The Woodward effect, also referred to as a Mach effect, is part of a hypothesis proposed by James F. Woodward in 1990.^[1] The hypothesis states that transient mass fluctuations arise in any object that absorbs internal energy while undergoing a proper acceleration. Harnessing this effect could generate a reactionless thrust, which Woodward and others claim to measure in various experiments.^[2][3] If proven to exist, the Woodward effect could be used in the design of spacecraft engines of a field propulsion engine that would not have to expel matter to accelerate. Such an engine, called a Mach effect thruster (MET) or a Mach Effect Gravitational Assist (MEGA) drive, would be a breakthrough in space travel.^[4][5] So far, no conclusive proof of the existence of this effect has been presented.^[6] Experiments to confirm and utilize this effect by Woodward and others continue.^[7] The anomalous thrust detected in some RF resonant cavity thruster (EmDrive/Cannae drive) experiments may be explained by the same type of Mach effect proposed by Woodward.^[8][9][10]
The Space Studies Institute was selected as part of NASA's Innovative Advanced Concepts program as a Phase I proposal in April of 2017 for Mach Effect research.^{[11][12][13][14]}

Mach effects

According to Woodward, at least three Mach effects are theoretically possible: vectored impulse thrust, open curvature of spacetime, and closed curvature of spacetime.^[15]
The first effect, the Woodward effect, is the minimal energy effect of the hypothesis. The Woodward effect is focused primarily on proving the hypothesis and providing the basis of a Mach effect impulse thruster. In the first of three general Mach effects for propulsion or transport, the Woodward effect is an impulse effect usable for in-orbit satellite station-keeping, spacecraft reaction control systems, or at best, thrust within the solar system. The second and third effects are open and closed spacetime effects. Open curved spacetime effects can be applied in a field generation system to produce warp fields. Closed-curve spacetime effects would be part of a field generation system to generate wormholes.^{[citation needed]}
The third Mach effect is a closed-curve spacetime effect or closed timelike curve called a benign wormhole. Closed-curve space is generally known as a wormhole or black hole. Prompted by Carl Sagan for the scientific basis of wormhole transport in the movie Contact, Kip Thorne^[16] developed the theory of benign wormholes. The generation, stability, and traffic control of transport through a benign wormhole is only theoretical at present. One difficulty is the requirement for energy levels approximating a "Jupiter size mass".
Kenneth Nordtvedt showed in 1988 that gravitomagnetism, which is an effect predicted by general relativity but hasn't been observed yet at that time and was even challenged by the scientific community, is inevitably a real effect because it is a direct consequence of the gravitational vector potential. He subsequently shown that the gravitomagnetism interaction (not to be confused with the Nordtvedt effect), like inertial frame dragging and the Lense–Thirring precession, is typically a Mach effect.^[17]

Hypothesis

Mach's principle

The Woodward effect is based on the relativistic effects theoretically derived from Mach's principle on inertia within general relativity, attributed by Albert Einstein to Ernst Mach.^[18] Mach's Principle is generally defined as "the local inertia frame is completely determined by the dynamic fields in the Universe."^[19] The conjecture comes from a thought experiment:^[20]

Ernst Mach (1838–1916) was an Austrian physicist […] contemporary of Einstein, to whom he suggested a thought experiment: What if there was only one object in the universe? Mach argued that it could not have a velocity, because according to the theory of relativity, you need at least two objects before you can measure their velocity relative to each other.
Taking this thought experiment a step further, if an object was alone in the universe, and it had no velocity, it could not have a measurable mass, because mass varies with velocity.
Mach concluded that inertial mass only exists because the universe contains multiple objects. When a gyroscope is spinning, it resists being pushed around because it is interacting with the Earth, the stars, and distant galaxies. If those objects didn't exist, the gyroscope would have no inertia.
Einstein was intrigued by this concept, and named it "Mach's principle."

Gravity origin of inertia

A formulation of Mach's principle was first proposed as a vector theory of gravity, modeled on Maxwell's formalism for electrodynamics, by Dennis Sciama in 1953,^[21] who then reformulated it in a tensor formalism equivalent to general relativity in 1964.^[22]
In this paper, Sciama stated that instantaneous inertial forces in all accelerating objects are produced by a primordial gravity-based inertial radiative field created by distant cosmic matter and propagating both forwards and backwards in time at light speed:

Inertial forces are exerted by matter, not by absolute space. In this form the principle contains two ideas:

1. Inertial forces have a dynamical rather than a kinematical origin, and so must be derived from a field theory [or possibly an action-at-a-distance theory in the sense of J.A. Wheeler and R.P. Feynman…
2. The whole of the inertial field must be due to sources, so that in solving the inertial field equations the boundary conditions must be chosen appropriately.
   — Dennis W. Sciama, in "The Physical Structure of General Relativity", Reviews of Modern Physics (1964).

Sciama's inertial-induction idea has been shown to be correct in Einstein's general relativity for any Friedmann–Robertson–Walker cosmology.^[23][24] According to Woodward, the derivation of Mach effects is relativistically invariant, so the conservation laws are satisfied, and no "new physics" is involved besides general relativity.^[25]

Gravitational absorber theory

As previously formulated by Sciama, Woodward suggests that the Wheeler–Feynman absorber theory would be the correct way to understand the action of instantaneous inertial forces in Machian terms.^[26][27][28]

A first image to understand would be filming a sequence where a rock is thrown in the middle of a pond, making concentric ripples on the water propagating towards the shore.
Running the sequence backwards (thinking it as seeing events running backward in time) we then observe concentric waves propagating from the shore towards the center of the pond, where a rock emerges.
The thing to understand is that advanced waves coming back from the future never propagate farther into the past than the rock hitting the water that initiated all of the waves.

— James F. Woodward, in Making Starships and Stargates, Springer 2013, page 49.^[15]

The Wheeler-Feynman absorber theory is an interpretation of electrodynamics that starts from the idea that a solution to the electromagnetic field equations has to be symmetric with respect to time-inversion, as are the field equations themselves.^[21][29] Wheeler and Feynman showed that the propagating solutions to classical wave equations can either be retarded (i.e. propagate forward in time) or advanced (propagate backward in time). The absorber theory has been used to explain quantum entanglement and led to the transactional interpretation of quantum mechanics,^[30][31][32] as well as the Hoyle-Narlikar theory of gravity, a Machian version of Einstein's general relativity.^[33]Fred Hoyle and Jayant Narlikar originally developed their cosmological model as a quasi steady state model of the universe, adding a "Creation field" generating matter out of empty space, an hypothesis contradicted by recent observations.^[34] When the C-field is not used, ignoring the parts regarding mass creation, the theory is no longer steady state and becomes a Machian extension of general relativity. This modern development is known as the Gravitational Absorber Theory.^[35]
As the gravitational absorber theory reduces to general relativity in the limit of a smooth fluid model of particle distribution,^[36] both theories make the same predictions. Except in the Machian approach, a mass changing effect emerges from the general equation of motion, from which Woodward's transient mass equation can be derived.^[37] A resulting force suitable for Mach effect thrusters can then be calculated.^[38]
While the Hoyle-Narlikar derivation of the Mach effect transient terms is done from a fully nonlinear, covariant formulation, it has been shown Woordward's transient mass equation can also be retrieved from linearized general relativity.^[39][40]

Transient mass fluctuation

The following has been detailed by Woodward in various peer-reviewed papers throughout the last twenty years.^[41][42][43]
According to Woodward, a transient mass fluctuation arises in an object when it absorbs "internal" energy as it is accelerated. Several devices could be built to store internal energy during accelerations. A measurable effect needs to be driven at a high frequency, so macroscopic mechanical systems are out of question since the rate at which their internal energy could be modified is too limited. The only systems that could run at a high frequency are electromagnetic energy storage devices. For fast transient effects, batteries are ruled out. A magnetic energy storage device like an inductor using a high-permeability core material to transfer the magnetic energy could be especially built. But capacitors are preferable to inductors because compact devices storing energy at a very high energy density without electrical breakdown are readily available. Shielding electrical interferences are easier than shielding magnetic ones. Ferroelectric materials can be used to make high-frequency electro-mechanical actuators, and they are themselves capacitors so they can be used for both energy storage and acceleration. Finally, capacitors are cheap and available in various configurations. So Mach effect experiments have always relied on capacitors so far.
When the dielectric of a capacitor is submitted to a varying electric power (charge or discharge), Woodward's hypothesis predicts^[43] a transient mass fluctuation arises according to the transient mass equation (TME):

${\displaystyle \delta m_0=\frac{1}{4\pi G}[\frac{1}{{\rho }_0{c}^2}\frac{\mathrm{\partial }P}{\mathrm{\partial }t}-{\left(\frac{1}{{\rho }_0{c}^2}\right)}^2\frac{P^2}{V}]}$

where:

• ${\displaystyle m_0}$ is the proper mass of the dielectric,
• ${\displaystyle G}$ is the gravitational constant,
• ${\displaystyle c}$ is the speed of light in vacuum,
• ${\displaystyle {\rho }_0}$ is the proper density of the dielectric,
• ${\displaystyle V}$ is the volume of the dielectric,
• ${\displaystyle P}$ is the instantaneous power delivered to the system.

This equation is not the full Woodward equation as seen in the book. There is a third term, ${\displaystyle -{\left(\frac{\mathrm{\partial }}{\mathrm{\partial }t}\frac{\varphi }{c^2}\right)}^2}$, which Woodward discounts because his gauge sets ${\displaystyle \frac{\varphi }{c^2}\approx 1}$; the derivatives of this quantity must therefore be negligible.^[43]

Propellantless propulsion

The previous equation shows that when the dielectric material of a capacitor is cyclically charged then discharged while being accelerated, its mass density fluctuates, by around plus or minus its rest mass value. Therefore, a device can be made to oscillate either in a linear or orbital path, such that its mass density is higher while the mass is moving forward, and lower while moving backward, thus creating an acceleration of the device in the forward direction, i.e. a thrust. This effect, used repeatedly, does not expel any particle and thus would represent a type of apparent propellantless propulsion, which seems to be in contradiction with Newton's third law of motion. However, Woodward states there is no violation of momentum conservation in Mach effects:^[41]

If we produce a fluctuating mass in an object, we can, at least in principle, use it to produce a stationary force on the object, thereby producing a propulsive force thereon without having to expel propellant from the object. We simply push on the object when it is more massive, and pull back when it is less massive. The reaction forces during the two parts of the cycle will not be the same due to the mass fluctuation, so a time-averaged net force will be produced. This may seem to be a violation of momentum conservation. But the Lorentz invariance of the theory guarantees that no conservation law is broken. Local momentum conservation is preserved by the flux of momentum in the gravity field that is chiefly exchanged with the distant matter in the universe. [emphasis added]

Two terms are important for propulsion on the right-hand side of the previous equation:

• The first, linear term ${\displaystyle \frac{1}{{\rho }_0{c}^2}\frac{\mathrm{\partial }P}{\mathrm{\partial }t}}$ is called the impulse engine term because it expresses mass fluctuation depending on the derivative of the power, and scales linearly with the frequency. Past and current experiments about Mach effect thrusters are designed to demonstrate thrust and the control of one type of Mach effect.
• The second, quadratic term ${\displaystyle -{\left(\frac{1}{{\rho }_0{c}^2}\right)}^2\frac{P^2}{V}}$ is what Woodward calls the wormhole term, because it is always negative. Although this term appears to be many orders of magnitude weaker than the first term, which makes it usually negligible, theoretically, the second term's effect could become huge in some circumstances. The second term, the wormhole term, is indeed driven by the first impulse engine term, which fluctuates mass by around plus or minus the rest mass value. When fluctuations reach a very high amplitude and mass density is driven very close to zero, the equation shows that mass should achieve very large negative values very quickly, with a strong non-linear behavior. In this regard, the Woodward effect could generate exotic matter, although this still remains very speculative due to the lack of any available experiment that would highlight such an effect.

Applications of propellantless propulsion include straight-line thruster or impulse engine, open curved fields for starship warp drives, and even the possibility of closed curved fields such as traversable benign wormholes. ^[44]

Negative bare mass of the electron

The mass of the electron is positive according to the mass–energy equivalence E = mc² but this invariant mass is made from the bare mass of the electron "clothed" by a virtual photon cloud. According to quantum field theory, as those virtual particles have an energy more than twice the bare mass of the electron, mandatory for pair production in renormalization, the nonelectromagnetic bare mass of the "unclothed" electron has to be negative.^[45]
Using the ADM formalism, Woodward proposes that the physical interpretation of the "wormhole term" in his transient mass equation could be a way to expose the negative bare mass of the electron, in order to produce large quantities of exotic matter that could be used in a warp drive to propel a spacecraft or generate traversable wormholes.^[46]

Space travel

Current spacecraft achieve a change in velocity by the expulsion of propellant, the extraction of momentum from stellar radiation pressure or the stellar wind or the utilisation of a gravity assist ("slingshot") from a planet or moon. These methods are limiting in that rocket propellants have to be accelerated as well and are eventually depleted, and the stellar wind or the gravitational fields of planets can only be utilized locally in the Solar System. In interstellar space and bereft of the above resources, different forms of propulsion are needed to propel a spacecraft, and they are referred to as advanced or exotic.^[47][48]

Impulse engine

If the Woodward effect is confirmed and if an engine can be designed to use applied Mach effects, then a spacecraft may be possible that could maintain a steady acceleration into and through interstellar space without the need to carry along propellants. Woodward presented a paper about the concept at the NASA Breakthrough Propulsion Physics Program Workshop conference in 1997,^[49][50] and continued to publish on this subject thereafter.^{[51][52][53][54]}
Even ignoring for the moment the impact on interstellar travel, future spacecraft driven by impulse engines based on Mach effects would represent an astounding breakthrough in terms of interplanetary spaceflight alone, enabling the rapid colonization of the entire solar system. Travel times being limited only by the specific power of the available power supplies and the acceleration human physiology can endure, they would allow crews to reach any moon or planet in our solar system in less than three weeks. For example, a typical one-way trip at an acceleration of 1 g from the Earth to the Moon would last only about 4 hours; to Mars, 2 to 5 days; to the asteroid belt, 5 to 6 days; and to Jupiter, 6 to 7 days.^[55]

Warp drives and wormholes

As shown by the transient mass fluctuation equation above, exotic matter could be theoretically created. A large quantity of negative energy density would be the key element needed to create warp drives^[56] as well as traversable wormholes.^[57] As such, if proven to be scientifically valid, practically feasible and scaling as predicted by the hypothesis, the Woodward effect could not only be used for interplanetary travel, but also for apparent faster-than-light interstellar travel:

• The negative mass could be used to warp spacetime around a spaceship according to an Alcubierre metric.^[42][56]
• Enough exotic matter could also be concentrated into a point of space to create a wormhole, and prevent it from collapsing. Woodward and others also state that exotic matter could defocus energy at the outer mouth of the wormhole (making it a white hole) and shape the throat of such a gravitational singularity flat enough to avoid horizon and tidal stresses, resulting in an "absurdly benign traversable wormhole" linking two regions of distant spacetime, a concept well spread in science fiction as stargates, which could be used for instant intergalactic travel or time travel.^{[15][42][57][58][46]}

Patents and practical devices

Two patents have been issued to Woodward and associates based on how the Woodward effect might be used in practical devices for producing thrust:

• In 1994, the first patent was granted, titled: "Method for transiently altering the mass of objects to facilitate their transport or change their stationary apparent weights".^[59]
• In 2002, a second patent was granted, titled: "Method And Apparatus For Generating Propulsive Forces Without The Ejection Of Propellant".^[60]
• In 2016, a third patent was granted and assigned to the Space Studies Institute, covering the realistic realizations of Mach effects.^[61]

Woodward and his associates have claimed since the 1990s to have successfully measured forces at levels great enough for practical use and also claim to be working on the development of a practical prototype thruster. No practical working devices have yet been publicly demonstrated.^{[2][3][6][41]}
The NIAC contract awarded in 2017 by NASA for the development of Mach effect thrusters is a primary three-task effort, two experimental and one analytical:^[12]

1. Improvement of the current laboratory-scale devices, in order to provide long duration thrust at levels required for practical propulsion applications.
2. Design and development of a power supply and electrical systems to provide feedback and control of the input AC voltage, and resonant frequency, that determine the efficiency of the MET.
3. Improve theoretical thrust predictions and build a reliable model of the device to assist in perfecting the design. Predict maximum