The Raychaudhuri equation for a quantized timelike geodesic congruence

Abstract

A recent attempt to arrive at a quantum version of Raychaudhuri’s equation is looked at critically. It is shown that the method, and even the idea, has some inherent problems. The issues are pointed out here. We have also shown that it is possible to salvage the method in some limited domain of applicability. Although no generality can be claimed, a quantum version of the equation should be useful in the context of ascertaining the existence of a singularity in the quantum regime. The equation presented in the present work holds for arbitrary $$n+1$$ n + 1 dimensions. An important feature of the Hamiltonian in the operator form is that it admits a self-adjoint extension quite generally. Thus, the conservation of probability is ensured.

Singularity attenuation with quantum‐mechanically revisited metric tensor

“…Landau-Raychaudhuri evolution along geodesic congruence for g 𝜶𝜷 For the classical fundamental tensor g 𝛼𝛽 , the Landau-Raychaudhuri equations, which describe the evolution along a geodesic congruence for the kinematic variables, the quadratic invariant expansion (volume scalar) Θ, shearing (anisotropy) 𝜎 2 = 𝜎 𝜇𝜈 𝜎 𝜇𝜈 , rotation (vorticity) 𝜔 2 = 𝜔 𝜇𝜈 𝜔 𝜇𝜈 , and Ricci identity (local gravitational field) R 𝜇𝜈 u 𝜇 u 𝜈 (Gupta Choudhury et al 2021;Raychaudhuri 1957;Landau et al 1975) read…”

“…For the classical fundamental tensor $$$ {g}_{\alpha \beta} $$$, the Landau–Raychaudhuri equations, which describe the evolution along a geodesic congruence for the kinematic variables, the quadratic invariant expansion (volume scalar) $$$ \Theta $$$, shearing (anisotropy) $$$ {\sigma}^2={\sigma}_{\mu \nu}{\sigma}^{\mu \nu} $$$, rotation (vorticity) $$$ {\omega}^2={\omega}_{\mu \nu}{\omega}^{\mu \nu} $$$, and Ricci identity (local gravitational field) $$$ {R}_{\mu \nu}{u}^{\mu }{u}^{\nu } $$$ (Gupta Choudhury et al 2021; Raychaudhuri 1957; Landau et al 1975) read $$$$ \frac{d\Theta}{d\tau}=-\frac{1}{3}{\Theta}^2-{\sigma}^2+{\omega}^2-{R}_{\mu \nu}{u}^{\mu }{u}^{\nu }. $$$$ …”

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

“…It should be noted that as R 2 2 s + is a function of Λ alone, so we denote it by Φ 0 (Λ). Then the corresponding Lagrangian is given by [33]…”