Non-linear energy conservation theorem in the framework of Special Relativity

TL;DR

This paper derives a non-linear energy conservation law within Special Relativity that highlights gravity's non-linear nature, connects to Newtonian and post-Newtonian results, and suggests gravity's geometric interpretation outside pure Special Relativity.

Contribution

It introduces a relativistic non-linear energy conservation theorem consistent with the equivalence principle and develops a covariant Lagrangian formalism linking gravity to curved spacetime geometry.

Findings

Reproduces Newtonian energy conservation in low-energy limit.

Derives the gravitational redshift formula from the conservation law.

Shows that gravity in Special Relativity can be described by a curved Lorentzian manifold.

Abstract

In this work we revisit the study of the gravitational interaction in the context of the Special Theory of Relativity. It is found that, as long as the equivalence principle is respected, a relativistic non-linear energy conservation theorem arises in a natural way. We interpret that this non-linear conservation law stresses the non-linear character of the gravitational interaction.The theorem reproduces the energy conservation theorem of Newtonian mechanics in the corresponding low energy limit, but also allows to derive some standard results of post-Newtonian gravity, such as the formula of the gravitational redshift. Guided by this conservation law, we develop a Lagrangian formalism for a particle in a gravitational field. We realize that the Lagrangian can be written in an explicit covariant fashion, and turns out to be the geodesic Lagrangian of a curved Lorentzian manifold. Therefore, any attempt to describe gravity within the Special Theory, leads outside their own domains towards a curved space-time. Thus, the pedagogical content of the paper may be useful as a starting point to discuss the problem of Gravitation in the context of the Special Theory, as a preliminary step before introducing General Relativity.