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Thrust from Symmetric Capacitors using

Quantised Inertia: Version 2.

M.E. McCulloch

∗

August 6, 2021

Abstract

It has been found, by some experiments, that during internal eld

emission, capacitors thrust anomalously towards their anodes. It is shown

here, using an argument based around the well-known Casimir eect, that

this thrust is predicted by quantised inertia, a theory that also predicts

galaxy rotation without dark matter. One experiment has claimed that

the capacitors' thrust was as large as 120 N/kW, and quantised inertia

predicts that this is further enhanceable, making this a potential launch

system.

1 Introduction

Special relativity predicts that accelerating objects see a Rindler horizon behind

them because of the limitations of information transfer to the speed of light, and

this horizon splits virtual particles from the vacuum forming so called Unruh

radiation (Unruh, 1976).

A theory has been proposed suggesting that the horizon damps the Unruh ra-

diation on one side of the object causing an Unruh radiation imbalance that

pushes the object back against its acceleration (McCulloch, 2007, 2013). This

theory, called quantised inertia, and models inertia as an asymmetric Casimir

eect, provides the rst mechanistic explanation for inertia and predicts galaxy

rotation without dark matter (McCulloch, 2017). Quantised inertia also pre-

dicts that instead of horizons, metal structures can be used to damp the Unruh

radiation in such a way as to produce a thrust that does not require propellant

(McCulloch, 2018).

It was observed by Canning

et al

. (1990), T. Musha (2008) and Becker and

Bhatt (2018) that capacitors which undergo highly-accelerated discharges or, in

Becker and Bhatt's case eld emission, thrust towards their anodes. The thrust

∗

mike.mcculloch@plymouth.ac.uk. m.e.mcculloch@protonmail.com. Plymouth University,

PL4 8AA, UK

1

is many orders of magnitude larger than that expected from ion drift and many

times more than possible given that no ablation of the plates was seen. A good

summary is available in Martins and Pinheiro (2011).

Becker and Bhatt (2018) attempted, after some liaison with the author, to

demonstrate that quantised inertia predicts the thrust they saw. However, al-

though it was useful qualitatively, their derivation was unfortunately awed

because they used an uncertainty relation with

h

not

~/2

which gives an answer

12.5 times too large and they used another erroneous assumption resulting in

a formula that predicted well because these two errors cancelled (no oense to

Becker and Bhatt who should be commended for their technical skill & eort).

In this paper a derivation is presented and it is shown that quantised inertia

predicts the thrust from the capacitors and its dependence on plate separtion

and power very well, though some aspects, such as the reduction of thrust seen

when the cathode was thickened remain unmodelled.

2 Method

Figures 1a and 1b are schematics showing the dierence between the behaviour

of an electron, in quantised inertia, in both open space (1a, rst panel) and in

a capacitor (1b, second panel). Figure 1a shows that normally, in quantised

inertia, an accelerated electron sees more Unruh radiation in front (the orange

area) because there is no horizon there damping it, and less Unruh radiation

behind (the yellow area) because it is damped by the Rindler horizon (the

vertical black line). So the electron sees a radiation imbalance pushing it back

against its acceleration and it has what we refer to as inertia.

Figure 1b show the situation in the capacitor. The electrons are accelerated to

the right by a potential dierence between the plates. Again they see a Rindler

horizon, shown on the left by the vertical black line. The electron cloud will see

Unruh waves, but these will be deselected between the capacitor plates so that

the normal gradient is reversed and now there are more Unruh waves behind

(the yellow area) so the electrons get an extra kick forward which is translated

to the anode. In a sense the electrons' inertial mass reverses. The corresponding

gradient to the right outside the capacitor is far less important because, given

the acceleration, most of the electrons are on the left hand side of the gap and

the dynamics of the deceleration when hitting the anode are dierent: there is

a far higher deceleration than acceleration (see discussion).

In the derivation that follows we start from the Casimir eect (Quantised in-

ertia's technical name has always been a Hubble-scale Casimir Eect, and the

Hubble horizon can be replaced by metal plates). The Casimir force (from the

normal vacuum) pushing together two parallel plates is

F=~cπ2A

240d4

(1)

2

where A is the plate's area and d is their separation. Now instead of the normal

zero point eld that the Casimir eect assumes we take account for the fact that

the electrons see a much more energetic zero point eld due to their acceleration,

but which is still damped between the plates to pull the electrons forward. We

can model this by replacing the

~

(units Js) in the original Casimir eect by

the energy-time we are putting in electrically. The assumption is that all this

energy accelerates the electrons, and ends up as Unruh radiation which is then

damped between the plates

~→Eτ =Ed

v=P τ d

v=P d2

v2

(2)

where E is the energy between the plates,

τ

is the time it stays between the

plates and P is the power applied. The electrons' acceleration is

a=qeV/med

so their nal speed across the gap is

v=at =qeV t/med=qeV/(me(v

/2))

(v/2

is the average speed across the gap). So that

v=p2qeV/me

) and so their

average speed is half that, so

v2=1

/4(2qeV/me)

~→2P d2me

qeV

(3)

So the force is

F=2P d2me

qeV

cπ2A

240d4

(4)

Since P=IV we have

F=0.082IAmec

qed2∼0.00014IA

d2

(5)

This is the thrust predicted by quantised inertia and will be transmitted by

electron impacts to the anode and the capacitor structure. The electron accel-

eration occurs as the electron passes between the plates. The deceleration at the

anode is much larger, but the Rindler horizon in that case is very close (higher

acceleration) so the Unruh waves to the right are fewer or similar to those seen

between the plates, so there will be an extra thrust rightwards (see Fig.1, fourth

panel). Note that Eq. 5 has no arbitrary parameters.

3 Results

Figure 2 shows the results of the experiment of Becker and Bhatt (2018) as

the black diamonds. The observed thrust is on the y axis as a function of the

separation of the capacitor plates on the x axis. The error bars are also shown.

Figure shows the predictions of quantised inertia by the squares. The error

bars on the predictions, of 30%, assume an error in the current of 20% plus an

3

error in the estimate separation of 10% (Becker and Bhatt, pers. comm.). The

predictions agree within the error bars of the data.

It can be seen that the thrust increases exponentially as the plate separation is

reduced and the largest observed force (on the left) was about

6×10−3N

for a

voltage of 5000V and a current of

10µA

which implies a power of 0.05W. This

means that this capacitor has a thrust of 120 N/kW. If a thruster was built using

for example a Lithium-air battery that is capable of 1.8kW/kg then an upwards

force of 180N would be achieved, much larger than the downwards force from the

battery (1kg=10N) and thruster (a weight of up to 1kg = 10N). These results

therefore show that this thruster could be used to launch signicant payloads

into space. From Eq. 5 is is clear that the best way to enhance the thrust would

be to reduce the separation of the plates.

4 Discussion

Eq.5 predicts that the thrust depends most strongly on plate separation, but

also linearly on current and power. The predicted thrust does not depend on

the voltage, because although a higher voltage increases the amount of power

between the plates it reduces the time the electrons spend there by the same

proportion. Indeed Becker and Bhatt (2018) found a linear dependence with

both current and power, which suggests that indeed voltage was not important.

Eq. 5 also predicts a dependence on capacitor area.

It should be noted that this is only an approximate model, that assumes, for

example, that metal plates completely block Unruh radiation. A factor which

is simply unknown. The model cannot cope with more complex arrangements.

For example, when the cathode was thickened, the result was that the thrust

reduced and when a neutral conductor was positioned between the cathode and

the Rindler horizon the force reversed (Becker and Bhatt, 2018). The reason

for this is not clear.

In the case of the capacitor itself, it was assumed that the plates completely

enforce a node on the Unruh waves at the position of the plate. In the case of

arbitrary arrangement of metal plates this is more dicult to model since more

than one node will be enforced and the thickness of the plates and the damping

factor will have to be considered.

Finally, a metal plate placed to the right of the calculated position of the Rindler

horizon caused a reversal, but when the plate was placed to the left of the

Rindler horizon it had no eect. This makes sense in quantised inertia since

events beyond the horizon cannot be 'known' by the accelerated electrons by

denition. Therefore this test by Becker and Bhatt (2018) may be the rst

direct observation of a Rindler horizon.

4

5 Conclusion

Capacitors with electron eld emissions show unexplained thrust towards their

anodes.

It is shown, by analogy to the Casimir eect, that quantised inertia (QI) can

predict this thrust.

The experimental results show the thruster is potentially a launch system.

QI predicts exponentially more thrust for closer plates.

Acknowledgements

Thank you to F. Becker and A. Bhatt who valiantly did the experiment, and

kindly provided the data, J.R. Hall, Mike Fiddy, and NosaerFodnus on twitter,

for comments on previous drafts and to DARPA for funding grant HR001118C0125.

References

Becker, F.M. and A.S. Bhatt, 2018. Electrostatic accelerated electrons within

symmetric capacitors during eld emission condition events exert bidirectional

propellant-less thrust. Arxiv: https://arxiv.org/abs/1810.04368

Canning, F.X., C. Melcher, W. Winet, 2004. Asymmetrical capacitors for

propulsion. NASA/CR-2004-213312

Martins, A.A., and M.J. Pinheiro, 2011. On the propulsive force developed

by asymmetric capacitors in a vacuum. SPESIF-2011.,

Physics Procedia

, 2011:

20:112-119.

McCulloch, M.E., 2007. Modelling the Pioneer anomaly as modied inertia.

MNRAS

, 376(1), 338-342

McCulloch, M.E., 2013. Inertia from an asymmetric Casimir eect.

EPL

, 101,

59001.

McCulloch, M.E., 2017. Galaxy rotations from quantised inertia and visible

matter only.

Astro. Sp. Sci.

, 362,149.

McCulloch, M.E., 2018. Propellant-less propulsion from quantised inertia.

J.

Space Exploration

. 7(3).

Musha, T., 2008. Explanation of Dynamical Biefeld-Brown eect from the

standpoint of zpf eld.

JBIS

, 61, 379-384.

Unruh, W.G., 1976. Notes on black hole evaporation,

Phys. Rev. D.

, 14, 870.

5

Figures

Figure 1. Schematic showing (a) an electron accelerating to the right under

quantised inertia. It sees more Unruh radiation in front of it (orange) since

there are no horizons there. It sees less Unruh radiation to the left (yellow) as it

is damped by the horizon (black line). This pushes the electron back against it

acceleration - inertia. In the capacitor (b) the opposite occurs. The accelerating

electron sees few Unruh waves in front as they are damped by the capacitor

plates (white area) and more waves behind (yellow) so the electron is pushed

forwards more than expected and pushes the anode plate. The deceleration is

more abrupt when the electron hits the anode (c) which enhances the rightward

thrust.

6

Figure 2. Results. The observed thrust from Becker and Bhatt (2018) (black

diamonds), and that predicted by QI using Eq. 5 (squares).

7