Abstract
Classical chaos is often characterized as exponential divergence of nearby trajectories. In many interesting cases these trajectories can be identified with geodesic curves. We define here the entropy by with being the distance between two nearby geodesics. We derive an equation for the entropy, which by transformation to a Riccati-type equation becomes similar to the Jacobi equation. We further show that the geodesic equation for a null geodesic in a double-warped spacetime leads to the same entropy equation. By applying a Robertson–Walker metric for a flat three-dimensional Euclidean space expanding as a function of time, we again reach the entropy equation stressing the connection between the chosen entropy measure and time. We finally turn to the Raychaudhuri equation for expansion, which also is a Riccati equation similar to the transformed entropy equation. Those Riccati-type equations have solutions of the same form as the Jacobi equation. The Raychaudhuri equation can be transformed to a harmonic oscillator equation, and it has been shown that the geodesic deviation equation of Jacobi is essentially equivalent to that of a harmonic oscillator. The Raychaudhuri equations are strong geometrical tools in the study of general relativity and cosmology. We suggest a refined entropy measure applicable in cosmology and defined by the average deviation of the geodesics in a congruence.
Highlights
Classical chaos is generally defined as exponential divergence of nearby trajectories causing instability of the orbits with respect to initial conditions or quite as high sensitivity to initial conditions
We provide a relation between the Jacobi equation, the entropy and the geodesic equation itself
It is noteworthy that a resemblance between the geodesic equation and the entropy equation is obtained by inserting the null condition into the time part of the geodesic equation and not the spatial part, which underlines the connection between the present definition of entropy with time rather than space
Summary
Classical chaos is generally defined as exponential divergence of nearby trajectories causing instability of the orbits with respect to initial conditions or quite as high sensitivity to initial conditions. The extent of divergence is quantified in terms of Lyapunov exponents measuring the mean rate of exponential separation of neighboring trajectories. The Kolmogorov entropy is related to the Lyapunov exponents. It gives a measure of the amount of information lost or gained by the system as it evolves [1]. Casartelli et al [2] argued that this quantity is deeply related to the Kolmogorov entropy and exhibits strong sensitivity to the initial conditions. We provide a relation between the Jacobi equation, the entropy (as defined above) and the geodesic equation itself. For a flat space K = 0, and Equation (3) takes the form: S(τ ) + (S( ̇ τ ))2 = 0,. In particular if Γρσ vanishes for ρ not equal to σ
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