Application of Lagrange Mechanics for Analysis of the Light-Like Particle Motion in Pseudo-Riemann Space

Wladimir Belayev

doi:10.5923/j.ijtmp.20120202.03

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 principle for the stationary gravity field. The produced equations are solved for the metrics of Schwarzschild, FLRW model for the flat space and GÃ¶del. For these spaces effective mass of light-like particle is established. Relativistic analogue of inertial mass for photon is determined in central gravity field in empty space.

Highlights

One of postulates of general relativity is claim that in gravity field in the absence of other forces the word lines of the material particles and the light rays are geodesics
We examine choosing of energy so in order that application of variational principle to its integral for deriving of the isotropic critical curves equations would not lead to considering non-null paths
The proposed form of energy allows applying of the Lagrange's mechanics for analysis of the light-like particle motion

Summary

Introduction

One of postulates of general relativity is claim that in gravity field in the absence of other forces the word lines of the material particles and the light rays are geodesics. The other terms of series in (2), containing variations of coordinates and their derivatives by μ in more high powers or their products and being able to have nonzero values, don't take into account Such method admits violation of the condition η = 0 , which means that with certain coordinates variations the interval a prior becomes time-like or space-like. Approximating time-like interval conforming in general relativity to the material particle motion between fixed points to null leads in physical sense to unlimited increase of its momentum, and the space-like interval doesn't conform to move of any object In this connection it should pay attention on speculation that discreteness at the Planck scale reveals maximum value of momentum for the fundamental particles [8]. We examine choosing of energy so in order that application of variational principle to its integral for deriving of the isotropic critical curves equations would not lead to considering non-null paths

Definition of Energy and Its Variation

Equations of Isotropic Critical Curve

Extremal Isotropic Curves in FLRW Space-Time

Extremal Isotropic Curves in Gödel Space-Time

Conclusions