Geometric aspects of Raychaudhuri equation and implications in nonsingular universe

Raychaudhuri equation (RE) is a very important equation in gravity and cosmology as it is the key ingredient of the seminal singularity theorems. In this paper, we present a novel method of deriving the RE and restating the famous singularity theorems in GR using the matrix representation of tensor fields. For the first time in the literature, we introduce a Raychaudhuri matrix, whose evolution equation gives the matrix RE. Further, the paper deals with the consequences of RE in the context of nonsingular model of the universe like wormhole and emergent universe to highlight the fact that RE is not only important in hinting the singular nature of Einstein gravity but also bears importance in framing nonsingular models. The paper also identifies some transformations which can be used to restate the Focusing theorem. Further, it shows how RE is analogs to the differential equation of a real Harmonic oscillator and convergence condition is associated with the time varying frequency of the oscillator. Moreover, RE hints that exotic matter is needed for emergent model of the universe. Finally, the fate of time-like as well as null geodesics inside a wormhole and role of red-shift, shape function in convergence is analyzed from the point of view of RE.