How do you apply General Relativity to the EmDrive Problem?

Question

In the last couple of months a peer reviewed study from NASA has asserted that the results from EmDrive warrant further testing in space. In addition, in the last couple of days the Chinese government has claimed that they've been looking into the drive since 2010 and that they're already testing in space.

Thus, I think that the question of whether or not any of our current theories support such a device should be taken a bit more seriously. My question is whether or not using the full power of General Relativity might be sufficient to resolve the conservation of momentum violation that seemingly has to occur for the device to work.

I've been studying the theory over winter break and I'm about at the point where I'm starting to look at Einstein's Field Equations. It seems as though momentum isn't conserved in the global sense because it's impossible to add vectors at different locations due to the curvature of space-time. Rather, it seems as though the main conservation law is the equation for covariant derivative of the stress-energy tensor which is,

$$\nabla_v T^{uv}=0$$

This is of course only a local conservation law, thus the validity of some kind of global conservation isn't exactly obvious.

Now, in the proposed EmDrive, an electromagnetic wave is shot down from inside the base of an insulated cone towards the tip. I don't really know the order of the effect, but as the waves are compressed towards the tip, the components stress-energy tensor at that location should go up. Thus, since the curvature of space time is directly related to the stress-energy tensor, space time should subsequently be more strongly curved at the tip of the cone as opposed to the base. Thus, the whole structure will be warped. Now, because most of the mass of the EmDrive is towards the base, the structure as a whole should travel towards the more curved portion of space-time...which happens to be forward and thus generate thrust???

Can someone explain how to do the math of this properly? I'm reading Gravity and I'm in the section about linearizing the equations. Could that be done here?

Answer 1

In principle your thoughts are correct, but they do not explain the effect measured by NASA, because when you calculate the extent of the effects you name, they are too small to account for the effect that NASA supposedly measures.

You're correct that your equation only expresses a local conservation. There is, in general, no global conservation. And your sketch of how, in principle, you would generate thrust is correct (although I'm not sure the description "generate thrust" is the best phrase for this phanomen). Indeed, there are several thought experiments grounded on global nonconservation of momentum, the best known of which is probably the swimmer; see J. Wisdom, "Swimming in Spacetime: Motion by Cyclic Changes in Body Shape", Science Magazine, 299, $21^{st}$ March 2003. This effect is unbelievably small for Earthly spaceship-sized objects, being of the order of $10^{-23}{\rm m}$ for each cyclic shape deformation of meter scale objects in the spacetime curvature present in low Earth orbit. Note that it is the whole Earth giving rise to the curvature needed for this swim!

One can even generate thrust if the conservation laws hold, because one can use Newtonian tidal effects to transfer angular momentum from the Earth to a spacecraft's orbital angular momentum; see Michael J. Longo, "Swimming in Newtonian space–time: Orbital changes by cyclic changes in body shape", Am. J. Phys. 72 10 , October 2004. This is a much bigger effect, but still too small to account for the NASA experiments. One can still think of this as a curvature effect; see the relevant chapter, approximately Ch20, in Misner, Thorne, Wheeler, "Gravitation", where E. Cartan's geometric formulation of Newtonian gravitation is reviewed. This is the same effect that slowly raises the orbital energy of the Moon's orbit, such that the Moon's distance from Earth increase by about 3cm each year. This effect is much bigger than the relativistic one; Longo estimates a hundred meter scale spaceship can de-orbit, lowering its perigee by five centimeters a day, by this method.

A practical transfer of angular momentum is realized in the manoeuvre of Gravitational Slingshot, which is essential to the acceleration of spacecraft to Solar System escape velocities.