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.

The Raychaudhuri equations: A brief review

We examine physical features that could lead to formation of a naked singularity rather than black hole, as end state of spherical collapse. Generalizing earlier results on dust collapse to general type I matter fields, it is shown that collapse always creates black hole if shear vanishes or density is homogeneous. It follows that nonzero shear is a necessary condition for singularity to be visible to external observers, when trapped surface formation is delayed by shearing forces or inhomogeneity within the collapsing cloud.

Why do naked singularities form in gravitational collapse

We obtain a new static model of the TOV equation for an anisotropic fluid distribution by imposing the Karmarkar condition. In order to close the system of equations we postulate an interesting form for the $$g_{rr}$$
 gravitational potential, which allows us to solve for $$g_{tt}$$
 metric component via the Karmarkar condition. We demonstrate that the new interior solution has well-behaved physical attributes and can be utilized to model relativistic static fluid spheres. By using observational data sets for the radii and masses for compact stars such as 4U 1538-52, LMC X-4, and PSR J1614-2230 we show that our solution describes these objects to a very good degree of accuracy. The physical plausibility of the solution depends on a parameter c for a particular star. For 4U 1538-52, LMC X-4, and PSR J1614-2230 the solutions are well behaved for $$0.1574 \le c \le 0.46$$
 , $$0.1235 \le c \le 0.35$$
 and $$0.05 \le c \le 0.13$$
 , respectively. The behavior of the thermodynamical and physical variables of these compact objects leads us to conclude that the parameter c plays an important role in determining the equation of state of the stellar material and observed that smaller values of c lead to stiffer equation of states.

https://link.springer.com/content/pdf/10.1140%2Fepjc%2Fs10052-017-4612-4.pdf

Physical viability of fluid spheres satisfying the Karmarkar condition

Relativistic models for superconducting-superfluid mixtures

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

A singularity theorem based on spatial averages

A particular choice of the time function in the recently presented spherical solution by Dadhich1 leads to a singularity free cosmological model which oscillates between two regular states. The energy–stress tensor involves anisotropic pressure and a heat flux term but is consistent with the usual energy conditions (strong, weak and dominant). By choosing the parameters suitably one can make the model consistent with observational data. An interesting feature of the model is that it involves blue shifts as in the quasi steady state model2 but without violating general relativity.

Oscillating Non-Singular Relativistic Spherical Model

It is shown that if the time-like eigenvector of the Ricci tensor is hypersurface orthogonal so that the space–time allows a foliation into space sections, then the space average of each of the scalars that appears in the Raychaudhuri equation vanishes provided that the strong energy condition holds good. This result is presented in the form of a singularity theorem.

A new singularity theorem in relativistic cosmology

It is shown that if the timelike eigenvector of the Ricci tensor be hypersurface orthogonal so that the space time allows a foliation into space sections then the space average of each of the scalar that appear in the Raychaudhuri equation vanishes provided the strong energy condition holds good. This result is presented in the form of a singularity theorem.

A. K. Raychaudhuri

Papers

Oscillating Non-Singular Relativistic Spherical Model

A new singularity theorem in relativistic cosmology

A new singularity theorem in relativistic cosmology

A random walk in relativity and cosmology : essays in honour of P. C. Vaidya and A. K. Raychaudhuri