  Recently Gouy rotation was observed with focused non-relativistic electron
vortex beams. If the electrons in vortex beams are very fast we have to take
into account relativistic effects to completely describe the Gouy phase on
them. Exact Hermite-Gaussian solutions to the Klein-Gordon equation for
particle beams are obtained here that make explicit the 4-position of the focal
point of the beam. These are Bateman-Hillion solutions with modified phase
factors to take into account the rest mass of the particles. They enable a
relativistic expression for the Gouy phase to be determined. It is in fact
shown all the solutions are form invariant under Lorentz transformations. It is
further shown for the exact solutions to correspond to those of the
Schr\"odinger equation the relative time between the focal point and any point
in the beam must be constrained to be a specific function of the relative
spatial coordinates.
