Variation of the light-like particle energy and its critical curve equations

TL;DR

This paper investigates the variation of light-like particle energy in Riemann space-time, deriving new equations for critical curves that differ from standard geodesics but yield similar solutions in Schwarzschild space-time.

Contribution

It introduces a novel variational approach to derive equations for light-like particles that maintains null path conformity unlike traditional interval variations.

Findings

Derived new equations for light-like particle paths.

Showed solutions coincide with standard geodesics in Schwarzschild space-time.

Established a variational method consistent with null path constraints.

Abstract

We consider variation of energy of the light-like particle in 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 shown to not lead to violation of conformity of varied curve to the null path in contradistinction of the interval variation. Though found equations are differ from standard form of geodesics equations, for the Schwarzschild space-time their solutions coincide with each other to within parameter of differentiation.