Electron vortex beams (EVBs) have potential applications in nanoscale magnetic probes of condensed matter and nanoparticle manipulation as well as radiation physics. Recently, a relativistic electron vortex beam (REVB) has been proposed [Phys. Rev. Lett.107 174802 (2011)]. Compared with EVBs, except for orbital angular momentum, an REVB has intrinsic relativistic effect, i.e., spin angular momentum and spin–orbit coupling. We study the electromagnetic field of an REVB analytically. We show that the electromagnetic field can be separated into two parts, one is only related to orbital quantum number, and the other is related to spin–orbit coupling effect. Exploiting this separation property, the difference between the electromagnetic fields of the REVB in spin-up and spin-down states can be used as a demonstration of the relativistic quantum effect. The linear momentum and angular momentum of the generated electromagnetic field have been further studied and it is shown that the linear momentum is weakly dependent on the spin state; while the angular momentum is evidently dependent on the spin state and linearly increases with the topological charge of electron vortex beam. The electromagnetic and mechanical properties of the REVB could be useful for studying the interaction between REVBs and materials.