Charrier1,2 recently proposed a novel propellantless thruster concept that makes use of retarded electromagnetic (EM) interactions3 to generate a net thrust force. The device1 (which appears similar to an earlier concept proposed by Goodwin4) consists of a coil (with or without a laminated steel core) coaxially attached to a copper disc placed at a distance L away. Because it takes EM signals a finite time to reach the disc (, where v is the speed of light in the core material), it was argued that by applying a fast current ramp (at t = 0) to the coil, a net thrust (due to a Lorentz force induced on the disc) can be produced on the device for times between approximately . For , a counter force is induced on the coil due to EM waves emitted from the disc, which result from eddy currents produced by the original EM waves from the coil. By then applying a series of current ramps, a continuous thrust can be generated.
Within the theoretical model used to describe the thrust force in Refs. 1 and 2, there is a subtle yet important assumption which is not discussed, and which we highlight here: when the Lorentz force is first induced on the copper disc, it is assumed that this force is instantly transmitted to the coil as well. This is equivalent to assuming that the device can be treated as a perfectly rigid body. However, within the context of special relativity the concept of a rigid body breaks down,5 and it would take a finite time for this force to be physically transmitted to the coil. The minimum time taken would be equal to L/v, although in practice force signals in solids propagate at speeds of the order of the sound speed in the solid, us, and since , the counter force on the coil will always appear before the force signal from the disc has arrived. Thus, the coil and disc will each slightly stretch in opposite directions relative to some “central” point located in between. Hence, if any net force is produced on the device, it must be significantly smaller than that predicted in Refs. 1 and 2.
The above discussion can perhaps be more easily illustrated with a simple thought experiment. Consider a steel rod of length L with two observers, A and B, located near the right-hand side (RHS) and left-hand side (LHS) ends of the rod, respectively. At t = 0, observer A applies a known pulling force (either mechanically or electromagnetically) to the RHS end of the rod, while simultaneously sending a light signal (perhaps with a laser) towards observer B. When observer B receives the light signal he then also applies a pulling force (with an equal magnitude agreed upon by both observers prior to the experiment) on the LHS end of the rod. In the context of the present thought experiment, the theoretical model used by Charrier in Refs. 1 and 2 essentially assumes that the rod is perfectly rigid, and hence that when observer A applies the force to the RHS, it is instantly transmitted to the LHS, so that the whole rod is immediately displaced to the right before observer B has received the light signal (which takes a time L/c, where c is the speed of light in vacuum). Thus, the special theory of relativity is violated. As with many similar apparent paradoxes in relativistic mechanics, the resolution lies with the fact that there is no such thing as a perfectly rigid body. In reality, when observer A applies the force to the RHS, it takes a finite amount of time before the force reaches the LHS. As mentioned above, force signals typically propagate at speeds of the order of the sound speed, us, in solids, and again, since , the light signal will always reach observer B before the force applied by observer A. A similar discussion applies when observer A turns off the force at the RHS (again simultaneously sending a light signal towards observer B telling him to do the same). Thus, while the rod will very slightly stretch by an amount, , when the force is applied, the center-of-mass of the rod will not be displaced after a complete on-off cycle. We point out here that despite the fact that , this does not mean that the rigid body concept is valid (as we have seen above).
Because of the non-symmetric thruster geometry, it is expected that the device in Ref. 1 can still generate a small net thrust simply due to the asymmetric emission of EM radiation (which carries momentum3). In this case, the device would behave essentially identically to the more well-known propellantless “photon” thruster concepts (see, for example, Ref. 6 and references therein). Such photon thrusters have also recently been technically demonstrated.7 The major disadvantage with these systems, however, is that even for a 100% efficient onboard power supply and a perfectly collimated radiation beam, the thrust-to-power ratio, T/P, is extremely low, and given exactly6,7 as . From the data supplied in Ref. 1, we can roughly estimate the device power by multiplying the applied voltage (20 V) with the maximum coil current (0.3 mA; found by multiplying the current ramp gradient, 14.4 A/ms, with the ramp duration, 22 ns), which gives about 6 mW. This, together with the measured thrust force of 4 μN, gives ; a value 5 orders of magnitude larger than that achievable by radiation/photon pressure alone. Note that the power calculation above likely overestimates the actual radiated power, and so the thrust-to-power ratio of the device will deviate even further from the theoretical limit of 3.33 nN/W. Thus, the measured forces in Ref. 1 must either be due to an interaction of the device with its surroundings or some other (as yet unspecified) phenomenon. This conclusion is further supported by the fact that the measured displacements in Ref. 1 were found to be similar both with and without the steel core present, yet calculations of the forces with the equations supplied in Ref. 1 (and also the discussion in Ref. 2) suggest that with the core removed the thrust should be many orders of magnitude smaller.
Finally, on a lesser note, we point out that Eq. (4) in Ref. 1 appears to contain an error, since it has the wrong units, [Nm], for force.
