The Abraham–Minkowski controversy is a physics debate concerning electromagnetic momentum within dielectric media. Traditionally, it is argued that in the presence of matter the electromagnetic stress-energy tensor by itself is not conserved (divergenceless). Only the total stress-energy tensor carries unambiguous physical significance, and how one apportions it between an “electromagnetic” part and a “matter” part depends on context and convenience. In other words, the electromagnetic part and the matter part in the total momentum can be arbitrarily distributed as long as the total momentum is kept the same. There are two incompatible equations to describe momentum transfer between matter and electromagnetic fields. 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 version of momentum “does indeed represent the true momentum density of electromagnetic fields” for electromagnetic waves, while Minkowski version of momentum is “pseudomomentum” or “wave momentum”.
The two equations for the photon momentum in a dielectric with refractive index n are:
- the Minkowski version:
- the Abraham version:
Abraham photon momentum is inversely proportional to the refractive index of the medium, while Minkowski’s is directly proportional to the index. Barnett and Loudon assert that the early experiments by Walker et al. “provide evidence that is no less convincing in favor of the Abraham form”, but Feigel insists that “as far as we know, there are no experimental data that demonstrate the inverse dependence of the radiation pressure on the refractive index”; in other words, no experimental observations of light momentum are quantitatively in agreement with the formulation given by Abraham. However the direct fiber-recoiling observation by She at al. reportedly suggest that “Abraham’s momentum is correct”.
In 2005, the experiment by Campbell and coworkers suggests that in a dilute gas of atoms, the recoil momentum of atoms caused by the absorption of a photon is the Minkowski momentum . In 2006, Leonhardt noted that, “whenever the wave aspects of atoms dominate, as in Campbell and colleagues’ interference experiment, the Minkowski momentum appears, but when the particle aspects are probed, the Abraham momentum is relevant.”
Leonhardt ascribed the 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. 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.
Leonhardt's insight inspired Barnett's 2010 resolution, wherein the Abraham version is the kinetic momentum and the Minkowski version is the canonical momentum; “the kinetic momentum of a body is simply the product of its mass and velocity”, while “the canonical momentum of a body is simply Planck’s constant divided by its de Broglie wavelength”. 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. 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. However Wang disagrees, criticizing 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?”
How to understand “the kinetic momentum of a body is simply the product of its mass and velocity”? Wang argues that in the definition of kinetic momentum, the “mass” should be the “momentum-associated mass” (), instead of “energy-associated mass” (), and the photon’s momentum and energy must constitute a Lorentz four-vector so that the global momentum-energy conservation law can be satisfied in the Einstein-box thought experiment within the principle-of-relativity frame.
Alternatively, analyzed by EM boundary-condition matching approach, Einstein-box thought experiment suggests that
- “in the Maxwell-equation frame, the medium Einstein-box thought experiment supports both light momentum formulations, instead of just Abraham’s. However, the two formulations cannot be ‘both correct’; otherwise it is not determinate whether the medium box gets a pulling force or a pushing force when a specific photon goes into the medium from vacuum. In other words, without resort to the principle of relativity, this thought experiment cannot be used to identify the correctness of light momentum definitions.”
Sheppard and Kemp differently identified the difference between canonical (Minkowski) and kinetic (Abraham) momentums, explaining that the canonical momentum or wave momentum “represents the combination of both field and material momentum values”, while the kinetic momentum “represents the photon momentum void of material contributions”. This explanation is completely consistent with Alexander 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”. According to this theory, Abraham momentum is the quantized field momentum (= field part of total momentumphoton number), while the Minkowski momentum is the quantized wave momentum (= total momentum including both field part and material partphoton number).
However, Wang indicates 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; 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.
Wang 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”. Because Minkowski photon momentum and energy constitute a Lorentz four-vector, given by (Lorentz invariant) Planck constant multiplied by wave four-vector , the Minkowski momentum is parallel to the wave vector in all inertial frames, and thus it meets light-momentum criterion.
Wang  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.
It should be noted that there is another different understanding for canonical momentum of photon. Barnett’s definition of canonical momentum is clear, reading:
- “the canonical momentum of a body is simply Planck’s constant divided by its de Broglie wavelength”.
According to this definition, canonical momentum is an observable quantity (at least in principle). Alternatively, Milonni and Boyd provide a different understanding for the canonical momentum, arguing:
- Canonical momentum “differs in general from kinetic momentum. For a particle of charge and mass in an electromagnetic field, for example, the kinetic momentum is , whereas the canonical momentum , where is the particle velocity and is the vector potential.” 
According to Milonni-Boyd explanation, the canonical momentum may not be an observable quantity, because gauge freedom is an unavoidable presence, and “the gradient of an arbitrary scalar function can be added to without changing the result”; thus the vector potential is not unique, although “it has observable effects as in the Aharonov–Bohm effect”.
The two equations for the electromagnetic momentum in a dielectric are:
- the Minkowski version:
- the Abraham version:
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.
Pfeifer and coworkers claim that the "division of the total energy–momentum tensor into electromagnetic (EM) and material components is arbitrary". 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".
Conventionally, the Poynting vector E × H as EM power flow has been thought to be a well-established basic concept in textbooks.      In view of the existence of a certain mathematical ambiguity for this conventional basic concept, some scientists suggested it to be a "postulate", while some others suggested it to be a "hypothesis", "until a clash with new experimental evidence shall call for its revision". 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", and “this conclusion is clearly supported by Fermat’s principle and special theory of relativity”.
In addition to the Poynting vector criterion, 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. The Laue–Møller criterion supports Minkowski EM tensor, because the Minkowski tensor is a real four-tensor while Abraham's is not, as re-discovered by Veselago and Shchavlev recently. 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 ". Some scientists also criticized the justifications of the energy–velocity definition and the imposed four-vector covariance in Laue–Møller criterion. 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. Regarding the imposed four-velocity covariance, which was probably prompted by the relativistic velocity addition rule applied to illustrating Fizeau running water experiment, 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." 
Wang also indicates that
- “In fact, there is another interesting question in Laue–Møller theory. The Laue–Møller theory assumes the Poynting vector as the EM power flow (energy flow). Because the photon is the carrier of the EM energy and momentum, the Minkowski momentum which the theory solely supports is supposed to be parallel to the Poynting vector. However, the Minkowski momentum and Poynting vector are not parallel in general in a moving medium; resulting in a serious contradiction between the basic assumption and conclusion.”
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”. 
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.” 
Penfield and Haus state that
- ”Abraham’s tensor has the virtue that it is symmetric (at least for fluids), whereas Minkowski’s tensor is nonsymmetric.” 
Robinson states that
- ”We may also remark that, because they [Penfield and Haus] involve a symmetric field stress tensor and identify the electromagnetic momentum density with the energy flux vector, they fit much more naturally into the general scheme of relativistic electrodynamics.”
Landau and Lifshitz state that
- “the energy-momentum tensor must be symmetric”.
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; thus questioning the claim by Bethune-Waddell and Chau  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)”. 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. 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 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.
In regard to why the EM momentum-energy stress tensor is not enough to correctly define light momentum, the study 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.”
Landau-Lifshitz, Weinberg’s, and Møller’s versions of von Laue’s theorem are well known in the dynamics of relativity, and they are often invoked to resolve the Abraham-Minkowski controversy. For example, Landau and Lifshitz presented their version of Laue’s theorem in their textbook while Jackson and Griffiths use this version of Laue’s theorem to construct a Lorentz four-vector; Weinberg presented his version of Laue’s theorem in his textbook while Ramos, Rubilar, and Obukhov use the Weinberg’s version of Laue’s theorem to obtain both Abraham 4-momentum and Minkowski 4-momentum for electromagnetic field; Møller presented his version of Laue’s theorem in his textbook while Brevik and Ellingsen use Møller’s version of Laue’s theorem to conclude that the Minkowski energy-momentum tensor “is divergence-free in a homogeneous medium without external charges implying that the four components of energy and momentum make up a four-vector”.
However, a recent study indicates that “the Landau-Lifshitz version of Laue’s theorem (where the divergence-less of a four-tensor is taken as a sufficient condition) and Weinberg’s version of Laue’s theorem (where the divergence-less plus a symmetry is taken as a sufficient condition) are both flawed”, while “Møller’s version of Laue’s theorem, where the divergence-less plus a zero-boundary condition is taken as a sufficient condition, has a very limited application”.
In a beautiful 1970 original research work, Brevik and Lautrup argue that for a pure radiation field, the space integrals of the time column elements of a canonical energy-momentum tensor constitutes a Lorentz four-momentum; in the 2012 work, Brevik and Ellingsen invoke Møller’s version of Laue’s theorem to support his original argument for Minkowski tensor, because the Minkowski tensor is thought to be a canonical energy-momentum tensor and it is divergence-free for a pure radiation field (while the Abraham tensor is not divergence-free); in the recent 2013 work, Brevik emphasizes that “it is the Minkowski energy-momentum tensor which is the most convenient alternative to work with, as this tensor is divergence-free causing the total radiation momentum and energy to make up a four-vector”; and in the most recent 2016 work, Brevik further emphasizes that “the Minkowski tensor is divergence-free for a pure radiation field, thus leading to a four-vector property of the total energy and momentum”. However, in all those publications, Brevik did not provide any explanations why the canonical energy-momentum tensor or Minkowski tensor for a pure radiation field satisfies the zero-boundary condition required by Møller’s version of Laue’s theorem; thus leaving readers an open question: Is Møller’s version of Laue’s theorem applicable to the Minkowski tensor for a pure radiation field?
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. 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 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?
- Can the Abraham photon momentum and energy constitute a Lorentz four-vector?
The results through the years have been mixed, at best. However, a report on a 2012 experiment claims that unidirectional thrust is produced by electromagnetic fields in dielectric materials. 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:
- “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:
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, 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.
Theories of Reactionless drives
At least one report has suggested Minkowski's formulation, if correct, would provide the physical base for a reactionless drive. 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."
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