A look inside the theory of the linear approximation

TL;DR

This paper explores the linear approximation in General relativity, distinguishing between general and special gauge transformations, and develops a formalism to define gravitational energy-momentum and analyze gravitational fields.

Contribution

It introduces a new formalism within linearized gravity that differentiates gauge transformations, enabling the definition of gravitational energy-momentum and analysis of gravitational fields.

Findings

Defined gravitational energy-momentum tensor in linear approximation

Analyzed gravitational fields of point masses and distributions

Strengthened analogy between electromagnetism and gravity in special relativity

Abstract

We introduce in the framework of the linear approximation of General relativity a natural distinction between General gauge transformations generated by any vector field and those Special ones for which this vector field is a gradient. This allows to introduce geometrical objects that are not invariant under General gauge transformations but they are under Special ones. We develop then a formalism that strengthens the analogy of the formalisms of the electromagnetic and the gravitational theories in a Special relativity framework. We are thus able to define the energy-momentum tensor of the gravitational field and to fully analyze the gravitational field of isolated point masses or continuous distributions of them obtained by linear superpositions.