^{1}

^{*}

^{2}

^{*}

^{1}

^{*}

^{1}

^{*}

According to Einstein, General Relativity contains the essence of Mach’s ideas. Mach’s principle can be summarized by stating that the inertia of a body is determined by the rest of the mass-energy content of the universe. Inertia here arises from mass-energy there. The latter, was a statement made by John Wheeler in his 1995 book, Gravitation and Inertia, coauthored by Ciufolini. Einstein believed that to be fully Machian, gravity would need a radiative component, an action-at-a-distance character, so that gravitational influences on a body from far away could be felt immediately. In 1960’s, Hoyle and Narlikar (HN) developed such a theory which was a gravitational version of the Absorber theory derived by Wheeler-Feynman for classical electrodynamics and later expanded upon by Davies and Narlikar for quantum electrodynamics. The HN-field equation has the same type of mass fluctuation terms as in the Woodward Mach effect thruster theory. The force equation, used to predict the thrust in our device, can be derived from the mass fluctuation. We outline a new method for deriving the force equation. We present new experimental tests of the thruster to show that the thrust seen in our device is not due to either heating or Dean Drive effects. Successful replications have been performed by groups in Austria and Canada, but their work is still pending in the peer review literature.

Mach’s principle was the name Einstein gave, in 1918, to the proposition that the inertia of a body is the result of the gravitational interaction between the body and the rest of the mass-energy in the universe. In 1912, Einstein considered the gravitational interaction of a spherical shell of material and a point mass located at the center of the shell [

Moreover, Einstein had come to appreciate that as inertial forces are acceleration dependent forces, they had at least one of the signatures of radiative interactions. This had led him to draw a distinction between, as he called it, “the relativity of inertia” and “Mach’s principle”. Einstein took Mach’s principle to encompass the radiative nature of the presumed interaction between test particles and the rest of the “matter” in the universe. For this interaction matter, Einstein claimed it was based on “action-at-a-distance”, as inertial forces were experienced instantly on the application of “external” forces. The relativity of inertia merely required that inertia and inertial forces be dependent on the presence of a field to act on accelerating objects, and thus could be encompassed by his theory of gravity, general relativity. The type of action-at-a-distance that Einstein was talking about was the Newtonian type: instantaneous communication of effects over finite distances. The modern concept of action-at-a-distance is one that is consistent with the principle of relativity.

This paper is a follow on to paper I [

The modern version of action-at-a-distance was first introduced in the 1920s by Hugo Tetrode [

Absorber theory, which is a direct particle interaction theory, is never caught on. Wheeler never abandoned it and was still writing about direct particle interactions in his 1995 book, Gravitation and Inertia, 1995. Feynman recounted that when he presented the theory to a Princeton Physics Department colloquium [

Wolfgang Pauli who was sitting next to Einstein, said: “I do not think this theory can be right because of this, that and the other thing” …At the end of this criticism, Pauli said to Einstein, “Don’t you agree, Professor Einstein? I don’t believe this is right, don’t you agree, Professor Einstein?” Einstein said, “No,” in a soft German voice that sounded very pleasant to me, very polite. “I find only that it would be very difficult to make a corresponding theory [i.e., an action at a distance theory] for gravitational interactions.”

Einstein didn’t need to try to construct such a theory, for he was certain that inertia was already accounted for as a gravitational phenomenon in general relativity. General relativity does not encompass the version of Mach’s principle that includes inertial actions as radiative in nature.

Fred Hoyle and Jayant Narlikar [

Hawking [

In the conformal theory of Hoyle and Narlikar [

a further conformal transformation is needed to convert this equation into the Einstein field equation, see below.

where

To connect with the Woodward’s mass change equations, which were originally derived from Einstein’s and Maxwell’s results, we consider the extra terms in Equation (1) alone. The extra terms are on the right side of the equation along with the energy stress tensor. The 4-momentum density can be written,

as measured by an observer in a Lorentz frame with 4-velocity

where the

when we divide by

Apart from a 4/3 numerical factor, these are the mass fluctuation terms, originally derived by one of us JFW [

These results were reported in the last Joint Propulsion Conference Ohio 2014 [

Our standard derivation for the force is not ideal by any means. It should be considered an estimate only, and appears to give predictions within an order of magnitude of the actual result. For example see Woodward’s book [

where energy

After some simple trignometric manipulation we get,

where we have used expansions for

where

On resonance when

where we have substituted back for

The

tional to

In previous work, we always took the second line interpretation, and proceeded in terms of bulk acceleration. The previous assumption was that if no bulk acceleration of the object takes place, there would be no Mach type mass fluctuation. We previously took a sum over all the masses that make up the device. Using

The resulting mass fluctuation is caused only by the center of mass (COM) motion of the device,

where (1/m)

parts. The simplest Mach effect depends on the square of the acceleration of the COM of the body in which it is produced. [

The chief response of the PZT stack is piezoelectric. The stack responds to a periodic voltage,

The PZT stack has both a piezo electric and an electrostrictive response. The easiest way to include the piezo and electrostrictive constants is via the displacements in the material. The electrostrictive response corresponds to

where N is the number of discs in the stack,

we have neglected smaller terms here for convenience only. Using only electrostriction for the external applied force, we get,

here we have left in the arbitrary phase

If the phase difference between the piezoelectric and the electrostriction is zero, then we get for the magnitude of the thrust/force;

The trigonometric terms time average to zero, but the unity term does not and leads to the constant thrust seen in our device to first approximation. However, when there is a phase difference

which oscillates and time averages to zero. Clearly, we require the mass fluctuation (caused primarily by the piezoelectric effect) and the applied force (caused by electrosctriction) to be in phase, to see thrust.

In summary, the force equation prediction for the Mach effect is given by Woodward in his book [

where

A quick look at the online literature shows that the usual model used by the engineering community for piezoelectric materials is the Mason Model [

The system mass is equivalent to inductance l, a large value drags the resonant frequency down. The system stiffness k (like a spring constant) is equal to the inverse capacitance C. A high value increases the resonance of the device. The system damping

This can easily be solved by assuming that

which when differentiated with respect to time gives,

Off resonance, if

However, if

Here we describe the work, in the last 6 months, in the Woodward laboratory at California State University Fullerton, Physics Department.

We have thermistors embedded in the aluminum end cap and the brass reaction mass. Thrust forces in the devices are very small, on the order of a few micro-Newtons. A very sensitive torsional thrust balance was constructed to measure the predicted forces. It consists of a beam supported by “c-flex” flexural bearings that provide a small restoring torque when the beam is displaced from its rest position.

Details of the device and experimental set up can be found in previous works [

As one might guess, there is a linear dependence on voltage for the heating of the device, see [

Two thermal effects are important: First, heating or cooling of the device during operation will cause the expansion/contraction of parts of the device, causing the center of mass of the device to move, resulting in rezeroing of the position of the balance beam. Second, motion induced by heating/cooling will produce forces on the balance when the velocity induced by heating/cooling changes. That is, thermally induced accelerations of parts of the device are present. Qualitatively, these are expected when the heating rate changes. In

Dean drive effects depend on the coefficient of friction in relatively moving parts differing as the relative speed of the parts changes-allowing a displacement in one direction to accumulate more rapidly than average, as the vibration continues. Asymmetry effects, rather than being accumulated displacements, depend on the expansion/contraction of the device being prolonged/shortened in one part of each cycle, leading to a displace- ment of the time-averaged center of mass of the device. This displacement of the center of mass produces a shift in the zero position of the balance beam when the device is powered that masquerades as thrust. In

The obvious question is: why do we get thrust with device 1 and not with a very similar constructed device 4?

The answer is simply that the first and second harmonics need to be in phase for the Mach effect to work and to get thrust. In device 1 both first and second harmonics are in phase as you can see in

The force equation prediction for the Mach effect is given by Woodward in his book [

The mathematica fit to the data gave a cubic and a quartic of the form,

for V in volts and F in Newtons. Using the following constants, ω =228 Krad/s, corresponding to 36.3 KHz frequency, m_{0} = 0.025 Kg, ^{3},

observation of power loss of -13 dB from first to second harmonics,

which agrees quite well with the Mathematica curve fit for the quartic. The resonant frequency

Einstein understood Mach’s principle, as a gravitational interaction between a test particle and the rest of the mass-energy of the universe, to be of a radiative nature and to act instantaneously. This is made possible if the gravitational interaction is carried by advanced waves as in the HN-theory. It has been shown that the Wood- ward result [

We are currently modeling the device with COMSOL and ANSYS software. There have been two successful replications seeing the thrust from devices supplied by JFW. One result is still pending publication, the other was by Nembo Baldrini in Austria [

JFW thanks N. Herbert for suggesting the voltage scaling test because of its unusual quartic signature in these devices. We thank A. Zachar for helpful discussion during the writing of this paper. One of us, NvR, thanks the Space Studies Institute (ssi.org, Mojave, California), for a research stipend which is paid for accommodation and expenses for a summer trip to CSU Fullerton to work on this project 2015.

HeidiFearn,Nolan vanRossum,KeithWanser,James F.Woodward, (2015) Theory of a Mach Effect Thruster II. Journal of Modern Physics,06,1868-1880. doi: 10.4236/jmp.2015.613192