Application of Lagrange mechanics for analysis of the light-like particle motion in pseudo-Riemann space

TL;DR

This paper applies Lagrange mechanics to analyze light-like particle motion in pseudo-Riemann space, deriving equations of motion, energy, and momentum, and compares this approach with classical principles across various spacetime metrics.

Contribution

It introduces a Lagrangian-based method for analyzing light-like particles in pseudo-Riemann space, providing new equations and solutions for different gravitational metrics.

Findings

Derived equations for energy and momentum transfer to gravity.

Solved critical curve equations for Schwarzschild, FLRW, and G"odel metrics.

Identified the gravitation mass of the photon in the Newtonian limit.

Abstract

We consider variation of energy of the light-like particle in the pseudo-Riemann space-time, find Lagrangian, canonical momenta and forces. Equations of the critical curve are obtained by the nonzero energy integral variation in accordance with principles of the calculus of variations in mechanics. This method is compared with the Fermat's and geodesics principles. Equations for energy and momentum of the particle transferred to the gravity field are defined. Equations of the critical curve are solved for the metrics of Schwarzschild, FLRW model for the flat space and Goedel. The gravitation mass of the photon is found in central gravity field in the Newtonian limit.