# A mechanical force of electromagnetic origin

The Abraham force is the force exerted by a time-dependent electromagnetic field on neutral, polarizable matter, and has been debated for over a hundred years. There have been two incompatible equations to describe momentum transfer between electromagnetic fields and matter. These two equations were first suggested by Hermann Minkowski (1908) and Max Abraham (1909), from which the controversy’s name derives. Both were claimed to be supported by experimental data. Theoretically, it is usually argued that Abraham’s version of momentum does indeed represent the true momentum density of electromagnetic fields, while Minkowski’s version of momentum is a pseudo-momentum. Several papers have now claimed to have resolved this controversy. This controversy also has inspired various theories proposing the existence of reactionless, or more realistically, **propellantless drives**.

**Experiments have been reported** which verified that at low frequencies a time-varying polarisation P in a dielectric, if taking place in a magnetic field H, results in a **mechanical force.** This force is not predicted by the Minkowski energy-momentum tensor, but the Abraham form of that tensor gives a force density in a homogeneous isotropic body, usually called **the Abraham force,** which in a dielectric can be written as *Ṗ*×μ_{0}*H *+ *P*×μ_{0}*Ḣ *.

### It is not surprising that *Ṗ*×μ_{0}*H* should give a mechanical force, since *Ṗ* corresponds to a movement of electric charge associated with polarisation, but it is **difficult to understand** why *P*×μ_{0}*Ḣ* should give a force.

## The **Abraham force** constitutes the physical basis for explaining the **Biefeld-Brown effect****.**

### Observation of the intrinsic Abraham force in time-varying magnetic and electric fields

The experimental observation of the Abraham force induced by an oscillating electric field and a static magnetic field was reported by James and by Walker et al in solid dielectrics, and recently by Rikken and Van Tiggelen in gases. These observations clearly invalidated the Minkowski version, although modifications of the Minkowski energy-momentum tensor have been proposed to make it consistent with these results. However, the Abraham force due to an oscillating magnetic field and a static electric field has so far never been observed and was even reported unobservable in a specifically designed experiment.

The Abraham force exerted by a time-dependent electromagnetic field on neutral, polarizable matter has two contributions. The one induced by a time-varying magnetic field and a static electric field is reported here for the first time. We discuss our results in the context of the radiative momentum in matter.

### ON THE **FEIGEL EFFECT** — EXTRACTION OF MOMENTUM FROM VACUUM

The Green-function formalism for the electromagnetic field in a magnetoelectric (ME) medium is constructed, as a generalization of conventional Casimir theory. It is shown how the formalism predicts electromagnetic momentum to be extracted from the vacuum field, just analogous to how energy is extracted in the Casimir case. The possibility of extracting momentum from vacuum was discussed recently by Prof. Alexander Feigel. By contrast to **Feigel’s approach,** we assume that the ME coupling occurs naturally, rather than being produced by external strong fields. We also find the same effect qualitatively via another route, by considering one single electromagnetic mode.

In the case of high frequency fields, in particular optical fields, the Minkowski theory appears to be both simple and capable of describing all experiments. However, at low frequencies where the effect of the oscillations are themselves observable – notably in the Walker-Lahoz experiment – the experiments agree not with the Minkowski but rather with the Abraham force, at these frequencies.

Before leaving this idea, let us however note the following point: Assume that strong crossed fields E and H are applied between the conducting plates. Then, during the time when the external fields increase in strength, there acts an Abraham force in the fluid in the interior. Integrating over time, we see that the following mechanical momentum density is imparted to the fluid. This is the dominant momentum given to the fluid between the plates. In addition comes the momentum transferred from the wave modes. The momentum is actually very similar to the momentum, or more strictly to the angular momentum, transferred to the suspended dielectric cylindrical shell in the Walker-Lahoz experiment in ordinary electrodynamics.

### A new form of the energy-momentum tensor of the interaction of an electromagnetic field with a non-conducting medium

Wave equation for momentum density describes simultaneous **transfer of momentum, and angular momentum,** regardless of radiation polarization. From the tensor follow the balance equations of the electromagnetic forces for the momentum density in a continuous medium, and lead to the conclusion about the equality and mutual complementation of Minkowski’s and Abraham’s forms of momentum density. It is shown that the Abraham force exists only in the medium where the vectors D and E, H and B, are not parallel.

### Observation of static electromagnetic angular momentum *in vacua*

Our programme of measurement of forces related to electromagnetic momentum at low frequencies in matter has culminated in the first direct observation of free electromagnetic angular momentum created by quasistatic and independent electromagnetic fields E and B in the vacuum gap of a **cylindrical capacitor.** A resonant suspension is used to detect its motion. The observed changes in angular momentum agree with the classical theory within the error of ∼20%. This implies that the vacuum is the seat of something in motion whenever static fields are set up with non-vanishing Poynting vector, as Maxwell and Poynting foresaw.

### The energy-momentum tensor of electromagnetic fields in matter

In this paper we present a complete resolution of the Abraham-Minkowski controversy about the energy-momentum tensor of the electromagnetic field in matter. This is done by introducing in our approach several new aspects which invalidate previous discussions. These are:

- We show that for polarized matter the center of mass theorem is no longer valid in its usual form. A contribution related to microscopic spin should be considered. This allows to disregard the influencing argument done by Balacz in favor of Abraham’s momentum which discuss the motion of the center of mass of light interacting with a dielectric block.
**The electromagnetic dipolar energy density contributes to the inertia of matter**and should be incorporated covariantly to the the energy-momentum tensor of matter.**This implies that there is also an electromagnetic component in matter’s momentum density**which accounts for the difference between Abraham and Minkowski’s momentum densities. The variation of this contribution explains the results of G. B.**Walker,**D. G.**Lahoz**and G. Walker’s experiment which until now was the only undisputed support to**Abraham’s force.**- A careful averaging of microscopic Lorentz’s force results in the unambiguos expression for the force density which the field exerts on matter. This force density is different to the ones used in most of the discussions and is consistent with all the experimental evidence.
- Newton’s Third law or equivalently momentum conservation determines the electromagnetic energy-momentum tensor as the only one consistent with Maxwell’s equations whose divergence is minus the force density, where we have the standard tensor of the field in vacuum. This tensor is different from Abraham’s and Minkowski’s tensors, but one recovers Minkowski’s expression for the momentum density. In particular the energy density is different from Poynting’s expression but Poynting’s vector remains the same.

Our tensor is non-symmetric which allows the field to exert a distributed torque on matter. As a result of the specific form of the antisymmetric part of the constructed tensor, the spin density of the electromagnetic field decouples from the polarization of matter. To give further support to the proposed tensor, we also deduce its form using an alternative method based on direct a veraging of the microscopic equations and on imposing consistency of the dipolar coupling. We use our results to discuss momentum and angular momentum exchange in various situations of physical interest. We find complete consistency of our equations in the description of the systems considered. We also show that several alternative expressions of the field energy-momentum tensor and force-density cannot be successfully used in all our examples. In particular we verify in two of these examples that the center of mass and spin introduced by us moves with constant velocity, but that the standard center of mass does not.

In this paper we attain a complete resolution of the Abraham-Minkowski controversy on the energy-momentum tensor of the electromagnetic field in matter by introducing a new energy-momentum tensor which is different from those previously proposed. Our approach emphasizes the local character of the force and the torque exerted by the field on matter and requires that the energy-momentum tensor be locally well determined. It cannot be arbitrarily modified by the addition of a divergenceless term, even if by doing so the total energy and momentum do not change.

Our discussion departs from the main lines of argumentation prevailing in the recent literature on the topic. It is based on four main points: First, the force density on matter is in principle a measurable quantity and should be computable as the divergence of the matter tensor. Momentum conservation determines in an unambiguous way the energy-momentum tensor of the electromagnetic field. Second, all terms which contribute to inertia, including those of electromagnetic origin, should be included in the matter tensor and consequently excluded from the electromagnetic one. Third, the force density should be compatible with the microscopic Lorentz force.

### Magnetization is a manifestation of spin and EM fields produce torques on polarized matter, hence a complete understanding of the mechanical behavior of polarizable matter in an electromagnetic field requires the dynamics of spin.

In this work the necessity of a relativistic invariant description of the matter-field interaction and the understanding of the true consequences of such a description play a key role. Throughout the discussion we insisted on the idea that the polarizations tensor which includes the electric polarization and the magnetization in a relativistic covariant object, should be used to describe the polarization of matter. We avoid arguments that rely on the properties of the electric and magnetic permeabilities which are defined only in the rest frame of matter (which not always exist). As an element which has more general consequences we discuss the serious limitations of the CMMT, which does not take into account the contribution of spin to the dynamics.

The uncritical use of the CMMT in this context has been the source of many confusions and even mistakes in previous papers on this subject.

The conditions imposed on the form of the force density on matter by the relativity principle, the validity of Maxwell’s microscopic and macroscopic equations and the consistency with Lorentz microscopic force allow us to determine uniquely an expression of the force density on polarizable matter. This expression is similar but different to some of the expressions discussed in the past. The total force that the electromagnetic field exerts on an isolated sample of matter computed with our expression is the same as the total Lorentz force computed on the bound charges and currents of the sample. It is also equal to the total force predicted by the use of Minkowski’s tensor but it is different to the one consistent with Abraham’s tensor. The term known as Abraham’s force is not present in our formulation. Our analysis of the force density uncovers also the portion of the energy momentum tensor of matter which depends on the field. This includes the electrostatic potential energy of the dipoles, a momentum contribution required for the polarization of matter and, depending on the external conditions on matter, a field dependent mechanical response of matter which modifies its stress tensor. The energy-momentum tensor of the field is fixed by the form of the force density and we are able to get its correct expression. The energy density of this tensor is not Poynting’s classic formula. It is shown to be the energy density of the electromagnetic field in vacuum minus the electrostatic energy of the dipoles. This term represents a electromagnetic contribution to inertia. Poynting’s vector maintains its interpretation as the energy density current once the correct expression of the power transmitted by the field to matter is taken into account.

The momentum density is shown to be Minkowski’s expression. It is shown to be the momentum in vacuum minus the field dependent momentum of matter. This puts an end to the long standing controversy.

### The energy-momentum tensor of the field is asymmetric. This has important consequences. This asymmetry is shown to be necessary for the coupling of the total spin density (magnetization) and the orbital angular momentum of matter. The asymmetry cancels exactly the effect of the torque density in the dynamical equation of the field spin density with the general result that the spin of the field is decoupled from the polarization of matter.

We presented a second approach to deduce the form of the electromagnetic energy-momentum tensor in matter. For this we perform a space-time average starting from the microscopic equations and postulate a monopolar and a dipolar coupling of matter with the macroscopic field. By imposing consistency with Maxwell’s equations we identify the **monopolar** current with the free charges, and show that the **dipolar** coupling is described by the polarization tensor in such a way that we recover exactly the formulation of the first approach. Although it may be foreseeable, **this result has a deep meaning:**

### Relativistic invariance and the physical hypotheses discussed in section V uniquely determine the equations of the **macroscopic field up to the dipolar** approximation. It will be interesting to investigate if the **quadrupolar** approximation [complex magnetic fields] maintains such robustness.

The arguments presented in this paper are of theoretical nature. Of course, the ultimate test of the validity of our approach should be the direct measurement of the force density in different situations. For some of the experiments that have been done in the past, it is not completely straightforward to conciliate the measurements obtained with a definite choice of the force density or momentum expressions. In consequence there have been confronting opinions on how to interpret correctly the results of those experiments. Nevertheless there are some situations in which in our opinion our analysis proves to be clearly better. The applications and examples presented in the text were devised with this in mind.

In section V we discuss some of the implications of our approach for the analysis of the experiments which have been performed to determine the electromagnetic force density in matter. In particular we show that the result of the Walker, Lahoz and Walker experiment, which is usually presented as an evidence for the existence of Abraham’s force term, may be better understood as a manifestation of the field dependent terms of the matter energy-momentum tensor. Our discussion of the electromagnetic forces on solid and liquid dielectrics strongly supports the validity of the force density that we are proposing.

We use our force density, the energy and momentum definitions obtained from and Maxwell’s equations to verify energy and momentum conservation in the interaction of a packet of electromagnetic waves with a dielectric medium. In opposition to the statement of the influential argument first advanced by Balazs, that for sixty years has been considered a strong support in favor of Abraham’s point of view, we show that for n>1 the wave packet effectively pulls the material when it enters a medium. This in fact is a most comfortable result after observing that it has a simple physical explanation.

**Dielectric and paramagnetic materials are attracted while diamagnetic materials are repelled in the direction to high field regions. When the wave packet is entering the medium it pulls the material unless diamagnetism prevails.** For the same reason when the wave leaves, it drags the block forward. We complete our picture of this process by computing the evolution of the **spin** density (magnetization). This calculation shows that the boost momentum gives the precise contribution necessary for our center of mass and **spin** to behave inertially.

The example of the charge-magnet system presented in section X, gives in a very simple setup an illustration of the consequences of the main ideas behind our formulation. In the same spirit, for the magnetized cylinder analyzed in section XI we show how our approach gives an appealing and consistent

### description of a system where the orbital angular momentum stored in the **EM fields** is **transferred** to the orbital angular momentum of **matter.**

As a by-product of the analysis presented in this paper the weakness of the CMMT was exposed in specific physical situations, and the validity of the improved theorem which includes the spin contributions was tested. These results also show that the hidden momentum hypothesis is unnecessary. Although we have not made a detailed check in each case (which would take a disproportionate effort), we think that the many analysis found in the literature which are based in this hypothesis should prove wrong or at least superfluous.

Our results open a range of theoretical and experimental possibilities of research in the study of the interaction of the electromagnetic fields with polarizable matter. Our construction of the force density suggests to make a methodological shift in the approach to the many applications in which it plays a role and concentrate in the determination of the physically acceptable polarization tensor of the system under study, preferably in the laboratory reference frame. **This may be applied to a variety of systems, which include ionized gases, dielectric liquids, and non linear magnetic matter, to give a few examples. In the past these were either ignored** or studied in terms of the permeabilities. We note that the polarization tensor of some of these systems may depend on the local velocity of matter, so that velocity dependent effects are not banned from our formulation. It is only Abraham’s like terms which are found unnecessary.

On the experimental side the determination of force density instead of the total force advocated by some authors and in particular by Brevik in the past acquires new importance. A critical reanalysis an even a remaking of the experiments which perform a direct measurement of the momentum of photon should be useful for establishing without further doubts Minkowski’s momentum as the classical and quantum momentum of light. Experiments exploring the inertiality of the center of mass and spin in this context which may include the real versions of the idealized situations discussed in sections IX, X and XI will also contribute to the complete experimental clarification of this subject.

In our presentation, we also note that although in a general situation Minkowski’s tensor is not particularly useful, for a material with non-dispersive linear polarizabilities it may be interpreted as the energy-momentum tensor of the electromagnetic field plus the fraction of the energy of matter arising from the polarization. Working with care it can be used in this case. Nevertheless its divergence is not the reaction of the force acting on matter. This explains the absence of magnetostriction and electrostriction in Minkowski’s approach. When an electromagnetic wave propagates on such a medium Minkowski’s tensor describes the whole wave and hence satisfies von Laue-Møller criterion.

## A Blueprint for a Quantum Propulsion Machine

The quantum vacuum has fascinated physicists ever since Hendrik Casimir and Dirk Polder suggested in 1948 that it would exert a force on a pair of narrowly separated conducting plates. Their idea was eventually confirmed when the force was measured in 1997. Just how to exploit this force is still not clear, however.

In recent years, a new way of thinking about the quantum vacuum has emerged which has vastly more potential. And today, one physicist describes how it could be used to create propulsion.

Before we discuss that, let’s track back a little. According to quantum mechanics, any vacuum will be filled with electromagnetic waves leaping in and out of existence. It turns out that these waves can have various measurable effects, such as the Casimir-Polder force.

The new approach focuses on the momentum associated with these electromagnetic fields rather than the force they exert. The question is whether it is possible to modify this momentum because, if you can, you should receive an equal and opposite kick. **That’s what rocket scientists call propulsion.**

Today, Prof. Alexander Feigel at the Soreq Nuclear Research Center, a government lab in Yavne Israel, suggests an **entirely new way to modify the momentum of the quantum vacuum and how this can be exploited to generate propulsion. **Feigel’s approach combines two well-established ideas.

The first is the Lorentz force experienced by a charged particle in electric and magnetic fields that are crossed. The second is **the magneto-electric effect, the phenomenon in which an external magnetic field induces a polarised internal electric field in certain materials, and vice versa.**

The question that Feigel asks is in what circumstances the electromagnetic fields in a quantum vacuum can exert a Lorentz force. The answer is that **the quantum vacuum constantly interacts with magneto-electric materials generating Lorentz forces. Most of the time these forces sum to zero.**

However, Feigel says there are four cases in which the forces do not sum to zero. Two of these are already known, for example confining the quantum field between two plates, which excludes longer wavelength waves.

But Feigel says the two others offer entirely new ways to exploit the quantum vacuum using magneto-electric nanoparticles to interact with the electromagnetic fields it contains.

The first method is to rapidly aggregate a number of magneto-electric nanoparticles, a process which influences the boundary conditions for higher frequency electromagnetic waves, generating a force.

### The second one, is to **simply rotate, like a gyro, **a group of magneto-electric nanoparticles, which also generates the Lorentz force [ and the Abraham-Magnus force ].

Either way, the result is a change in velocity. As Feigel puts it: *“mechanical action of quantum vacuum on magneto-electric objects may be observable and have a significant value.”*

**The beauty of Feigel’s idea is that it can be easily tested.** He suggests building an addressable array of magneto-electric nanoparticles, perhaps made of a material such as FeGaO3 which has a magneto-electric constant of 10^-4 in a weak magnetic field.

These nanoparticles **simply have to be rotated** in the required way to generate a force. Feigel calls it a magneto-electric quantum wheel.

Of course, nobody is getting a free lunch here. ** “Although the proposed engine will consume energy for manipulation of the particles, the propulsion will occur without any loss of mass,”** says Feigel. He even suggests, with masterful understatement, that this might have practical implications.

### So here is a high-risk idea with a huge potential payoff. The question is: **Who has the balls to try it?**