Ultrafast visualization of an electric field under the Lorentz transformation

Abstract

In special relativity, four-vectors, such as space–time, energy–momentum and electromagnetic potential vectors, follow the Lorentz transformation. The transformations of space–time and energy–momentum vectors have been tested by time dilation and rest mass energy experiments, respectively. However, the Lorentz transformation of electromagnetic potentials has never been directly demonstrated in experiments due to the limitations of the diagnostics. Here we show the spatiotemporal electric-field profiles around a highly energetic electron beam with a subpicosecond temporal resolution obtained by a terahertz technique based on electro-optic sampling. We observe the Coulomb field contraction in the propagation direction of the beam under the Lorentz transformation. After passing the beam through a metallic plate, we investigate the birth of the Coulomb field around it and follow the spatiotemporal evolution of the spherical wavefront, which is derived by the Liénard–Wiechert potentials. The measured electric-field profile in the far propagation distance is experimentally confirmed to coincide with the calculation of the Liénard–Wiechert potentials and the Lorentz transformation. Our ultrafast measurement of a spatiotemporal relativistic electric field provides both additional experimental evidence of special relativity and a platform for detailed experimental research of energetic charged particle beams and electromagnetic radiation physics.

Students’ confusions about the electric field of a uniformly moving charge

“…But wait! A pure E x field from (8) seems to be decreased by 1/γ 2 , instead of remaining the same, as in (1). What gives?…”

“…In this work, we wish to keep things as simple as possible, for maximum clarity. For this reason we consider only the electric field of a point charge in a uniform motion (a Coulomb field), entirely excluding all magnetic fields from a discussion 1 . Furthermore, whenever a decomposition of vectors into components is required, we will use a two-dimensional coordinate system.…”

“…For now we only require the visual suggestions from any of these displays, as they present themselves throughout the literature. 1 We would, of course, have to account for a magnetic field of a moving charge if we were interested in any specific effect upon other charges. But in this work we are only interested in an electric field itself.…”

“…Relativistic transformation of an electric field-particularly that of a point charge in a uniform motion (a Coulomb field)-is a regular topic of (under)graduate physics courses such as General Physics and dedicated courses on Classical Electrodynamics. In light of a recent groundbreaking direct experimental confirmation of a Lorentz contraction of Coulomb field [1], we expect that this subject will become even more prominent in teaching the special relativistic aspects of Classical Electrodynamics. We use this opportunity to address some seemingly widespread confusions (if not misconceptions) that are often encountered in teaching these.…”

“…The goal of our discussion is twofold: (1) to introduce the confusing issues in a compelling and meaningful way, so as to arouse the students' interest in what may otherwise be seen as a dry subject; (2) and to resolve these issues in an insightful way that should yield a greater appreciation of a deep meaning behind seemingly uninformative equations. At the same time, bringing these issues to the explicit attention of teachers should help them in recognizing and appreciating the students' struggles with the basic facts about the dynamic electric fields.…”