The Raychaudhuri equation is derived by assuming geometric flow in space–time M of $n+1$ dimensions. The equation turns into a harmonic oscillator form under suitable transformations. Thereby, a relation between geometrical entropy and mean geodesic deviation is established. This has a connection to chaos theory where the trajectories diverge exponentially. We discuss its application to cosmology and black holes. Thus, we establish a connection between chaos theory and general relativity.

In this paper we review the fundamental concepts of entropy bounds put forward by Bousso and its relation to the holographic principle. We relate covariant entropy with logarithmic distance of separation of nearby geodesics. We also give sufficient arguments to show that the origin of entropy bounds is not indeed thermodynamic, but statistical.

The energy of a particle in the Rainbow Rindler metric is derived using Hamilton's variational principle. Saha Ionization equation and Pair production in Rainbow Rindler metric have been investigated. Saha ionization equation and Pair production in Rainbow Rindler metric depend on corrected acceleration which is energy-dependent. This means energy ($E$) and metric tensor ($g_{\mu \nu}(E)$) depends on the inertial observer. The physical implications of energy-dependent acceleration, observer-dependent(inertial) energy and metric tensor are discussed. On comparing the above results with that of Rindler metric we get the result obtained by Sanchari De and Somenath Chakarabarthy by applying the correspondence principle.

We studied the Raychaudhuri equation in Kaluza-Klein space-time. We derived an additional term that is solely due to Kaluza-Klein's unification. This term affects the defocus of world lines near the singularity of charged higher-dimensional black holes. We illustrated it with a static higher-dimensional black hole generated by Einstein-Maxwell equations. It has been found that the term derived modifies the repulsive effect of kaluza Klein scalar on world lines. We proved some mathematical theorems which give an insight into the interior of the black holes.