Amal Kumar Raychaudhuri died on June 18, 2005. This essay follows the lecture which I gave in honour of this great Indian scientist and teacher on December 26, 2005 in Puri, India.

The Raychaudhuri equation is central to the understanding of gravitational attraction in astrophysics and cosmology, and in particular underlies the famous singularity theorems of general relativity theory. This paper reviews the derivation of the equation, and its significance in cosmology.

I first recount Raychaudhuri's deep involvement with the singularity problem in general relativity. I then argue that precisely the same situation has arisen today in loop quantum cosmology as obtained when Raychaudhuri discovered his celebrated equation. We thus need a new analogue of the Raychaudhuri equation in quantum gravity.

Inspired by Raychaudhuri's work, and using the equation named after him as a basic ingredient, a new singularity theorem is proved. Open non-rotating Universes, expanding everywhere with a non-vanishing spatial average of the matter variables, show severe geodesic incompletness in the past. Another way of stating the result is that, under the same conditions, any singularity-free model must have a vanishing spatial average of the energy density (and other physical variables). This is very satisfactory and provides a clear decisive difference between singular and non-singular cosmologies.

We present a brief review on the Raychaudhuri equations. Beginning with a summary of the essential features of the original article by Raychaudhuri and subsequent work of numerous authors, we move on to a discussion of the equations in the context of alternate non-Riemannian spacetimes as well as other theories of gravity, with a special mention on the equations in spacetimes with torsion (Einstein–Cartan–Sciama–Kibble theory). Finally, we give an overview of some recent applications of these equations in general relativity, quantum field theory, string theory and the theory of relativisitic membranes. We conclude with a summary and provide our own perspectives on directions of future research.

Basic features of dynamical black holes in full, non-linear general relativity are summarized in a pedagogical fashion. Qualitative properties of the evolution of various horizons follow directly from the celebrated Raychaudhuri equation.

In general relativity space-like or null singularities are common: they imply that `time' can have a beginning or end. Well-known examples are singularities inside black holes and initial or final singularities in expanding or contracting universes. In recent times, string theory is providing new perspectives of such singularities which may lead to an understanding of these in the standard framework of time evolution in quantum mechanics. In this article, we describe some of these approaches.

A simple, geometrical construction is given for three-dimensional spacetimes with negative cosmological constant that contain two particles colliding head-on. Depending on parameters like particle masses and distance, the combined geometry will be that of a particle, or of a black hole. In the black hole case the horizon is calculated. It is found that the horizon typically starts at a point and spreads into a closed curve with corners, which propagate along spacelike caustics and disappear as the horizon passes the particles.

We consider here the genericity aspects of spacetime singularities that occur in cosmology and in gravitational collapse. The singularity theorems (that predict the occurrence of singularities in general relativity) allow the singularities of gravitational collapse to be either visible to external observers or covered by an event horizon of gravity. It is shown that the visible singularities that develop as final states of spherical collapse are generic. Some consequences of this fact are discussed.

We generalise the Raychaudhuri equation for the evolution of a self gravitating fluid to include an Abelian and non-Abelian hybrid magneto fluid at a finite temperature. The aim is to utilise this equation for investigating the dynamics of astrophysical high temperature Abelian and non-Abelian plasmas.

The equation of Raychaudhuri is one of the key concepts in the formulation of the singularity theorems introduced by Penrose and Hawking. In the present article, taking into account QED vacuum polarization, we study the propagation of a bundle of rays in a background gravitational field through the perturbative deformation of Raychaudhuri's equation. In a sense, this could be seen as another semiclassical study in which geometry is treated classically but matter (which means the photon here) is allowed to exhibit quantum characteristics that are encoded in its coupling to the background curvature.