# Effect of Thomas Rotation on the Lorentz Transformation of Electromagnetic fields

## TL;DR

This paper investigates how Thomas rotation affects the Lorentz transformation of electromagnetic fields during successive relativistic boosts, highlighting its significance in precise electromagnetic calculations involving relativistic particles.

## Contribution

It provides a detailed calculation of electromagnetic field tensors considering Thomas rotation for general three-dimensional boosts, enhancing the understanding of relativistic electromagnetic phenomena.

## Key findings

- Thomas rotation influences electromagnetic field transformations during relativistic boosts.
- Comparison of direct and successive boost calculations confirms the role of Thomas rotation.
- Framework applicable to frequency shift calculations in relativistic spin precession scenarios.

## Abstract

A relativistic particle undergoing successive boosts which are non collinear will experience a rotation of its coordinate axes with respect to the boosted frame. This rotation of coordinate axes is caused by a relativistic phenomenon called Thomas Rotation. We assess the importance of Thomas rotation in the calculation of physical quantities like electromagnetic fields in the relativistic regime. We calculate the electromagnetic field tensor for general three dimensional successive boosts in the particle's rest frame as well as the laboratory frame. We then compare the electromagnetic field tensors obtained by a direct boost $\vec{\beta} + \delta \vec{\beta}$ and successive boosts $\vec{\beta}$ and $\Delta \vec{\beta}$ and check their consistency with Thomas rotation. This framework might be important to situations such as the calculation of frequency shifts for relativistic spin-1/2 particles undergoing Larmor precession in electromagnetic fields with small field non-uniformities.