Killing Vector Fields, Maxwell Equations and Lorentzian Spacetimes

TL;DR

This paper explores Maxwell equations in Lorentzian spacetimes with potentials aligned to Killing vector fields, revealing their wave nature, superconducting-like currents, and the interplay with Einstein equations in coupled electromagnetic, fluid, and gravitational systems.

Contribution

It demonstrates that potentials proportional to Killing vectors satisfy wave equations and produce superconducting currents, linking Maxwell, Einstein, and fluid dynamics in a unified framework.

Findings

Maxwell potentials aligned with Killing vectors obey wave equations.

The current associated with such potentials is of superconducting type.

Einstein equations reduce to Maxwell equations with a specific current in coupled systems.

Abstract

In this paper we first analyze the structure of Maxwell equations in a Lorentzian spacetime where the potential A is proportional to 1-form K physically equivalent to a Killing vector field (supposed to exist). We show that such A obeys the Lorenz gauge and also a wave equation that can be written in terms of the covariant D'Alembertian or the Ricci operator. Moreover, we determine the correct current defined by that potential showing that it is of superconducting type, being two times the product of the components of A by the Ricci 1-form fields. We also study the structure of the spacetime generated by the coupled system consisting of a electromagnetic field F = dA (A, as above), an ideal charged fluid with dynamics described by an action function S and the gravitational field. We show that Einstein equations in this situation is then equivalent to Maxwell equations with a current givn by fFAF (the product meaning the Clifford product of the corresponding form fields), where f is a scalar function which satisfies a well determined algebraic quadratic equation.