Ultrafast visualization of an electric field under the Lorentz transformation

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.