Weinstein Gilbert, Strauss Yosef, Bondarenko Sergey, Yahalom Asher, Lewkowicz Meir, Horwitz Lawrence Paul, Levitan Jacob
Physics Department, Ariel University, Ariel 40700, Israel.
Department of Mathematics, Ariel University, Ariel 40700, Israel.
Entropy (Basel). 2017 Dec 25;20(1):6. doi: 10.3390/e20010006.
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 S = ln χ ( x ) with χ ( x ) 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.
经典混沌通常被描述为附近轨迹的指数发散。在许多有趣的情况下，这些轨迹可以用测地线来识别。我们在此通过(S = \ln\chi(x))定义熵，其中(\chi(x))是两条附近测地线之间的距离。我们推导了一个熵的方程，通过变换为里卡蒂型方程后，它变得类似于雅可比方程。我们进一步表明，双扭曲时空中零测地线的测地线方程会导致相同的熵方程。通过对作为时间函数而膨胀的平坦三维欧几里得空间应用罗伯逊 - 沃克度规，我们再次得到了熵方程，强调了所选熵度量与时间之间的联系。我们最后转向用于膨胀的瑞查德符里方程，它也是一个类似于变换后的熵方程的里卡蒂方程。那些里卡蒂型方程具有与雅可比方程相同形式的解。瑞查德符里方程可以变换为一个谐振子方程，并且已经表明雅可比的测地线偏差方程本质上等同于一个谐振子的方程。瑞查德符里方程是广义相对论和宇宙学研究中的强大几何工具。我们提出一种适用于宇宙学的精细熵度量，它由共动中测地线的平均偏差定义。
