EmDrive
EmDrive (also Relativity Drive) is the name of a spacecraft propulsion system proposed, and reportedly developed, by Roger Shawyer.  New Scientist ran a cover story on EmDrive in its September 8, 2006 issue. The device is a Magnetron with a specially shaped fully enclosed tapering resonator cavity whose area is greater at one end. The inventor claims that the device generates thrust even though no detectable energy leaves the device. The inventor proposes to use it as a spacecraft propulsion system that uses no fuel (other than electricity).
On his homepage the inventor claims that independent peer-review is underway, but so far no publication is available.
The device's operation as described seemingly violates several basic laws of physics, notably conservation of momentum, though the inventor insists to the contrary. John Costella, an expert in relativistic electrodynamics describes the EmDrive as a 'fraud'.

# Principle of operation
The device exploits an idea first suggested by Allen Cullen in the 1950's, an electrical engineer then at University College London, that involves forces created by reflecting microwaves between opposite walls of a cavity. The "idea" is to try to design a cavity in such a manner that forces on one side are greater than the other.
The drive comprises a resonant cavity flooded with microwave radiation. The radiation exerts radiation pressure on the walls of the cavity, and normal Newtonian mechanics would indicate that, no matter what shape the cavity is, the forces exerted upon it from within must balance to zero. Shawyer claims that relativistic effects cause a cavity shaped like a truncated cone to experience a larger force against the large end than the small end, due to the group velocity of the wave changing as the local diameter of the cavity varies.
The increased confinement of the tapered end of the cone leads to a higher effective propagation constant (phase velocity). It also leads to local reflections which account for the apparent force imbalance when considering only the end walls. However, since it is the phase of the light rather than the actual photons bouncing off the walls, each force acts quasi independently from another - much like in a ring laser gyroscope where the beams act as if having an external frame of reference (which they have, since the speed of light is constant). The same principle applies to the EmDrive.
No microwaves or anything else are allowed to leave the device. Since nothing leaves the drive for propulsive purposes an EmDrive can be classed as a reactionless drive. Thus the principle by which the EmDrive is supposed to operate seems to violate conservation of momentum. It is known that the physics equations describing microwaves, Maxwell's equations, conserve momentum, and this would seem to cast doubt on Shawyer's derivation of a thrust effect. In his paper (attached), Shawyer thus takes the following view: any thrust extracted from his device is directly withdrawn from the energy stored in his cavity (due to the Q reflections an average wave encounters when inserted into the cavity, the energy levels quickly build up). In other words: the apparent force on the wider diameter of the cone seems to wane. The extent to which that happens perfectly matches the amount predicted by the law of conservation of momentum.

# Criticism
The EmDrive was the cover story for the September 8, 2006 issue of New Scientist.
After receiving criticism that no peer-reviewed publications on the subject had been made, Mr. Shawyer submitted a theory paper to New Scientist (which is not a peer reviewed scientific journal)  Shawyer's paper was almost immediately challenged by Dr. John Costella, a theoretical physicist and electrical engineer who works for the Australian Department of Defence, whose Ph.D. is in relativistic electrodynamics, the field of physics that Mr. Shawyer relies on to support his theory.

# Prototypes and tests
In September 2006 it was reported that Shawyer had constructed a prototype unit weighing 9 kilogrammes that consumes 700 watts of power and produces 88 millinewtons of force.
In May 2007 Eureka magazine reported that a second unit has been built for demonstration purposes, weighing 100 kilogrammes, consuming 300 watts for microwave production (and an unspecified amount for ancillary purposes such as cooling) and producing 9.8 grams (96.1 millinewtons) of force during testing in October 2006. Unlike the prototype unit, which can only be run for short periods before burning out its magnetron, the demonstration unit can be run continuously. 
The limiting factor for performance is claimed to be the Q factor of the cavity, as microwave energy lost to heating the cavity reduces the field strength within, so Shawyer is experimenting with a cavity lined in a superconducting material that may produce Q factors sufficient to build a device capable of generating 30 newtons per watt.
These results have neither been reproduced by other scientists or engineers, nor have they been published in a peer-reviewed scientific journal.

# New Scientist article
EmDrive was featured on the cover of issue 2568 of New Scientist, a weekly science magazine. The article portrayed the device as plausible, and emphasized the arguments of those who held that point of view, although it did quote one engineer as saying "it's a load of bloody rubbish." The article included the following arguments from proponents of the theory:
- With a grant from the UK government's Department of Trade and Industry of 250,000, (actually two grants; one a feasibility study of 45,000 and a second of 81,000 to build a demonstration engine - source SPR Ltd.) a commercial regulatory and support agency, Shawyer has built two prototypes that reportedly produce 16mN and 300mN of thrust respectively; each using 1kW of electrical power. A condition of the funding was independent analysis, which was recently completed by John Spiller who says "The thruster's design is practical and could be adapted fairly easily to work in outer space". Shawyer claims that he has been visited by representatives from China and the US Air Force, but ESA has not yet shown much interest. He estimates that his design could save the aerospace community $15 billion over the next ten years.
- Engineers in Germany have created superconducting resonators (for use in particle accelerators) with Q values of several billion, which Shawyer claims would equate to a thrust of 30kN per kilowatt, "enough to lift a large car". Shawyer states that the thruster works best while stationary relative to their thrust.
New Scientist has drawn great criticism from the scientific community due to the uncritical treatment of EmDrive in its article. Science fiction writer Greg Egan distributed a public letter stating that "a sensationalist bent and a lack of basic knowledge by its writers" was making the magazine's coverage sufficiently unreliable "to constitute a real threat to the public understanding of science".  In particular, Egan found himself "gobsmacked by the level of scientific illiteracy" in the magazine's coverage of the EmDrive, where New Scientist allowed the publication of "meaningless double-talk" designed to bypass a fatal objection to Sawyer's proposed space drive, namely that it violates the conservation of momentum.  Egan urged those reading his letter to write to New Scientist and pressure the magazine to raise its standards, instead of "squandering the opportunity that the magazine's circulation and prestige provides" for genuine science education.  The letter was endorsed by mathematical physicist John C. Baez and posted on his blog.
Egan has also recommended that New Scientist publish Costella's refutation of Shawyer's theory paper.

# Analysis
Any claim of a reactionless drive is treated with skepticism by the physics community, since it violates well-established principles such as the conservation of momentum and conservation of energy, both of which have enormous experimental support.
Since there are no known phenomena that do not conserve energy, any calculation based on standard physical theory that predicts a violation of energy conservation almost certainly is in error. This is a non-controversial and fundamental fact regarding the mathematical structure of the theories, regardless of whether the theories themselves are or are not correct descriptions of the physical world. Accordingly, the results reported regarding the EmDrive, if true, would demonstrate that existing physical theory (or its application in engineering) is incorrect or incomplete.
The EmDrive has been compared to the previous Dean Drive, in that an oscillatory motion is set up so that it has a different effect in each direction of the stroke, in the hope that momentum transfer will differ in each direction, except in this case the oscillations are said to be electromagnetic.
Conservation of momentum is also required and maintained in Maxwell's equations, Newtonian mechanics, Special relativity and quantum mechanics (and their combination, quantum electrodynamics), so this claim cannot be valid unless these well-established physical theories are false or can be otherwise explained in terms within these existing theories.
Shawyer's calculations may be in error. He may have incorrectly identified the forces on the sides of the waveguide. If an error is present, it is most likely that the 'thrust' is eliminated and the drive then cannot accelerate. Despite some criticism Shawyer still claims his machines work.
Any dispute will be settled when independent observations are able to conclude whether or not the machine works in the way it is claimed.