Lagrangian formalism in the theory of relativistic vector fields

TL;DR

This paper develops a covariant Lagrangian formalism for relativistic vector fields in curved spacetime, deriving equations for matter, fields, and conserved quantities, including electromagnetic and gravitational contributions.

Contribution

It introduces a covariant Lagrangian approach directly involving the Lagrangian and derivatives, applied specifically to vector fields in curved spacetime, with new formulas for physical quantities.

Findings

Derived covariant equations for vector fields in curved spacetime

Formulated expressions for energy-momentum and angular momentum pseudotensors

Established covariant formulas for the center of momentum in physical systems

Abstract

The Lagrangian formalism is used to derive covariant equations that are suitable for use in continuously distributed matter in curved spacetime. Special attention is given to theoretical representation, in which the Lagrangian and its derivatives are directly involved. The obtained results, including equation for metric, equation of motion, equations for fields, are applied to purely vector fields. As a consequence, formulas are determined for calculating the basic quantities necessary to describe physical systems. In this case, not only the pressure field and the acceleration field are taken into account, but also the electromagnetic and gravitational fields outside the matter, which contribute to the four-momentum and to the four-dimensional angular momentum pseudotensor of each system. It is shown that the canonical representation of the angular momentum pseudotensor is its representation with covariant indices. The radius-vector of the center of momentum of a physical system is determined in covariant form.