The correct and unusual coordinate transformation rules for electromagnetic quadrupoles

TL;DR

This paper derives the correct, novel transformation rules for electromagnetic quadrupoles involving second derivatives and integrals, enabling their proper definition in general relativity and coordinate systems.

Contribution

It introduces the first correct transformation rules for quadrupoles, involving second derivatives and integrals, allowing their consistent use in curved spacetime.

Findings

Transformation rules involve second derivatives and integrals.

Quadrupoles can be defined without metric dependence.

Coordinate systems can alter dipole and quadrupole components.

Abstract

Despite being studied for over a century, the use of quadrupoles have been limited to Cartesian coordinates in flat spacetime due to the incorrect transformation rules used to define them. Here the correct transformation rules are derived, which are particularly unusual as they involve second derivatives of the coordinate transformation and an integral. Transformations involving integrals have not been seen before. This is significantly different from the familiar transformation rules for a dipole, where the components transform as tensors. It enables quadrupoles to be correctly defined in general relativity and to prescribe the equations of motion for a quadrupole in a coordinate system adapted to its motion and then transform them to the laboratory coordinates. An example is given of another unusual feature: a quadrupole which is free of dipole terms in polar coordinates has dipole terms in Cartesian coordinates. It is shown that dipoles, electric dipoles, quadrupoles and electric quadrupoles can be defined without reference to a metric and in a coordinates free manner. This is particularly useful given their complicated coordinate transformation.