  Laboratory optics, typically dealing with monochromatic light beams in a
single reference frame, exhibits numerous spin-orbit interaction phenomena due
to the coupling between the spin and orbital degrees of freedom of light.
Similar phenomena appear for electrons and other spinning particles. Here we
examine transformations of paraxial photon and relativistic-electron states
carrying the spin and orbital angular momenta (AM) under the Lorentz boosts
between different reference frames. We show that transverse boosts inevitably
produce a rather nontrivial conversion from spin to orbital AM. The converted
part is then separated between the intrinsic (vortex) and extrinsic (transverse
shift or Hall effect) contributions. Although the spin, intrinsic-orbital, and
extrinsic-orbital parts all point in different directions, such complex
behavior is necessary for the proper Lorentz transformation of the total AM of
the particle. Relativistic spin-orbit interactions can be important in
scattering processes involving photons, electrons, and other relativistic
spinning particles, as well as when studying light emitted by fast-moving
bodies.
