Vector Field Dynamics: Field Equations and Energy Tensor

TL;DR

This paper develops the relativistic field theory for a vector field in curved spacetime, deriving field equations, conserved currents, and the energy-momentum tensor, and analyzing their properties and special cases.

Contribution

It provides a comprehensive derivation of the field equations and energy tensor for vector fields with quadratic Lagrangians in curved spacetime, including analysis of irreducible components.

Findings

Derived general Euler-Lagrange equations for vector fields.

Established existence of conserved currents.

Analyzed the Hilbert energy-momentum tensor and special parameter cases.

Abstract

Relativistic field theory for a vector field on a curved space-time is considered assuming that the Lagrangian field density is quadratic and contains field derivatives of first order at most. By applying standard variational calculus, the general Euler-Lagrange equations for the field are derived and the existence of a conserved current is achieved. The field equations are also analyzed from an eikonal-like point of view. The Hilbert energy-momentum tensor of the field is also derived and the influence of each one of the irreducible pieces appearing in the Lagrangian is studied. Particular values of the free parameters allow to retrieve known results.