Raychaudhuri Equation, Geometrical Flows and Geometrical Entropy

Abstract

The Raychaudhuri equation is derived by assuming geometric flow in space–time M of n+1 dimensions. The equation turns into a harmonic oscillator form under suitable transformations. Thereby, a relation between geometrical entropy and mean geodesic deviation is established. This has a connection to chaos theory where the trajectories diverge exponentially. We discuss its application to cosmology and black holes. Thus, we establish a connection between chaos theory and general relativity.

The Raychaudhuri equation in inhomogeneous FLRW space-time: A f(R)-gravity model

“…The RE, a first order non-linear equation, is of central importance in the context of the Singularity Theorems [16,17]. Further, mathematically the RE is known as a Riccati equation, and it becomes a second order linear equation in the form of a harmonic oscillator equation with varying frequency as [20,23]…”

“…On the other hand, after the detection of gravitational waves, general relativity (GR) [13,14] is a universally accepted theory of gravity, despite the inherent existence of singularity in it as predicted by the famous singularity theorems of Hawking and Penrose [15][16][17]. The RE [18][19][20][21][22][23][24] is the main ingredient behind these singularity theorems. The RE as it stands is purely a geometric identity in Riemannian geometry.…”

The classical and quantum implications of the Raychaudhuri equation in f(T)-gravity

“…This is the famous RE [35][36][37][38][39][40][41], named after Prof. Amal Kumar Raychaudhuri and is the main ingredient for the celebrated Singularity Theorems [32][33][34] by Hawking and Penrose. Moreover it is a fundamental result to study exact solutions of Einstein's equations in GR and has much to contribute in modern cosmology.…”

“…The singularity theorems [32][33][34] use the notion of geodesic incompleteness as a stand-in for the presence of infinite curvature. It is interesting to note that the Raychaudhuri equation (RE) [35][36][37][38][39][40][41] is the main ingredient behind these singularity theorems. RE dictates the dynamical evolution of the Universe, describing the time evolution of the expansion scalar.…”

Raychaudhuri equation and the dynamics of cosmic evolution

“…For ref. see [15][16][17][18][19][20][21]. The importance of RE lies not only in the singularity analysis in Einstein gravity but also in extended theories of gravity [22][23][24].…”