Derivation of the Raychaudhuri Equation

As a homage to A K Raychaudhuri, I derive in a straightforward way his famous equation and also indicate the problems he was last engaged in. PACS numbers: 04.20.Jb, 04.2.Cv, 98.80 Dr Let us consider a collection of bodies falling freely under their own gravity. They would all attract each other and would tend to converge. If they are expanding or contracting and their gravitational potential energy is greater in magnitude than the kinetic energy, they would all meet in future while if they are expanding with velocity greater than a critical (escape) value, it could be inferred by extrapolation back in time that they would have all been together at a point in the past. In general we can have a distribution of matter in any form, yet the same result will be expected. At the point of convergence, there will be divergence of density. That is what will characterize singularity signaling breakdown of the gravitational theory there. Following this Newtonian argument, we could say that if matter distribution is homogeneous and isotropic (all points and all directions in space are equivalent and there is no way to distinguish one from the other), gravitational force at any point will vanish. Isotropy of space demands that in any direction there will be equal and opposite force and hence its sum will vanish because force is a vector quantity. Homogeneity will ask for this to happen at all points. This means in a perfectly homogeneous and isotropic Universe, there is no gravitational force on any body. That is gravity is completely annulled