Mach's Principle and Impulse Engines:
Toward a Viable Physics of Star Trek?

Presented at The NASA Breakthrough Propulsion Physics Workshop
Cleveland, Ohio
August 12-14, 1997

James F. Woodward
Departments of History and Physics
California State University
Fullerton, California 92634

Mach's principle and local Lorentz-invariance together yield the prediction of transient rest mass fluctuations in accelerated objects. These restmass fluctuations, in both principle and practice, can be quite large and, in principle at least, negative. They suggest that exotic spacetime transport devices may be feasible, the least exotic being "impulse engines", devices that can produce accelerations without ejecting any material exhaust. A scheme of this sort is presented and issues raised relating to conservation principles are examined.



Aerospace propulsion technology to date has rested firmly on simple applications of the reaction principle: creating motion by expelling propellant mass from a vehicle. We can do better. A peculiar, overlooked relativistic effect makes it possible to induce large, transient rest mass fluctuations in electrical circuit components [Woodward, 1990; 1992]. Such fluctuations may be combined with a synchronized, pulsed thrust to greatly increase the acceleration attainable from a given amount of ejected reaction mass. A yet more innovative implementation of the effect suggests it may be possible to make engines that accelerate without the expulsion of any material whatsoever. These "impulse engines" are achieved without any moving parts (in the conventional sense). The concepts involved are supported by experimental results already in hand. Moreover, due to the nonlinearity of the effect, Morris and Thorne's [1988] traversable wormholes and Alcubierre's [1994] "warp drive" may be attainable with known technology (while remaining fully in line with the established laws of physics, despite their "Star Trek" nature). Here, however, I deal only with impulse engines.


The effect to be derived is predicated upon two assumptions.

  1. Inertial reaction forces in objects subjected to accelerations are produced by the interaction of the accelerated objects with a field -- they are not the immediate consequence only of some inherent property of the object.
  2. Any acceptable physical theory must be locally Lorentz invariant; that is, in sufficiently small regions of spacetime special relativity theory (SRT) must obtain.

We ask: In the simplest of all possible circumstances -- the acceleration of a test particle in a universe of otherwise constant matter density -- what, in the simplest possible approximation, is the field equation for inertial forces implied by these propositions? SRT allows us to stipulate the inertial reaction force F on our test particle stimulated by the external accelerating force figure as:

figure (2.1)

with P = figure and figure. Bold capital letters denote four-vectors and bold lower-case letters denote three-vectors, P and p are the four- and three-momenta of the test particle respectively, is the proper time of the test particle, v the instantaneous velocity of the test particle with respect to us, and c the speed of light.

We specialize to the frame of instantaneous rest of the test particle. In this frame we can ignore the difference between coordinate and proper time, and figure since they are equal to one. (We will not recover a generally valid field equation in this way, but that is not our objective.) In this frame Eq. (2.1) becomes F = - dP/dt = figure, with f = dp/dt . Since we seek the equation for the field (i.e., force per unit mass) that produces F, we normalize F by dividing byfigure. Defining figure, we get,

figure (2.2)

To recover a field equation of standard form we let the test particle have some small extension and a proper matter density figure. Eq. (2.2) then is figure From SRT we know that figure being the proper energy density, so we may write:

figure (2.3)

To get the field equation that corresponds to F in terms of its local source density we take the four-divergence of F getting,

figure (2.4)

We write the source density as figure, leaving its physical identity unspecified for the moment. f is irrotational in the case of our translationally accelerated test particle, so we may write figure figure being a scalar field (or the timelike part of a vector potential field), and Eq. (2.4) is

figure (2.5)

Now we must write figure in such a way that we get a wave equation that is consistent with local Lorentz-invariance. Given the coefficient of figure, only one choice is possible: figure. This choice for figure yields:

figure (2.6)

If we ignore the terms of order figure and those involving derivatives of figure, we have in Eq. (2.6) the usual wave equation for figure in terms of a source charge densityfigure. Since figure is the potential of a field that acts on all matter in direct proportion to its mass and is insensitive to direct interaction with all other types of charge, it follows that the source of figure must be mass. That is, figure. Thus the field that produces inertial reaction forces is the gravitational field [as expected in general relativity theory (GRT)].

Considering the stationary case, where all terms involving time derivatives vanish, Eq. (2.6) reduces to Laplace's equation, and the solution for figure is just the sum of the contributions to the potential due to all of the matter in the causally connected part of the Universe, that is, within the particle horizon. This turns out to be roughly GM/R, where M is the mass of the Universe and R is about c times the age of the Universe. Using reasonable values for M and R, GM/R is about figure. In the time-dependent case we must take account of the terms involving time derivatives on the RHS of Eq. (2.6). Note that these terms either are, or in some circumstances can become, negative. It is the fact that these terms can also be made very large in practicable devices with extant technology that makes them of interest for rapid spacetime transport.

Although standard techniques are used to obtain Eq. (2.6), one may be suspicious of the transient source terms. After all, they are unusual to say the least. Acceleration-dependent transient rest mass fluctuations are not commonplace, especially when they are potentially so large. Indeed, they seem almost too good to be true. Remark, however, that they have a well-known counterpart in standard GRT: the Nordtvedt effect. In the Nordtvedt effect the masses of the constituent parts of accelerated extended bodies are transiently changed (due to the dragging of spacetime by the body) [Nordtvedt, 1988]. The magnitude of the mass-shift in each part of the body is proportional to the product of the acceleration and the Newtonian gravitational potential of the rest of the body at its location. In the case of our accelerated test particle, in its instantaneous rest frame the remainder of the universe appears as an enveloping accelerated body. Accordingly, we might expect it to induce a transient mass fluctuation in the test particle. Eq. (2.6), nevertheless, is not validated by the occurrence of analogous effects in GRT or other theoretical speculations. Its validity is a matter of fact determined by experiments.


Since the predicted mass shift is transient, large effects can only be produced by very rapidly changing proper matter (or energy) densities. This means that the duration of any substantial effect will be so short that it cannot be measured by usual weighing techniques. If, however, we drive a periodic mass fluctuation and couple it to a synchronous pulsed thrust, it is possible to produce a measurable stationary effect. Consider, for example, the generic apparatus in Fig. 1.


Here a mass fluctuation is produced in an array of capacitors (CA) by driving them with an AC voltage. While the mass of the CA fluctuates, a piezoelectric force transducer (PZT) causes a synchronous, oscillatory acceleration of the CA. The inertial reaction force F felt by the PZT [and the enclosure (E) in which it is mounted] will be the product of the instantaneous mass of the CA times the acceleration of the CA induced by the PZT. If the mass fluctuation and acceleration are both sinusoidal and phase-locked at the same frequency, then their product yields a phase-dependent, time-independent term -- a stationary force.

We make this quantitative. The leading transient term in Eq. (2.6) gives a transient proper mass density:

figure (3.1)

where figure has been used to express the proper mass density as a proper energy density. The total transient mass fluctuation figure induced in the volume V of the dielectric in the capacitors then is:

figure (3.2)


In this case, since figure is the power density being stored in the capacitor array at any instant, and the integral over the volume of the capacitors is just the instantaneous power P being delivered to the capacitors, the integral on the RHS of Eq. (3.2) is figure. Thus,

figure (3.3)

When a sinusoidal voltage of angular frequency figure is applied to the capacitors, we may write figure and Eq. (3.3) becomes:

figure (3.4)

Substitution of realistic, laboratory scale values yields mass transient amplitudes on the order of tens of milligrams.

If we drive an oscillation in a PZT arranged like that in Fig. 1 with amplitude figure at figure, assume that the mass of the CA is small compared to that of the enclosure E so that the excursion of the PZT accelerates the CA only, and allow for a phase angle figure between figure and figure, we find for the time averaged inertial reaction force figure:

figure (3.5)

If figure is a few angstroms (easily achieved with normal PZTs), then when figure forces on the order of several dynes or more can be produced in the laboratory. I have done this in fact employing apparatus shown in general, schematic form in Fig. 1.


The enclosure is mounted via a shaft on a stainless steel diaphragm (D), a spring that supports the enclosure and its contents. An exceedingly sensitive vertical position sensor (S) detects the location of the shaft. It enables one to measure the weight, and thus the mass, of the suspended apparatus. (To change <F> into an equivalent weight it must be divided by the local acceleration of gravity.) Fig. 2 displays a photo of one of the capacitor arrays mounted on a PZT in the bottom part of its enclosure. (Further details of this apparatus can be found in Woodward, 1996b.)


In actual practice one takes the difference between runs adjusted so that figure and those where figure. Recent results obtained at 14 kHz with this device are shown in Fig. 3. In the 7 to 12 second interval the CA and PZT are activated producing the displayed differential weight shift. These results are those predicted to better than order of magnitude. (Earlier results accompanied by extensive validity checks and analysis are in Woodward, 1996b.) No weight shift like that in Fig. 3 occurs when either the CA or PZT is run separately. This behavior is shown in Figures 4 and 5 respectively. These and other checks leave little doubt that the signal seen is that sought. Can we use this effect to make impulse engines? Perhaps.





It seems, on the face of it, that impulse engines should be possible. Consider, for example, the case of a kid on a skateboard with a brick. The brick's mass magically fluctuates periodically. The kid throws the brick in the direction opposite to where s/he wants to go when its mass is enhanced, and an attached bungee cord returns the brick to him/her in the mass reduced state. The kid-brick-skateboard system accelerates in the desired direction. You may be inclined to think that even if transient mass fluctuations can really be induced, if the power source driving the fluctuation were loaded onto the skateboard, mass fluctuation effects occurring in it would cancel the acceleration produced by repetitively throwing the brick. Were this not the case, it would seem that we would be confronted by a violation of the conservation of momentum. Since we have introduced no "new physics", violations of momentum conservation shouldn't occur.

The acceleration revealed in the magic brick heuristic, nonetheless, should happen. This is easily shown by making the system a bit more complicated: using two magic bricks instead of one. Our magic bricks will represent either two capacitors, or better yet a capacitor (C) and an inductor (L). We drive mass fluctuations in these circuit elements that have 180 degrees relative phase. (This phase relationship is what makes a capacitor and inductor desirable. They can be made components of a resonant circuit. Since the phase of the instantaneous power [the voltage times the current] in these components differs by 180 degrees, the mass fluctuations will automatically have the desired phase relationship.) Note that the mass fluctuations in the L and C elements sum to zero [at least when the mass fluctuations are small so that the coefficients of the transient terms on the RHS of Eq. (2.6) can be taken as constants], so energy conservation in this circuit per se isn't violated. We interpose a force transducer (a PZT say) between them that expands and contracts at the mass fluctuation frequency. A device of this sort is shown schematically in Fig. 6.


When a device like that displayed in Fig. 6 is activated a stationary force is produced by each of the mass-fluctuating elements on the ends of the force transducer. Each the forces will be given roughly by Equation (3.5). (Even were we to assume that the mass of the force transducer to be effectively infinite -- as we assumed the enclosure of Figure 1 to be in Fig. 6: Impulse Engine Operation calculating the acceleration of the CA in obtaining Equation (3.5) -- a factor that reduces figure, and thus <F>, must be included to allow for the fact that the displacement involved in the acceleration of each of the elements is only a fraction of figure) We now remark that the phase difference in the mass fluctuations of the L and C circuit elements compensates for the fact that their accelerations induced by the force transducer are in opposite directions. Accordingly, the stationary forces produced by accelerating L and C as their masses fluctuate are both in the same direction; the L/C/PZT system -- an impulse engine -- experiences a steady, unidirectional accelerating force [which can be estimated with Equation (3.5)]. Should we now attach the power sources to our device, they too will be carried along by our impulse engine, even if they contain fluctuating masses.

It seems that we have constructed a device that blatantly violates the conservation of momentum. Perhaps we have ignored something important. For example, consider an object (a capacitor, inductor, magic brick, whatever), moving with some velocity v with respect to us, whose mass can be made to fluctuate. When the mass changes, does the velocity change too? Ostensibly no external force acts to change the momentum. So conservation of momentum seems to suggest that the velocity must change. If the local momentum conservation implicit in this inference is true, then we can solve our problem. Local momentum conservation guarantees that momentum must be conserved somehow point-by-point throughout our impulse engine. Thus it may wiggle a lot, but it goes nowhere. The assumption of point-by-point momentum conservation in this case, however, violates the principle of relativity, so it must be wrong.

Let us suppose that, viewed in our inertial frame of reference moving with respect to the brick, when the mass of the brick changes, its velocity changes too so that its momentum remains unchanged. (The cause of the velocity change is mysterious. After all, driving a power fluctuation in the brick to excite a mass fluctuation need not itself exert any net force on the brick. But we'll let that pass.) We see the brick accelerate. Now we ask what we see when we are located in the rest frame of the brick. The mass fluctuates, but in this frame the brick doesn't accelerate since its momentum was initially, and remains, zero. This, by the principle of relativity, is physically impossible. If the brick is observed to accelerate in any inertial frame of reference, then it must accelerate in all inertial frames. We thus conclude that mass fluctuations result in violations of local momentum conservation if the principle of relativity is right.

The appearance of momentum conservation violation in our impulse engine doesn't mean that momentum isn't conserved. It means that we can't treat the impulse engine as an isolated system. Since the effect responsible for the apparent violation of the conservation principle is inertial/gravitational, this should come as no surprise at all. As Mach's principle makes plain, anytime a process involves gravity/inertia, the only meaningful isolated system is the entire universe. Since inertial reaction forces appear instantaneous [see Woodward, 1996a and Cramer, 1997 in this connection], evidently our impulse engine is engaging in some "non-local" momentum transfer with the distant matter in the universe. With suitable choice of gauge, this momentum transfer can be envisaged as transpiring via retarded and advanced disturbances in the gravitational field that propagate with speed c.

Gauge freedom muddies up discussions of inertial reaction effects [Woodward, 1996a]. Choosing a gauge where all physical influences propagate at speeds figure has the advantage that lightcones in space-time have an invariant meaning, whereas the surfaces of simultaneity that appear in other gauges (e.g., the Coulomb gauge) do not. As just mentioned, in the Lorentz [or Einstein-Hilbert] gauge the inertial reaction effect, and thus our impulse engine, consists of a retarded/advanced coupling between the engine and the distant matter in the universe that lies along the future light cone. The introduction of the force transducer in the engine allows us to extract a net momentum flux here and now from the potentially largely thermalized matter in the far future. The net momentum flux is accompanied by a net energy flux, so although our impulse engine, considered locally, appears to violate energy conservation, that need not necessarily be the case. The extraction of useful work from matter that may be completely thermalized raises interesting questions. Boosting, rather than borrowing, from the future, however, seems to be the nature of the process involved.

Is any of this really right? Well, one way to get a fix on this is to run the experimental apparatus described above when it is rotated by 90 degrees -- that is, oriented horizontally rather than vertically. If the observed effect is some spurious local effect or couples to local gravity fields, the observed effect should change when the local orientation of the apparatus is altered. But if the effect is caused by the proposed non-local interaction with cosmological matter, it should be independent of the local orientation of the apparatus. Results obtained with the apparatus oriented horizontally are displayed in Fig. 7. At the level of experimental accuracy there is no significant difference in the magnitude of the effect for the two orientations.



It seems that at least one part of the physics of Star Trek -- impulse engines -- may lie within our grasp. Indeed, the transient Machian inertial reaction effect that makes impulse engines possible may also make "stargates" and time machines based on traversable wormholes feasible [Woodward, 1997]. This is a consequence of the strong nonlinearity of the total proper matter density as it approaches zero and negative values. (Negative mass has interesting properties. See: Forward [1989] and Price [1993].) The feasibility of such schemes, however, also depends on the magnitude of the bare masses of elementary particles and the nature of the vacuum. These matters are, at the very best, conjectural. Accordingly, the schemes are a good deal more speculative than impulse engines. But they, along with impulse engines, can be explored experimentally with present technology at reasonable cost.


Arguments and questions posed by Thomas Mahood and James van Meter, and John Cramer's recent Analog [1997] article have been most helpful in developing the ideas relating to impulse engines presented here. TM also suggested stylistic improvements (including the suppression of a tasteless remark or two). The experiments described here were supported in part by several CSU Fullerton Foundation grants. The impulse engine method described herein is a specific implementation of the general method of U.S. Patent 5,280,864.


Copyright © 1998, James F. Woodward. This work, whole or in part, may not be reproduced by any means for material or financial gain without the written permission of the author.