# Abraham–Minkowski controversy

The AbrahamMinkowski controversy is a physics debate concerning electromagnetic momentum within dielectric media. Related theories have been put forward that, should their principles be demonstrated to be true, they may allow the design of a reactionless drive.

## Theoretical basis

Two equations exist describing momentum transfer between matter and electromagnetic fields.[1] Both seem to be supported by contradictory experimental data. The two existing equations were first suggested by Hermann Minkowski (1908)[2] and Max Abraham (1909),[3][4] from which the controversy name derives.

Both define the momentum of an electromagnetic field permeating matter. Abraham's equation suggests that in materials through which light travels more slowly, electromagnetic fields should have lower momentum, while Minkowski suggests it should have a greater momentum. Using relativity, Alexander Feigel of Rockefeller University in New York found that "the Abraham definition accounts for the momentum of the electric and magnetic fields alone, while the Minkowski definition also takes into account the momentum of the material".[5] Feigel argues that “while Abraham’s expression is indeed the momentum of the field, the measured momentum also includes the matter contribution and its value coincides with Minkowski’s result.”[6] More recent work suggests that this characterization is incorrect.[7]

At least one report has suggested Minkowski's formulation, if correct, would provide the physical base for a reactionless drive.[8] However, an independent review from the United States Air Force Academy concluded that there would be no expected net propulsive forces, and a NASA report determined that "The signal levels are not sufficiently above the noise as to be conclusive proof of a propulsive effect."[9]

The two equations for the photon momentum in a dielectric with refractive index n are:

• the Minkowski version:
${\displaystyle p_{\mathrm {M} }={\frac {nh\nu }{c}};}$
• the Abraham version:
${\displaystyle p_{\mathrm {A} }={\frac {h\nu }{nc}},}$

where h is the Planck constant, ν is the frequency of the light and c is the speed of light in vacuum.

Leonhardt ascribed the preceding Minkowski and Abraham formulas to the wave-particle duality of light: Minkowski momentum is a wave-characteristics momentum, deduced from the combination of de-Broglieʼs relation with Einstein’s light-quantum theory; Abraham momentum is a particle-characteristics momentum, deduced from the combination of Newton’s law with Einstein’s energy-mass equivalence formula.[10] In his reasoning, Leonhardt implicitly used a plane-wave model, where a plane wave propagates in a lossless, non-conducting, uniform medium so that the wave phase velocity and the photon moving velocity are both equal to c/n. However this assignment of wave-particle duality is questioned by the result in a recent study, which claims that both the Minkowski and Abraham formulas can be directly obtained only from Einstein’s light-quantum theory (applied to the plane wave), without any need to invoke de-Broglieʼs relation, Newton’s law, and Einstein’s energy-mass equivalence formula.[11]

An elegant 2010 theoretical study by Barnett of the University of Strathclyde suggested that both equations are correct, with the Abraham version being the kinetic momentum and the Minkowski version being the canonical momentum, and claims to explain the contradictory experimental results using this interpretation.[12] Barnett argues that the medium Einstein-box thought experiment (also known as “Balazs thought experiment”) supports Abraham momentum while the photon–atom Doppler resonance absorption experiment supports Minkowski momentum.[11] In other words, the photon takes Abraham momentum in the Einstein-box thought experiment, while it takes Minkowski momentum in the photon–atom Doppler resonance absorption experiment; with both Abraham and Minkowski momentums being correct photon momentums.

In contrast to Barnett,[12] Sheppard and Kemp differently identified the difference between canonical (Minkowski) and kinetic (Abraham) momentums, explaining that the canonical momentum “represents the combination of both field and material momentum values”, while the kinetic momentum “represents the photon momentum void of material contributions”.[13] This explanation is completely consistent with Feigel's finding that “the Abraham definition accounts for the momentum of the electric and magnetic fields alone, while the Minkowski definition also takes into account the momentum of the material”.[5] According to Sheppard and Kemp,[13] the Abraham momentum is the correct photon momentum while the Minkowski momentum is not the photon momentum.

However, a recent study showed that in the principle of relativity frame the Abraham momentum would break the global momentum–energy conservation law in the medium Einstein-box thought experiment and it argues that the justification of Minkowski momentum as the correct light momentum is completely required by (i) the principle of relativity, (ii) Einstein light-quantum hypothesis, and (iii) the momentum–energy conservation law, which are all fundamental postulates of physics.[11][14]

The study[11] claims that based on the principle of relativity and Fermat’s principle a light-momentum criterion is set up, stating that “the momentum of light in a medium is parallel to the wave vector in all inertial frames of reference”, and “this light-momentum criterion provides a necessary physical condition to find out whether a mathematical expression can represent the correct momentum of light”. The study[11] argues that, because Minkowski photon momentum and energy constitute a Lorentz four-vector, given by (Lorentz invariant) Planck constant ${\displaystyle \hbar }$ multiplied by wave four-vector ${\displaystyle K^{\mu }}$, the Minkowski momentum is parallel to the wave vector in all inertial frames, and thus it meets light-momentum criterion.

The study [11] further argues that a material medium is made up of massive particles, and the kinetic momentum and energy of each massive particle constitute a momentum-energy four-vector; thus the photon momentum and energy must constitute a Lorentz four-vector in order to satisfy global momentum-energy conservation law within the relativity-principle frame in the Einstein-box thought experiment. "In other words, in a system consisting of massive particles and photons, the momentums and energies of all individual massive particles and photons constitute Lorentz four-vectors no matter whether they have interactions or not." Minkowski photon momentum and energy constitute a Lorentz four-vector and it is consistent with Einstein light-quantum hypothesis and momentum-energy conservation law within the relativity-principle frame; accordingly, the Minkowski momentum represents the unique correct photon momentum.

The study[11] criticizes that “In Barnett’s theory, the argument for supporting Abraham momentum is based on the analysis of the Einstein-box thought experiment by the ‘center-of-mass-energy’ approach, where the global momentum-energy conservation law is employed to obtain Abraham photon momentum and energy in the medium box in the laboratory frame. At first sight, such an approach is indeed impeccable; however, upon more careful investigation, one may find that the approach itself has implicitly assumed the Abraham momentum to be the correct momentum; thus leaving readers an open question: Do the Abraham momentum and energy obtained still satisfy the global momentum-energy conservation law in all inertial frames of reference so that the argument is consistent with the principle of relativity?”

The two equations for the electromagnetic momentum in a dielectric are:

• the Minkowski version:
${\displaystyle \mathbf {g} _{\mathrm {M} }=\mathbf {D} \times \mathbf {B} ;}$
• the Abraham version:
${\displaystyle \mathbf {g} _{\mathrm {A} }={\frac {1}{\mathrm {c} ^{2}}}\mathbf {E} \times \mathbf {H} ,}$

where D is the electric displacement field, B is the magnetic flux density, E is the electric field, and H is the magnetic field. The photon momentum is thought to be the direct result of Einstein light-quantized electromagnetic momentum.[14]

Pfeifer and coworkers claim that the "division of the total energy–momentum tensor into electromagnetic (EM) and material components is arbitrary".[1] In other words, the EM part and the material part in the total momentum can be arbitrarily distributed as long as the total momentum is kept the same. But some others don’t agree, and they suggested a Poynting vector criterion. They say for EM radiation waves the Poynting vector E × H denotes EM power flow in any system of materials, and they claim that the Abraham momentum E × H/c2 is "the sole electromagnetic momentum in any system of materials distributed throughout the free space".[15]

Conventionally, the Poynting vector E × H as EM power flow has been thought to be a well-established basic concept in textbooks.[16][17][18][19][20][21] In view of the existence of a certain mathematical ambiguity for this conventional basic concept, some scientists suggested it to be a "postulate",[15] while some others suggested it to be a "hypothesis", "until a clash with new experimental evidence shall call for its revision".[21] However, this basic concept is challenged in a recent study, which claims "Poynting vector may not denote the real EM power flow in an anisotropic medium",[14] and “this conclusion is clearly supported by Fermat’s principle and special theory of relativity”.[22]

In addition to the Poynting vector criterion,[15] Laue and Møller suggested a criterion of four-vector covariance imposed on the propagation velocity of EM energy in a moving medium, just like the velocity of a massive particle.[23] The Laue–Møller criterion supports Minkowski EM tensor, because the Minkowski tensor is a real four-tensor while Abraham's is not,[19] as re-discovered by Veselago and Shchavlev recently.[24] But some scientists disagree, criticizing that "it is widely recognized now that Abraham's tensor is also capable of describing optical experiments," and such a criterion of this type is only "a test of a tensor's convenience rather than its correctness ".[23] Some scientists also criticized the justifications of the energy–velocity definition and the imposed four-vector covariance in Laue–Møller criterion.[19] Regarding the energy–velocity definition which is given by Poynting vector divided by EM energy density in Laue–Møller criterion, they say "the Poynting vector does not necessarily denote the direction of real power flowing" in a moving medium.[14] Regarding the imposed four-velocity covariance, which was probably prompted by the relativistic velocity addition rule applied to illustrating Fizeau running water experiment,[25] they say "one essential difference between massive particles and photons is that any massive particle has its four-velocity, while the photon (the carrier of EM energy) does not."

Conventionally, the EM momentum-energy stress tensor (energy-momentum tensor) is used to define the EM momentum of light in a medium. Minkowski first developed an EM tensor, corresponding to Minkowski momentum D × B, and later, Abraham also suggested an EM tensor, corresponding to Abraham momentum E × H/c2. Bethune-Waddell and Chau claim that

the symmetry of an energy-momentum tensor is “a necessary condition to satisfy conservation of angular momentum and center-of-mass velocity”, while the Abraham energy-momentum tensor “is diagonally symmetric and therefore, consistent with angular momentum conservation”; thus “convincing theoretical arguments have been developed in support of the Abraham momentum density”. [26]

Pfeifer and coworkers state that

“The electromagnetic energy-momentum tensor of Minkowski was not diagonally symmetric, and this drew considerable criticism as it was held to be incompatible with the conservation of angular momentum.”[1]

Accordingly, it is a widely-accepted basic concept that the symmetry of an energy-momentum tensor is a necessary condition to satisfy conservation of angular momentum. However a study indicates that such a concept was set up from an incorrect mathematical conjecture in textbooks;[27] thus questioning the claim by Bethune-Waddell and Chau [26] that “convincing theoretical arguments have been developed in support of the Abraham momentum density”.

It is generally argued that Maxwell equations are manifestly Lorentz covariant while the electromagnetic stress–energy tensor follows from the Maxwell equations; thus the EM momentum defined from the EM tensor certainly respects the principle of relativity. For example, some scientists suggested that “the original (Abraham-Minkowski) debate is in regard to the 4 × 4 energy-momentum tensor (electromagnetic stress–energy tensor)”.[28] However a study indicates that “such an argument is based on an incomplete understanding of the relativity principle”, and states that the EM stress-energy tensor is not sufficient to define EM momentum correctly.[11] That is because, in a material medium, which is different from empty space, a covariant EM tensor usually does not guarantee that all the elements of the tensor have the same physical meanings in all inertial frames due to the existence of possible "intrinsic Lorentz violation", while the relativity principle requires that the mathematical equations describing a physical law must be the same in form in all inertial frames, and the specific physical implications of the equations must also be the same. For example, the principle of relativity requires: (i) the mathematical expressions of Maxwell equations be the same in form in all inertial frames, and (ii) the physical implications of all field quantities E, B, D, H, J and ρ appearing in the Maxwell equations be also the same.

The study[11] emphasizes that “the application of the relativity principle is very tricky, not just manipulating Lorentz transformations”. For example, when applying the relativity principle to the Maxwell equations in free space, one may directly obtain the constancy of light speed, without any need of Lorentz transformations.[29]

In regard to why the EM momentum-energy stress tensor is not enough to correctly define light momentum, the study[11] also provides a strong mathematical argument that the momentum conservation equations derived from EM stress-energy tensors are all differential equations, and they can be converted one to the other through Maxwell equations; thus “Maxwell equations support various forms of momentum conservation equations, which is a kind of indeterminacy. However it is this indeterminacy that results in the question of light momentum.” To remove the indeterminacy, the study argues, the principle of relativity is indispensable. “This principle is a restriction but also is a guide in formulating physical theories. According to this principle, there is no preferred inertial frame for descriptions of physical phenomena. For example, Maxwell equations, global momentum and energy conservation laws, Fermat’s principle, and Einstein’s light-quantum hypothesis are equally valid in all inertial frames, no matter whether the medium is moving or at rest, and no matter whether the space is fully or partially filled with a medium.”[11]

Theoretically speaking, the Abraham–Minkowski controversy is focused on the issues of how to understand some basic principles and concepts in special theory of relativity and classical electrodynamics.[11][13][28] For example, when there exist dielectric materials in space,

• Is the principle of relativity still valid?
• Are the Maxwell equations, momentum–energy conservation law, Einstein light-quantum hypothesis, and Fermat's principle[30] equally valid in all inertial frames of reference?
• Does the Poynting vector always represent EM power flow in any system of materials?
• Does the photon have a Lorentz four-velocity like a massive particle?
• Why is the EM momentum–energy stress tensor not enough to correctly define light momentum?
• Why is the principle of relativity needed to identify the justification of the light-momentum definition?
• Why must the photon momentum and energy constitute a Lorentz four-vector?[11]
• Can the Abraham photon momentum and energy constitute a Lorentz four-vector?[31]

## Experiments

The results through the years have been mixed, at best.[7][32] However, a report on a 2012 experiment claims that unidirectional thrust is produced by electromagnetic fields in dielectric materials.[33] A recent study shows that both Minkowski and Abraham pressure of light have been confirmed by experiments, and it has been published in May 2015. The researchers claim:[34]

“we illuminate a liquid … with an unfocused continuous-wave laser beam … we have observed a (reflected-light) focusing effect … in quantitative agreement with the Abraham momentum.”
“we focused the incident beam tightly … we observed a de-focusing reflection … in agreement with the Minkowski momentum transfer.”

In other words, their experiments have demonstrated that an unfocused laser beam corresponds to a response of Abraham momentum from the liquid, while a tightly-focused beam corresponds to a response of Minkowski momentum. But the researchers did not tell what the response will be for a less tightly-focused beam (between “unfocused” and “tightly-focused”), or whether there is any jump for the responses. The researchers concluded:[34]

We have obtained experimental evidence, backed up by hydrodynamic theory, that the momentum transfer of light in fluids is truly Janus–faced: the Minkowski or the Abraham momentum can emerge in similar experiments. The Abraham momentum, equation (2), emerges as the optomechanical momentum when the fluid is moving and the Minkowski momentum, equation (1), when the light is too focused or the container too small to set the fluid into motion. The momentum of light continues to surprise.

Thus the researchers’ claim that “the momentum transfer of light in fluids is truly Janus–faced” is an extrapolated conclusion, because the conclusion is drawn only based on the observed data of the cases with “unfocused” and “tightly-focused” beams (while excluding all other cases with beams between “unfocused” and “tightly-focused”) --- a line of reasoning similar to that used in the work for subwavelength imaging,[35] where

In the measured curves plotted in figure 4, the data on one side of the device were measured first, and the data on the other side were obtained by mirroring, under the symmetry assumption arising from the device structure.

## References

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