Raychaudhuri Equation from Lagrangian and Hamiltonian formulation : A Quantum Aspect

TL;DR

This paper reformulates the Raychaudhuri Equation using Lagrangian and Hamiltonian methods, explores its quantum aspects through Wheeler-DeWitt equation, and demonstrates quantum effects can eliminate classical singularities.

Contribution

It provides a novel Lagrangian and Hamiltonian formulation of the Raychaudhuri Equation and investigates its quantum implications in cosmology.

Findings

Quantum trajectories remove the initial big-bang singularity.

Wave function norm influences singularity behavior in quantum cosmology.

A new analytic solution for the Raychaudhuri Equation is derived.

Abstract

The paper deals with a suitable transformation related to the metric scalar of the hyper-surface so that the Raychaudhuri Equation (RE) can be written as a second order nonlinear differential equation. A first integral of this second order differential equation gives a possible analytic solution of the RE. Also, it is shown that construction of a Lagrangian (and hence a Hamiltonian) is possible, from which the RE can be derived. Wheeler-Dewitt equation has been formulated in canonical quantization scheme and norm of it's solution (wave function of the universe) is shown to affect the singularity analysis in the quantum regime for any spatially homogeneous and isotropic cosmology. Finally Bohmian trajectories are formulated with causal interpretation and these quantum trajectories unlike classical geodesics obliterate the initial big-bang singularity when the quantum potential is included.