Entropy measures as geometrical tools in the study of cosmology
  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 $S = \ln \chi (x)$ with
$\chi(x)$ being the distance between two nearby geodesics. We derive an
equation for the entropy which by transformation to a Ricatti-type equation
becomes similar to the Jacobi equation. We further show that the geodesic
equation for a null geodesic in a double warped space time leads to the same
entropy equation. By applying a Robertson-Walker metric for a flat
three-dimensional Euclidian 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 Ricatti equation similar to the transformed entropy equation.
Those Ricatti-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.
