Space-Time Singularities and Raychaudhuri Equations

Raychaudhuri equations play important roles to describe the gravitational focusing and space-time singularities. Amal Kumar Raychaudhuri established it in 1955 to describe gravitational focusing properties in cosmology. When the star is heavier than a few solar masses, it could undergo an endless gravitational collapse without achieving any equilibrium state. The final outcome of gravitational collapse of a massive star must necessarily be a black hole which covers the resulting space-time singularity and causal message from the singularity cannot reach the external observer at infinity. In this article Raychaudhuri equations are derived with the help of general relativity and topological properties. An attempt has been taken here to describe gravitational focusing and space-time singularities in some detail with easier mathematical calculations.