  The linear dispersion relation of a finite amplitude, parallel, circularly
polarized Alfv\'en wave in a relativistic electron-positron plasma is derived.
In the nonrelativistic regime, the dispersion relation has two branches, one
electromagnetic wave, with a low frequency cutoff at
$\sqrt{1+2\omega_p^2/\Omega_p^2}$ (where $\omega_p=(4\pi n e^2/m)^{1/2}$ is the
electron/positron plasma frequency), and an Alfv\'en wave, with high frequency
cutoff at the positron gyrofrequency $\Omega_p$. There is only one forward
propagating mode for a given frequency. However, due to relativistic effects,
there is no low frequency cutoff for the electromagnetic branch, and there
appears a critical wave number above which the Alfv\'en wave ceases to exist.
This critical wave number is given by $ck_c/\Omega_p=a/\eta$, where
$a=\omega_p^2/\Omega_p^2$ and $\eta$ is the ratio between the Alfv\'en wave
magnetic field amplitude and the background magnetic field. In this case, for
each frequency in the Alfv\'en branch, two additional forward propagating modes
exist with equal frequency. A simple numerical example is studied: by
numerically solving the coupled system of fluid and Maxwell equations, normal
incidence of a finite amplitude Alfv\'en wave on an interface between two
electron-positron plasmas of different densities is considered.
