A few years ago we heard a lot of controversy surrounding an unlikely new propulsion system. The new system has many names: Impossible Engine, Shawyer Drive, Cannae Drive, RF Resonant Cavity Thruster, or simply EmDrive. It claims to provide a source of thrust without requiring a propellant. The source of the controversy stems from the fact that such a system would violate fundamental physics, namely the conservation of momentum.
Isaac Newton first described the conservation of momentum and it is often stated as "For every action, there is an equal and opposite reaction." In other words, you can't push against something without it pushing back. More specifically, if I want to impart momentum to an object I must simultaneously impart an equal amount of momentum to another object in the opposite direction. Momentum is a vector quantity, meaning it has both magnitude and direction. It is defined as the product of the velocity times the mass of an object.
Conventional rockets move by pushing mass out the back. I'll never forget the first demonstration I saw as a child on Newton's Apple (1988) where a gentleman sat in a tiny wagon and proceeded to hurl tennis balls from a basket on the wagon. He threw the balls as fast as he could and sure enough, the wagon began to move in the opposite direction. He explained that in order to throw the tennis ball, he needed to push it away and according to the conservation of momentum it pushed back. The big change in velocity given to the small mass of the tennis ball meant that there was also a small change in velocity given to the large mass of the wagon in the opposite direction. Thus the overall momentum of the system was still zero.
The EmDrive appears to violate this principle since it requires no reaction mass to push against in order to generate a change in momentum. I've heard it described as trying to make your car go faster by pushing on the steering wheel or literally pulling yourself up by your bootstraps. According to Newton, there should be no way to change the momentum of a mass without using some other mass. This lead to a great deal of skepticism concerning the EmDrive, and initially, it was dismissed as malarky.
It's important to note that Newton isn't the final word in physics. While all of his principles hold under conventional conditions, then often break down on very large and very small scales. The first Newtonian equation to need revision was the force equation F=ma. Einstein elegantly revised the equation to make mass a function of velocity and thus he was able to introduce the theory of relativity without offending anyone.
Despite the widespread dismissal of the Emdrive, a few scientists, most notably Harold White of NASA's Eagleworks, decided to test the device to see if there might be something to this claim of reactionless propulsion. So far the tests have all showed a preliminarily positive result, with heavy caveats concerning possible sources of error. That's science speak for it looks like it worked but we still aren't sure. Until the device is launched into outer space and shown to work in real-world applications, there will still be doubts. The Cannae Corporation is planning to launch their own proprietary RF resonant cavity thruster in a cube sat to finally put the question to rest. Let's hope they decide to share the results!
That hasn't stopped the theorists from trying to figure out the underlying principles of the device. Several papers have been published on the subject and they all have subtly different notions about how this device might work. Harold White and his team have made stabs at the principle by suggesting things like non-trivial vacuum interaction and calling on pilot wave theory, but they stopped short of penning any concrete theories. A recent paper for a Portuguese team from the University of Lisbon has made some recent waves, you can find a draft here. We here at FTL have done our own derivation that we feel might hold some water. Our theory aligns closely with the Lisbon paper, but we took a different approach.
What follows is a purely speculative exploration of the underlying physics behind the as yet unverified and potentially erroneous claims about the RF resonant cavity thruster. Let's get cracking!
We first speculated that there may be some kind of quantum tunneling effect that would cause an asymmetry in radiation pressure in the chamber. That was dismissed since there was no way to show a mechanism for a probabilistic affinity for tunneling photons, and they should be equally likely to go every which way. But that got us thinking.
Next, we explored the resonant modes of the cavity itself. The EmDrive employs a finely tuned wave generator that sets up a standing waveform inside the cavity. This standing wave resonance would be easy to set up if the cavity were uniform, but it isn't. The cavity resembles a tapered cone shape with the top cut off. The walls of the cavity narrow from one end to the other and this means that the resonant properties of the cavity change slightly across the length of the device. The waves in the cavity would begin to interfere with each other and through a process of constructive and destructive interference the radiation in the cavity would set up the best standing wave pattern it could in isolated sections of the cavity.
Two assumptions were made in this system. One, that the walls of the cavity constitute an infinite potential well, and two, that the wave amplitude must be zero at the walls. The implication of this is that the waves inside the cavity must have half wavelengths that match integer divisions of the length and width of the cavity. Light waves are spherical and thus follow the three-dimensional wave equation.
Thus these waves must set up lobes which conform to the contours of both the horizontal and vertical potential boundaries of the cavity. It would impossible to do this with a single frequency of radiation since the width of the cavity changes as the walls narrow. Thus there must be a gradient of frequencies within the chamber. The equation for the energy of a photon is defined as the frequency of that photon multiplied by Planck's constant.
Energy has another definition in terms of force. That is, energy, or work, is defined as force integrated over a distance.
Using these two equations, one from quantum mechanics and the other from Newtonian physics we composed the following derivation. We start by differentiating the equation for the energy of a photon by frequency.
Next, we perform a similar operation on the work-energy equation. This we do for a one dimensional case.
Now we set the two equations equal to one another.
We then solve for force F.
We then apply a limit and allow the frequency delta and space delta to go to zero.
A bit of handwaving later and we can say that in the three-dimensional case the equation should appear as the gradient of the frequency across the region.
Now that is admittedly our novice derivation of something that is hotly debated and we don't claim for a second that this explains the principle of the EmDrive. It does seem to suggest that a change in the frequency of light across a region would result in a force. We asked ourselves, where have we seen a force coupled with a change in frequency?
The answer turned out to be simple, a gravitational well. Gravity causes light rays to become blue shifted as they move deeper into a gravitational well as a consequence of General Relativity. Could it be that the reverse is true? The equivalence shown here would suggest that a shift in frequency would result in a force. Gravity is a force, photons are the mediators of force, and Compton Scattering operates under a similar principle of changing the photon's frequency and angle in order to impart a force on an electron.
Wikimedia Commons, User:Vlad2i
This is where our notions get fuzzy. Just what happens when a photon strikes an electron? Is this related to the Casimir Effect? Could gravity be thought of as a manifestation of varying the frequency of virtual photons in the non-trivial vacuum? I'm sure most of the more rigorous and skeptical readers already tuned out long ago, but we started to question our basic assumptions. Perhaps the Emdrive is a gravitational field generator. Time will tell... if it even works at all.