Dear Nicolas, your statement is correct: the momentum equation breaks down. When deriving the formula for Alfven speed, a crucial step is the invariance of mass density (which in fact appears just as a multiplicative factor at the denominator of the squared Alfven velocity). But mass density is an invariant with respect to Galileian transformations, not to Lorenz transformations. 

Hi Timothee,

When the Alfven speed approaches the speed of light, the typical MHD assumption of ignoring the displacement current in Ampere's law is no longer valid. When the effects of the displacement current are retained in the derivation of the Alfven wave, you find that the actual speed of the wave (let's call it v) is given by v-2 = VA-2 + c-2, which asymptotically gives you v = VA for VA > c.

The wave phase velocity (including Alfven waves) can be, in principle, larger than the speed of light. However, the group velocity cannot, since it is related to energy propgation/transport. The interesting problem here is that the MHD shear Alfven wave has phase velocity = group velocity = VA, once Kperp is explicity only at the dispersion relation of the kinetic Alfven wave, I mean that the MHD shear Alfven wave do not propagate energy perpendicular to the ambiente/background magnetic field.

On relativistic Alfven waves, I sugest also the paper by Heyvaerts et alli.

Non-linear simple relativistic Alfvén waves in astrophysical plasmas J. Heyvaerts, T. Lehner1, and F. Mottez

A&A 542, A128 (2012) DOI: 10.1051/0004-6361/201118630 c ESO 2012

The enclosed file might be useful!

In addition to all the ansawers - so far -, I would say that a full analisys of this problem depend on the general background target plasma condition. More insights on that can be obtained considering the enclosed paper.

Jesse R. Woodroffe Hi Jesse, the statement that "When the effects of the displacement current are retained in the derivation of the Alfven wave, you find that the actual speed of the wave (let's call it v) is given by v-2 = VA-2 + c-2, which asymptotically gives you v = VA for VA > c." do you have any citations, (e.g. papers or books).

This correction is discussed in section 2 of Stasiewicz, K. et al. (2000), Small scale Alfvenic structure in the Aurora, Space Science Reviews, 92:423-533.

http://www.igpp.ucla.edu/public/mkivelso/refs/PUBLICATIONS/staciewicz%20AW%202000%20SpaceSciRev.pdf

There is a need to apply this correction in models of the near-Earth space environment as well as the Jovian magnetosphere because these regions feature relatively strong magnetic fields and relatively low plasma densities.

It's actually quite straightforward to derive yourself -- just use the Maxwell-Ampere equation, the MHD equation of motion, and ideal MHD Ohm's law. From there the quickest way is to probably to assume a time-stationary background magnetic field and use the equations to find an expression that involves neither the current density nor the fluid velocity. (This is mathematically equivalent to adding a polarization current to the Maxwell-Ampere equation, and that is an alternative approach to finding the same result.)