In what sense is it a perpetual motion machine? As far as I understand it violates the conservation of momentum, not the conservation of energy. How would a theoretical perpetual motion machine based on this effect work?
>The cone allows Unruh radiation of a certain size at the large end but only a smaller wavelength at the other end. So the inertia of photons inside the cavity must change as they bounce back and forth. And to conserve momentum, this must generate a thrust.
The new idea introduced is the quantization of inertia at small accelerations. As far as I understand it, from one end of the cone to the other there isn't a smooth change of inertia. This depends on the idea of Unruh radiation, and the reason the article gives for quantization of inertia is that as accelerations get very small, Unruh radiation wavelength becomes larger than the observable universe, forcing unruh radiation to take whole-value wavelengths (quantization). Again as far as I understand it, the inertia of photons on one end of the cone takes a different quantized value of inertia than the photons on the other end. So the thrust isn't a violation of conservation of momentum. The thrust is necessary to not violate conservation of momentum.
A nonzero photon mass makes a mess of particle physics at high energies. The Standard Model falls apart due to loss of gauge invariance, and QED becomes really obviously non-renormalizable near limits we have already tested. So this would be a big thing.
(Note that if you bite the bullet and take a nonzero photon mass (as McCullogh says in his paper at page 3: "Normally, of course, photons are not supposed to have inertial mass in this way, but here this is assumed") you can probably get a non-constant photon speed too, along the lines of neutrino oscillation. But you can always write down non-physical theories that conflict with frequently and precisely tested areas of physics...)
None of the above departs from the symmetries of Special Relativity, and one subgroup of those (invariance under spatial translation) is what implies the (local) conservation of (linear) momentum per Noether's (first) theorem.
So there's a really big looming question about why the Standard Model (which has the Poincare Group baked into it) works as well as it does everywhere but in an EmDrive or a single space probe, and resorting to special shapes of objects on scales much larger than that of atoms further conflicts with Poincare invariance.
Finally, Unruh radiation is a difference in particle count and particle energy measured by differently accelerated observers. The cosmological horizon, when it formed, produced an acceleration between observers then and observers in the future. Unruh radiation from that is pretty uncontroversial. However the problem is that the acceleration is pretty small, so the temperature will be much lower than that of the CMB. Also, you'd expect anisotropies based on Earth's (and its surrounding Local Group's) peculiar motions relative to the horizon. Why isn't there a dipole anisotropy similar to the one we see in the CMB, and if there is, by how much does it offset the inertial argument in McCulloch's paper?
The local symmetries in question are represented by the Poincare Group, which is the isometry group of Minkowski spacetime, which in turn is the unremovable background of Special Relativity. (The Lorentz Group is a subgroup of the Poincare Group).
This is another way of saying that in a 3+1 flat spacetime, well-designed local probes of fundamental physics will not depend on time translation (i.e., experimenting today vs experimenting tomorrow), on spatial translation (e.g., the same experimental results here vs there), on spatial orientation (i.e., rotation about the 3 spacelike axes; so you get the same result when you turn the system under test 90 degrees to the left), or under Lorentz boosts, which are basically instantaneous changes in constant uniform motion along any of the axes. Additionally, small-scale (small compared to several times the size and mass-energy of the whole solar system) natural phenomena are essentially always "well-designed local probes of fundamental physics".
Conservation of linear momentum arises from invariance under spacelike translation; a violation of that conservation makes it very difficult to maintain spatial translation symmetry, and in particular flies in the face of the many direct tests of physical systems at different places on our planet and in our solar system, for example. So that's a big deal that needs explaining, and in particular the explanation should preserve the known and reproducible invariance under those translations as well as allowing the EmDrive's alleged violation.
The perpetual motion machine argument is slightly different because it is rooted in a claim about how the EmDrive behaves when in non-constant motion; in the current version of the theory paper "V 9.4" at equations 14-16, there is an implied violation of the Einstein Equivalence Principle. The equation is a bit odd, and you easily can read it to say that there is a power<>thrust relationship that varies with acceleration, and raise the Einstein elevator objection: if you put the EmDrive into an upwards-accelerating box, or leave it on the ground, do you still take this power<>thrust relationship seriously? If so, you get "free power" (although that's a bit subtle). If not, then you have to explain a violation of the Equivalence Principle, which is also something that has also been very well tested and has so far applied without fail.
You could think of it with a somewhat concrete example: to hover at some fixed height above the Earth's surface, the EmDrive would have to emit thrust similar to the amount of work you would do to hold it at that height with your hands; since you are near the surface, that should be about "g" expressed as N/kg (and indeed Eq 16 tells you how much electrical power you will require for the EmDrive). So far so good. But (eq 16) can be read to say that when you place the EmDrive on the ground, it will produce electrical power proportional to to "g".
I think this is pretty clearly just a mistake rather than a serious claim. Unfortunately eq 16 features prominently in the FAQ and all the "marketing" material about the EmDrive. (Even more unfortunately it's hard to see how to fix the equation without making the drive obviously inoperable).
Given an exotic enough explanation (negative energies/faster than light particles/spacetime warping/..) it feels like you can cheat your way around almost any invariant? Occam is left weeping in a corner of course.
The problem with reaching for exotic particles or the like is that we don't see them in searches which involve tens of orders of magnitude more power, and we certainly don't see them in, for example, electric toasters or radar sets. So what would need explaining is what's peculiar about the EmDrive with respect to (extremely) unusual particle interactions, and there is [a] nothing obvious and [b] nothing at all in the "theory" paper.
Likewise, reaching for non-local physics answers raises similar questions (why is EmDrive doing non-local whatever but my kettle isn't; or alternatively, why is EmDrive coupling with whatever much much much more strongly than my conventional oven is).
I have another pair of things that you can add to the list: [a] EmDrive somehow violates causality in a way that other similar arrangements of mass-energy-momentum does not and [b] EmDrive somehow escapes logic in a way that other similar systems under study does not. These are neither more far-fetched nor more unpalatable, compared to abandoning local physics (causality and logic preserved, but hidden non local variables proliferate) or abandoning the Standard Model as an accurate low-energy theory of matter (causality, logic and locality preserved, but now what happens in the molecules and atoms of cars, computer chips, and light bulbs? we no longer can be quite so sure!).
There is A LOT of known physics and almost exactly zero examples of violations of the known invariants at low particle energies. You are re-testing relevant parts of all that in reading this comment.