The small anomalous acceleration of the emdrive (which is like a microwave oven built into a copper cone) towards its narrow end, may or may not be a real effect, but is proving to be a lot of fun to think about. I've written before about 'thinking in the context of a real experiment' and I think that is lacking in mainstream theoretical physics where people incessantly drop objects into imaginary black holes. Looking for new physics one must look for 'direct' observations that disagree with the old physics, and the emdrive is a great example of that, and one that has been reproduced in three different labs. It turns out that MiHsC predicts it quite well (see this entry, and for MiHsC, see McCulloch, 2013, 2014, references below). Specifically MiHsC predicts a force of

F = PQ/f * ((1/w_big)-(1/w_small)) (1)

where P is power input (Watts), Q is the number of photon 'bounces' before they are absorbed, f is the input microwave frequency (Hz) and w_big and w_small are the diameters (metres) of the big and small emdrive end plates. Last night I worked out how to prove this formula from first principles so I'm now much happier about it. I've also tried replacing f with c/CavityLength=c/s, where c is the speed of light. This is because in the derivation the f is the frequency of Unruh radiation seen by the photons, and this is determined not by the microwave frequency, but by the cavity length which forces the microwave photons to change direction back and forth, and it is this that does the accelerating. This gives the similar formula

F = PQs/c * ((1/w_big)-(1/w_small)) (2)

Now, you may be saying 'for photons the rules are different', but ultimately, causes must be universal and photons do have inertial mass, the source of radiation pressure (it is only the photons' rest mass that is zero) and if they have inertia, then according to MiHsC this is caused by Unruh radiation (which is more than just em radiation) and for the huge accelerations of these photons as they bounce between the ends of the cavity the Unruh waves are short enough to 'see' the cavity and be damped by its walls (as electrons move to cancel the field there). They will be more damped at the narrow end, meaning they will have less inertial mass at that narrow end. This means that, for each photon bounce, more mass goes from the narrow to wide end than the other way around, and so to balance the books the conservation of momentum demands a force towards the narrow end of a size as in Eq. 1 or Eq. 2. The best defense of this model is that it works quite well. For the results of Eq. 1 see two blogs back, and Eq. 2 predicts the following for the five experiments for which complete information is available (excluding the Chinese experiments whose geometry is unknown). Table:

Experiment P Q s w_big w_small Observed Predicted

(see refs) W dl m m m (mN) (mN)

-------------------------------------------------------------------------------------------------------------

Shawyer (expt) 850 5900 0.156 0.16 0.11 16 7

Shawyer (demo) 1000 45000 0.345 0.28 0.17 147 123

Brady

*et al*. A 16.9 7320 0.332 0.397 0.244 0.09 0.22

Brady

*et al*. B 16.7 18100 0.332 0.397 0.244 0.05 0.53

Brady

*et al*. C 2.6 22000 0.332 0.397 0.244 0.06 0.10

The table shows that the MiHsC predictions (the last column) agree quite well with the data (the second to last column), except for the second Brady

*et al.*(2014) experiment. More experimental data is urgently needed, but this suggests tentatively that MiHsC can be applied to light in cavities, and this opens up a whole new area for testing. I have submitted a paper on this, so let's see what the reviewers say. Note: Thanks to Dr Jose Rodal, aero & John Fornaro (NSF forum) for clarifying the emdrive geometry.References

Brady, D.,

*et al*., 2014. Anomalous thrust production from an RF test device measured on a low-thrust torsion pendulum. Conference proceedings, see Table page 18. Link

Shawyer, R., 2008. Microwave propulsion - progress in the emdrive programme. See section 6, page 6. Link

McCulloch, M.E., 2013. Inertia from an asymmetric Casimir effect. EPL, 101, 59001. http://arxiv.org/abs/1302.2775

McCulloch, M.E., 2014. Physics from the Edge, published by World Scientific. Link

## 25 comments:

Interesting post by notsosureofit at the NASA Spaceflight EM Drive thread:

"Just an odd analogy that just popped up. Note the motion to the right of the resonating "Hawking Black Hole"

http://goo.gl/PUkfB5

OK, thanks for the new estimates from Mulletron. Would be useful to know which values to trust most. I'll redo the table with a range of predictions for the Brady expts...

John (at NASA SpaceFligth forum) will provide another AutoCAD assessment on Brady's dimensions. Better to wait until then to recalculate, since then we will have 3 assessments for Brady's dimensions.

Very interesting theory. I have been following it since it was posted in Next Big Future.

Glad to see there could be some experimental confirmation.

It is interesting that in Shawyer's superconducting EM Drive design (http://www.emdrive.com/iac2014presentation.pdf slide 4) he has considerably reduced the ratio of the radial length "L" of the cavity with respect to the "diameters of the bases".

Previous Shawyer's design and NASA Eagleworks frustum approximately L = BigDiameter and L > SmallDiameter

in Shawyer's new design L < BigDiameter and L < SmallDiameter

this now is closer to meeting the required acceleration for Unruh waves as one needs to meet:

(Diameter/(CavityLength) > 4

Although Shawyer does not describe the cavity mode shapes the cavity is resonating at, Brady et.al. does. Brady et.al (NASA) used COMSOL Finite Element analysis to analyze the mode shapes. It is interesting that Brady et.al (A and B in your table above) in your table correspond to the TRANSVERSE MAGNETIC field mode shape TM211, while Brady et.al. (C in your table above) corresponds to the TRANSVERSE ELECTRIC field mode shape TE012.

The TRANSVERSE ELECTRIC mode shape resulted in a much greater thrust force/PowerInput than the TRANSVERSE MAGNETIC mode shapes

These are the dimensions obtained by John Fornaro. John used a superior method, using the known dimensions of the cross section of the beam, so in my opinion they should be the one taken into account, and all the other measurements for Brady et.al. should be ignored

Brady et.al.

Length = 0.33245 m

BigDiameter = 0.39697 m

SmallDiameter = 0.24393 m

Many thanks. I'll update the table. I'm v. interested by Shawyer's new dimensions & that the transverse electric (TE) mode is more effective. It's great to have a controllable lab expt to test MiHsC with.

Ok. Can you explain once more to me, because I don't get it: where is the momentum transfer from the pod supposed to go?

If nowhere, then this an over-unity drive, as all reaction-less drives inherently are.

Would be nice, but a much simpler explanation remains: the effect is a measurement artifact.

The Casimir effect produces a real force out of 'nothing' by making the zpf vary in space. This is proven. If MiHsC is right & the emdrive expt is right, then the emdrive is similarly producing a force out of 'nothing' by making Unruh radiation vary in space. It seems odd only bcos physics has so far ignored the proven ability of the zpf/Unruh for making force.

Well. I like wild speculation, but your hypothesis (if I am able to grasp you meaning correctly) that vacuum ground state could be used to perform work sounds simply wrong.

So you are proposing a mechanism for perpetual motion by a miracle?

To my experience, nature don't like miracles.

Thank you for your answer, I am happy with that.

The Casimir effect has been observed in labs all around the world. No miracle.

Classical aspects of Hawking radiation verified in analogue gravity experiment:

http://arxiv.org/pdf/1302.0375.pdf

"When the thermal emission was originally discovered by Hawking,

it was believed to be a feature peculiar to black holes. Our experiments, and

prior numerical work [21, 7], demonstrate that this phenomenon seems to be

ubiquitous, and not something that relies on quantum gravity or Planck-scale

physics."

aero found this information in the Chinese paper by Juan Yang http://www.emdrive.com/NWPU2010translation.pdf that confirms that the Chinese (just like NASA Brady et.al.) found that the highest thrust forces and the highest Q are produced by the Transverse ELECTRIC resonant modes and not by the transverse magnetic modes:

"Using finite element numerical method to numerical analyse the classical Maxwell equation of electric field of the idealised conical resonator, to obtain the model and practical of the distribution of the electric field of the cavity under 1000W. By analyse the properties under different modes and the different properties. Calculation show that under the four modes, TE011, TE012, TE111 and TM011, the quality factor of TE012 is highest and with highest thrust, followed by TE011."

contintued, from the Chinese paper (TE = Transverse Electric mode

TM= Transverse Magnetic mode):

"The TE10 mode rectangular waveguide is suitable for use in modes TE011 and TE012, and coupling to the side wall of the resonator in where the magnetic field is relatively strong, coupling using coaxial cable is suitable in modes TE011 and TM011, in axis of the resonator where electric field is relatively large.

With 1000W microwave input, using brass as the material of the cavity, using the classical theory of electrodynamics, the maximum theoretical thrust produced in modes TE011 and TE012 is 411mN and 456mN respectively, and the practical measurements are 214mN and 315mN."

Thanks. Important clue, but I'm going to have to find out what all these different modes look like..

This is a closed-form solution for a truncated cone with hemispherical end surfaces. This is more like Shawyer's new superconducting drive.

Also take into account that while the experimental Shawyer is a truncated cone, the demo Shawyer is acylinder with a truncated cone joined to it while the NASA drive inner surface also looks like a cylinder joined to a truncated cone. Here is the truncated cone with hemispherical ends solution: http://gregegan.customer.netspace.net.au/SCIENCE/Cavity/Cavity.html

The Greg Egan link was useful. It shows that the E-field is always zero at the cavity wall (as assumed in MiHsC) but the M-field isn't..

Greg Egan also shows that using classical electromagnetics (Maxwell's equations) all the electromagnetic forces sum up to zero. There is some confusion in the literature. For example in the Chinese paper where they argue otherwise. I think this is because the Chinese use the asymmetric Minkowski stress tensor instead of Abraham's symmmetric stress tensor. The mass density must be symmetric (to have a center of mass). Mass = E / c^2 . Hence if the Chinese wanted to use the Minkowski stress tensor they should have included so-called "hidden momentum", otherwise they end up with incorrect results when using classical electromagnetics.

Considering the EM Drive as a closed system, if there is a force it has to be explained as a result of an external field, not just with classical electromagnetics.

I have compared the experimental measurements vs. your predictions and vs. the predictions of Shawyer's equations here: http://forum.nasaspaceflight.com/index.php?topic=29276.msg1276053#msg1276053

Shawyer's predictions are pretty close to your predictions except for Shawyer's experimental EM Drive (the smaller size drive) where Shawyer's prediction is approximately 6 times larger than yours. Overall your predictions are closer to the experimental results with the added benefit that your equation is much simpler than Shawyer's and in it involves less variables as in your present form the microwave operating frequency is not part of your equation.

Great work! I'd also say that MiHsC is a self-consistent theory, whereas Shawyer's result is based on a misunderstanding of relativity which by itself shld cons' m'tum. I'm working on a 3d formula (tricky geometry & volume integrals!) to answer your pointed cone question, & I can then hopefully suggest an experiment that MiHsC predicts would maximise the anomaly.

Due to Shawyer including a "group velocity correction" factor that blows up when the free-space-wavelength becomes equal to the Geometric Average of the diameters of the truncated cone's bases, Shawyer's formula goes to infinity for his "experimental" EM drive when the small diameter becomes about 50% of the big diameter. That's why his prediction for that case is 6 times greater than yours. For comparison, your 1-D formula blows up when the small diameter approaches zero. I have discussed this here: http://forum.nasaspaceflight.com/index.php?topic=29276.msg1276505#msg1276505 including more details and an image (using Mathematica) of this problem with Shawyer's formula.

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