Is the EmDrive, or "Relativity Drive" possible? In 2006, New Scientist magazine published an article titled Relativity drive: The end of wings and wheels1 [1] about the EmDrive [Wikipedia] which stirred up a fair degree of controversy and some claims that New Scientist was engaging in pseudo-science. 
Since the original article the inventor claims that a "Technology Transfer contract with a major US aerospace company was successfully completed", and that papers have been published by Professor Yang Juan of The North Western Polytechnical University, Xi'an, China. 2 
Furthermore, it was reported in Wired magazine that the Chinese were going to attempt to build the device.
Assuming that the inventor is operating in good faith and that the device actually works, is there another explanation of the claimed resulting propulsion?
Notes: 
1. Direct links to the article may not work as it seems to have been archived.
2. The abstracts provided on the EmDrive website claim that they are Chinese language journals which makes them very difficult to chase down and verify.
 A: No. In special relativity, 4 momentum is exactly conserved. The first component of 4 momentum is total mass/energy, but the next 3 are given by:
p = m*γ(v)*v
m is the invariant mass, how much inertia it has when you are moving at the same velocity of it.
This is Newton except now momentum is a non-linear function of velocity. Nonlinearity does not change anything. Mass and momentum still are constant (ignoring leaks),
making γ(v)*v, and thus the center-of-mass velocity v, constant.
So why do we measure force? Possibly currents in the waveguide walls induce currents in the metal support structure which creates small magnetic forces between them.
A: It is impossible to generate momentum in a closed object without emitting something, so the drive is either not generating thrust, or throwing something backwards. There is no doubt about this.
Assuming that the thrust measurement is accurate, that something could be radiation. This explanation is exceedingly unlikely, since to get mN of radiation pressure you need an enormous amount of energy, since in 1s you get 1 ${\rm gm s^{-1}}$ of momentum, which in radiation can only be carried by $3 \times 10^5$ J (multiply by c), so you need 30,000 Watts of energy to push with mN force, or at least a million Watts for 80 mN. So, it's not radiation.
But a leaky microwave cavity can heat the water-vapor in the air around the object, and the heat can lead to a current of air away from the object. With a air current, you can produce mN thrusts from a relatively small amount of energy, and with a barely noticible breeze. To get mN force, you need to accelerate $300 \ {\rm cm^3}$ of air (1 gram) to 1 m/s every second, or to get 80 mN, accelerate $1 {\rm m^3}$ of air (3000 g) to 0.2 m/s (barely perceptible) and this can be done with a hot-cold thermal gradient behind the device which is hard to notice. If the thrust measurements are not in error, this is the certain cause.
So at best, Shawyer has invented a very inefficient and expensive fan.

EDIT: The initial tests were at atmospheric pressure. To test the fan hypothesis, an easy way is to vary the pressure, another easy way is to put dust in the air to see the air-currents. The experimenters didn't do any of this (or at least didn't publish it if they did), instead, they ran the device inside a vacuum chamber but at ambient pressure after putting it through a vacuum cycle to simulate space. This is not a vacuum test, but it can mislead one on a first read.
In response to criticism of this faux-vacuum test, they did a second test in a real vacuum. This time, they used a torsion pendulum to find a teeny-tiny thrust of no relation to the first purported thrust. The second run in vacuum has completely different effects, possibly due to interactions between charge building up on the device and metallic components of the torsion pendulum, possibly due to deliberate misreporting by these folks, who didn't bother to explain what was going on in the first experiments they hyped up. Since they didn't bother to do a any systematic analysis of the effect on the first run, to vary air-pressure, look at air flows with dust, whatever, or if they did this they didn't bother to admit their initial error, this is not particularly honest experimental work, and there's not much point in talking about it any more. These folks are simply wasting people's time.
A: Shawyer's "analysis" is a mess, incoherent and deeply confused about fundamental aspects of relativity: he mixes up frames, assumes a universal rest frame, etc. The EmDrive supposedly works best when "stationary relative to the thrust", whatever that means, and Shawyer goes on to suggest using it for levitating vehicles with some kind of conventional propulsion for driving them forward: he apparently believes there is something special about gravitational acceleration.
According to his latest paper, the EmDrive supposedly acts as an electric motor, consuming energy when accelerating and producing it when decelerating. However, a deceleration is just an acceleration in a particular direction, so if it worked, the EmDrive could operate as an infinite energy machine just sitting on one end in a gravity field or while producing thrust for a spacecraft.
So to answer the question in the title: "No."
As for other explanations of the observed propulsion, there aren't many details of the measurement procedures or results. There are videos of an EmDrive test on a rotating platform, but there's numerous pieces of equipment that may contain fans, thick power cables going to the equipment that may apply torques, and even a laptop with a hard drive that may be spinning up or down. (And on top of everything else, the whole thing's apparently rotating in the wrong direction.) If this rig is typical of his testing methodology, it's probably safe to chalk up the rest to bad measurements.
A: Yes, it might actually be possible. I'm not sure it has been proven that the math shows that it can't be done. I know Einstein found a way of defining relativistic mass and momentum as a function of rest mass and velocity for any particle and laws that conserve relativistic mass and momentum and are conserved in all frames of references and that momentum is even conserved in the collision of a photon with an object assuming the particle nature of the photon. The microwave photons on the other hand don't behave much like a particle at all and have a wavelength that's a very significant fraction of the size of the cavity. All that's left to do is determine whether the math shows that the microwave radiation can provide thrust according to the model where it's only an electromagnetic wave and not a particle.
