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 overrightarrow{beta }+delta overrightarrow{beta } and successive boosts overrightarrow{beta } and Delta overrightarrow{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.
Highlights
This rotation of the space coordinates under the application of successive Lorentz boosts is called Thomas rotation
Successive boosts which are non collinear, in general, result in Thomas rotation of the space coordinates or in other words, the boosted frames which are accelerating in the sense that their direction is changing will
The work presented in this paper is another confirmation of the fact that two successive boosts are not equal to a single direct boost
Summary
The main idea of this paper is to see how the electromagnetic fields transform relativistically when there is an accelerated motion. The overall approach stays the same but all the boost matrices are needed to be transformed in the xy-frame before being used to calculate the electromagnetic field tensor. In order to calculate the electromagnetic field tensor for various boosts in the lab xy-frame, we will just use the field tensor F as defined in Eq (23). After calculating the boost matrix Eq (36), we can again use Eq (25) to calculate the electromagnetic field tensor in the direct boosted frame with respect to the laboratory frame whose detailed expressions are provided in the Supplementary Information (Section 3.1). Using Eq (38) we can calculate electromagnetic fields due to pure Lorentz boosts whose detailed expressions are provided in the Supplementary Information (Section 3.2)
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