend only on time. Thus, the mixmas ter universe is a homogeneous universe ( a,b,c independent of x,y,z ), but it is in general anisotropic. If we let a = b = c , we recover the FriedmanRobertsonWalker cosmology, which is of course isotropic. The vacuum Einstein equations lead directly to the following equations for the scaling parameters: 2(ln a ) 00 = ( b 2 c 2 ) 2 a 4 , cyclic( a,b,c ) (2) 2 where the prime denotes differentiation with respect to a time τ which is related to the cosmic time by dt = abcdτ ....
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This note was uploaded on 02/07/2011 for the course PHYS 101 taught by Professor Aster during the Spring '11 term at East Tennessee State University.
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