Abraham–Minkowski controversy

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.[2]

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.[18] "provide evidence that is no less convincing in favor of the Abraham form",[19] 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";[20] 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 et al.[21] 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 ${\displaystyle p_{\mathrm {M} }}$.[22] 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."[23]

In a recent Editors' Suggestion of Physical Review A,[24] Brevik criticizes that the momentum and energy in the mass-polariton (MP) quasiparticle model proposed by Partanen and coworkers[13] "are not the components of a four-vector", and further, he indicates that Leonhardt and Philbin have already developed "the correct general-relativistic description of light",[25] which was pioneered by Gordon.[26]

Based on their theory, 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.[23] 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.[27]

Leonhardt's insight inspired Barnett's 2010 resolution published in an Editors' Suggestion of Physical Review Letters, which is one of the most highly cited papers in the Abraham–Minkowski controversy. In Barnett's resolution, 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".[28] 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.[27] In other words, the photon takes Abraham momentum in the Einstein's box thought experiment, while it takes Minkowski momentum in the photon–atom Doppler resonance absorption experiment; with both Abraham and Minkowski momenta being correct photon momenta. However Wang disagrees, criticizing that Barnett's physical model "is not consistent with global momentum-energy conservation law in the principle-of-relativity frame".[29] "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?"[27]

How to understand "the kinetic momentum of a body is simply the product of its mass and velocity"?[28] Wang argues that in the definition of kinetic momentum, the "mass" should be the "momentum-associated mass" (${\displaystyle =|{\text{momentum}}|/|{\text{velocity}}|}$), instead of "energy-associated mass" (${\displaystyle ={\text{energy}}/c^{2}}$), 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.[16]

Sheppard and Kemp differently identified the difference between canonical (Minkowski) and kinetic (Abraham) momenta, explaining that the canonical momentum or wave momentum[8] "represents the combination of both field and material momentum values", while the kinetic momentum "represents the photon momentum void of material contributions".[30] This explanation is completely consistent with Alexander Feigel's finding that "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";[20] thus "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".[31] According to this theory, Abraham momentum ${\displaystyle p_{\mathrm {A} }}$ is the quantized field momentum (= field part of total momentum${\displaystyle /}$photon number), while the Minkowski momentum ${\displaystyle p_{\mathrm {M} }}$ is the quantized wave momentum (= total momentum including both field part and material part${\displaystyle /}$photon number).[32]

In his Featured-in-Physics PRL Letter,[20] Feigel uses relativistic Lagrange formalism to analyze momentum transfer between matter and EM field in a moving dielectric medium, which is isotropic, nondispersive, and uniform, observed in the medium-rest frame. In Feigel's formalism, Minkowski approximate constitutive relation is taken into account in invariant Lagrangian density ${\displaystyle 0.5(\mathbf {D} \cdot \mathbf {E} -\mathbf {B} \cdot \mathbf {H} )}$. However, Lagrange formalism "gives no new clue on the interpretation of the macroscopic Maxwell equations", as criticized by Tiggelen and Rikken,[33] because the validity of Lagrange formalism is identified with whether the Euler-Lagrange equations generated by the Lagrangian density ${\displaystyle 0.5(\mathbf {D} \cdot \mathbf {E} -\mathbf {B} \cdot \mathbf {H} )}$ under the principle of least action are the same as Maxwell equations.[34] That is to say, it is the Maxwell equations that are the first principles for descriptions of macroscopic EM phenomena, instead of Lagrange formalism. From this, Lagrange formalism is supposed to be equivalent to Maxwell equations; otherwise, Maxwell EM theory would not be complete. Thus in principle, the criticism by Tiggelen and Rikken makes sense, and the difficulty how to correctly define the momentum of light in the Maxwell-equation frame would not be disappearing in Lagrange formalism.

Recently, Picardi and coworkers emphasized the physical difference between the kinetic-Abraham and canonical-Minkowski quantities, arguing that "the former ones describe the properties of electromagnetic fields only, while the latter ones characterize properties of the whole wave mode (i.e., a polariton, which involves, on the microscopic level, oscillations of both fields and electrons in matter)".[35] However the EM fields can be divided into two kinds: radiation field (made up of photons) and non-radiation field (such as the self-field carried by a charged particle). Picardi and coworkers did not explain whether or not the "electromagnetic fields only" include the non-radiation field carried by "electrons in matter", thus leading to an ambiguous implication.

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.[27]

According to Wang's theory, Minkowski photon is a kind of quasi-photon, and its "four-momentum ${\displaystyle \hbar K^{\mu }}$ denotes the macroscopic average of the properties of photons absorbed and re-emitted by the material subsystem".[16] ${\displaystyle \hbar K^{\mu }}$ only denotes the momentum and energy of a pure radiation field, because the momentum and energy of a non-radiation field (owned by the material) cannot constitute a four-vector. This explanation is completely different from the argument by Feigel and Kemp,[20][32] where Minkowski momentum is thought to include both the field part and material part.

Wang[27] 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 ${\displaystyle \hbar }$ 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. However Partanen and coworkers disagree, criticizing: Wang's theory "neglects the transferred mass ${\displaystyle \delta m}$, thus leading to mathematical problems", and "the neglectance of the transferred mass ${\displaystyle \delta m}$ … in turn leads to complicated mathematics without providing transparent and physically insightful covariant theory of light".[13]

The wave four-vector ${\displaystyle K^{\mu }}$ is a corollary of the invariance of Maxwell equations (although widely ignored in the analysis of Abraham–Minkowski debate), and it was first shown by Einstein in his 1905 paper when setting up "theory of Doppler's principle and of aberration".[36] Since ${\displaystyle K^{\mu }}$ is a Lorentz four-vector, ${\displaystyle K^{\mu }K_{\mu }}$ must be a Lorentz invariant, leading to ${\displaystyle \omega ^{2}(1-n^{2})=}$ Lorentz invariant.[27] In a recent excellent work, Partanen and coworkers claim that the energy ${\displaystyle E_{MP}=n^{2}\hbar \omega }$ and momentum ${\displaystyle p_{MP}=n\hbar \omega /c}$ for their MP quasiparticle also constitute a Lorentz four-vector, leading to ${\displaystyle (n\omega )^{2}(n^{2}-1)=}$ Lorentz invariant.[13] Since ${\displaystyle \omega ^{2}(1-n^{2})}$ and ${\displaystyle (n\omega )^{2}(n^{2}-1)}$ are both Lorentz invariants, the frequency ${\displaystyle \omega }$ and the refractive index ${\displaystyle n}$ must also be Lorentz invariants for ${\displaystyle n\neq 1}$, which means that there is no Doppler effect in a dielectric medium. Such a conclusion could call Einstein's special relativity into question.[16]

In 1999, Leonhardt and Piwnicki proposed a formulation of the optics of nonuniformly moving [isotropic] media, arguing that the moving medium acts on light as an effective gravitational field, and the light rays are geodesic lines with respect to Gordon's metric. According to Leonhardt–Piwnicki theory, the ray velocity for a plane wave in a moving isotropic uniform medium is not parallel to the wave vector in general.[37] Apparently, this result from Leonhardt–Piwnicki theory[37] is essentially different from the result from wang's theory, where the light ray velocity or photon's velocity is argued, according to Fermat's principle and relativity principle, to be parallel to the wave vector, observed in all inertial frames.[27] This difference between the two theories comes from the different understandings of Fermat's principle. In Leonhardt–Piwnicki understanding, the light rays are the "zero-geodesic lines [between two points] with respect to Gordon's metric", and only "in the special case of a medium at rest, this result is equivalent to Fermat's principle",[38] namely Fermat's principle is valid only in the medium-rest frame, while in Wang's understanding, Fermat's principle is valid in all inertial frames, and the light rays are the paths with the minimum optical length between two equiphase planes (instead of two points).[27] Apparently, Leonhardt–Piwnicki theory[37] (where as a physical law, Fermat's principle is valid only in the medium-rest frame) would not support the principle of relativity. On the other hand, a moving isotropic medium becomes anisotropic.[39] For a plane light wave in a uniform anisotropic medium, the light power (energy) must flow along the wave vector, otherwise energy conservation will be broken; namely Fermat's principle is consistent with energy conservation law.[40] Thus according to Wang's analysis, the Gordon-metric geodesic lines defined as light rays in moving media in Leonhardt–Piwnicki theory[37] would contradict against energy conservation law.

Formulation of a physical theory is supposed to be consistent with physical postulates, such as global momentum and energy conservation laws, and the principle of relativity. Regarding how to correctly apply conservation principles to derive the correctness of competing momentum formulations, Brevik indicates:

• "the electromagnetic field in a medium is a subsystem, which has to be supplemented with the material subsystem to form a closed system for which the conservation principles are more powerful."[41]

In above, the "conservation principles", which Brevik refers to, are supposed to be the "conservation principles within the principle-of-relativity frame", in order to make the obtained results fitting the principle of relativity.

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 Wang argues:

• The photon momentum and energy must constitute a Lorentz four-vector in order to satisfy global momentum-energy conservation law within the principle-of-relativity frame in the Einstein-box thought experiment.[29][16]

Minkowski photon momentum and energy constitute a Lorentz four-vector, and thus it satisfies the global momentum-energy conservation law within the relativity-principle frame in the thought experiment; accordingly, the Minkowski momentum represents the unique correct photon momentum. In other words, the "global momentum-energy conservation law within the principle-of-relativity frame" picks out Minkowski momentum in the competing momentum formulations.

Pointing to the application of momentum-energy conservation law in the Einstein-box thought experiment in Barnett's 2010 PRL Editors' Suggestion,[28] which is widely recognized by experts in the community (especially, the PRL referees are excellent experts), Wang criticizes:

• Barnett's application itself has an implicit assumption that, "once the Abraham momentum and energy satisfy the global momentum-energy conservation law in one inertial frame of reference, then they will do in all inertial frames. Obviously, there is no basis to support such an implicit assumption in the Einstein-box thought experiment."[29]

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".[28]

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 ${\displaystyle q}$ and mass ${\displaystyle m}$ in an electromagnetic field, for example, the kinetic momentum is ${\displaystyle m\mathbf {v} }$, whereas the canonical momentum ${\displaystyle \mathbf {p} =m\mathbf {v} +q\mathbf {A} }$, where ${\displaystyle \mathbf {v} }$ is the particle velocity and ${\displaystyle \mathbf {A} }$ is the vector potential."[42]

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 ${\displaystyle \mathbf {A} }$ without changing the result";[43] thus the vector potential ${\displaystyle \mathbf {A} }$ is not unique, although "it has observable effects as in the Aharonov–Bohm effect".[43]

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.[27]

For a plane light wave in a uniform medium, observed in the medium-rest frame, the Abraham momentum ${\displaystyle \mathbf {g} _{A}=\mathbf {E} \times \mathbf {H} /c^{2}}$ is equivalent to the Planck's momentum ${\displaystyle \mathbf {g} _{Planck}=\mathbf {S} _{ef}/c^{2}}$ with ${\displaystyle \mathbf {S} _{ef}}$ the energy flux (= energy density multiplied by velocity), usually called Plank's principle[41] or Planck's theorem.[44] According to Ives,[45] the Planck's momentum was first (implicitly) derived by Poincarè in 1900, and later (in 1907) Planck utilized it to study the relation between inertial mass and quantity of heat for a body. Since ${\displaystyle energy/c^{2}=mass}$ is Einstein's mass-energy equivalence equation, the Planck's principle is essentially the same as the Newton's law (momentum = mass multiplied by velocity), which is often used in solving the Abraham–Minkowski problem, such as in Leonhardt's analysis based on the wave–particle duality of light,[23] and in Barnett's analysis based on Einstein's medium-box thought experiment.[28]

In his respected textbook,[34] Jackson indicates that, "although a treatment using the macroscopic Maxwell equations leads to an apparent electromagnetic momentum, ${\displaystyle \mathbf {g} =\mathbf {D} \times \mathbf {B} }$ …, the generally accepted expression for a medium at rest is ${\displaystyle \mathbf {g} =\mathbf {E} \times \mathbf {H} /c^{2}}$ …"; within the medium, in addition to the EM momentum, there is an additional co-traveling mechanical momentum contributed by the oscillating bound electrons driven by the EM wave. However Peierls argues that neither Minkowski's result nor Abraham's result is correct.[46]

Pfeifer and coworkers claim that the "division of the total energy–momentum tensor into electromagnetic (EM) and material components is arbitrary".[3] 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 Mansuripur and Zakharian 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".[47]

Conventionally, the Poynting vector E × H as EM power flow has been thought to be a well-established basic concept in textbooks.[48] [49] [50] [51] [52] [53] In view of the existence of a certain mathematical ambiguity for this conventional basic concept, Mansuripur and Zakharian suggested it to be a "postulate",[47] while Stratton suggested it to be a "hypothesis", "until a clash with new experimental evidence shall call for its revision".[53] 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",[54] and "this conclusion is clearly supported by Fermat's principle and special theory of relativity".[40]

In addition to the Poynting vector criterion,[47] 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.[55] The Laue–Møller criterion supports Minkowski EM tensor, because the Minkowski tensor is a real four-tensor while Abraham's is not,[51] as re-discovered by Veselago and Shchavlev recently.[56] In his highly respected review paper Brevik, on one hand, disapproves for Laue–Møller criterion of four-vector covariance, criticizing:

• "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 ".[55]

On the other hand, Brevik approves for Laue–Møller criterion, arguing:

• "The propagation velocity of the energy of the wave (the 'ray' velocity) [photon's velocity] … transforms like a particle velocity under Lorentz transformations. This property is not merely of mathematical significance, … because there is experimental evidence that the ray velocity actually behaves in this way under Lorentz transformations."[55]

The "experimental evidence", which Brevik claimed, refers to Fizeau running water experiment.

Wang also criticized the justifications of the energy–velocity definition and the imposed four-vector covariance in Laue–Møller criterion.[51] Regarding the energy–velocity definition which is given by Poynting vector divided by EM energy density in Laue–Møller criterion, Wang argues "the Poynting vector does not necessarily denote the direction of real power flowing" in a moving medium.[54] Regarding the imposed four-velocity covariance, which was probably prompted by the relativistic velocity addition rule applied to illustrating Fizeau running water experiment,[57] Wang argues that any massive particle has its four-velocity, while the photon (the carrier of EM energy) does not.[27] Since the photon does not have a four-velocity, the Fizeau running water experiment should be taken to be in support of the Minkowski momentum, instead of the experimental evidence of the relativistic four-velocity addition rule.[27]

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."[27]

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".[44]

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."[3]

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."[58]

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."[59]

Landau and Lifshitz state that

"the energy-momentum tensor must be symmetric".[60]

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;[61][62] thus questioning the claim by Bethune-Waddell and Chau[44] 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. This is not exactly true. As indicated by Sheppard and Kemp, "the original [Abraham–Minkowski] debate is in regard to the 4×4 energy-momentum tensor [electromagnetic stress–energy tensor]".[63] Minkowski tensor is a real Lorentz four-tensor, apparently leading to Minkowski momentum, although it is non-symmetric. According to Maxwell equations, Abraham constructed symmetric Abraham tensor by assuming that Abraham momentum is correct momentum. However Abraham tensor is not a Lorentz four-tensor at all, although treated as a tensor to get Abraham force,[51] which is seriously contradicting the principle of relativity.

Regarding the Abraham tensor, Møller indicated that for a plane light wave propagating in an isotropic uniform medium, observed in the medium-rest frame the Abraham tensor produces an Abraham force, given by ${\displaystyle \mathbf {f} ^{Abr}=[(n^{2}-1)/c^{2}]\partial (\mathbf {E} \times \mathbf {H} )/\partial t}$, but "the electromagnetic energy is conserved", namely there is no energy exchange between the light wave and the medium; however, observed in a moving inertial frame, there is "an exchange of energy between the electromagnetic and the mechanical system, i.e. a local absorption and re-emission of light energy by the body [medium material]". According to the principle of relativity, Møller argues that Minkowski tensor "is more natural" than Abraham tensor.[51] However Brevik disagrees, arguing that "the Abraham force fluctuates out" for an optical pulse;[24] and he proposed an interesting experiment to detect this Abraham force, predicting "if this idea could be realized experimentally, it would be the first case that the Abraham force is detected explicitly in optics".[41] Brevik's prediction implies that the Abraham momentum, which was first assumed by Abraham,[51] has never been confirmed by experiments so far, although experimental observations of Abraham momentum have been already claimed by several research groups.[18][21][64][65]

In fact, the EM stress-energy tensor is not sufficient to define EM momentum correctly,[27] because the way to construct EM tensors is not unique. According to Minkowski and Abraham, a general EM tensor could be defined as ${\displaystyle T^{\mu \nu }=(1-a)T_{Abr}^{\mu \nu }+aT_{Min}^{\mu \nu }}$, where ${\displaystyle T_{Abr}^{\mu \nu }}$ is the Abraham tensor, ${\displaystyle T_{Min}^{\mu \nu }}$ is the Minkowski tensor, and ${\displaystyle a}$ is an arbitrary constant. Thus there are infinite EM tensors within the Maxwell-equation frame; ${\displaystyle T^{\mu \nu }=T_{Abr}^{\mu \nu }}$ for ${\displaystyle a=0}$, ${\displaystyle T^{\mu \nu }=T_{Min}^{\mu \nu }}$ for ${\displaystyle a=1}$, and ${\displaystyle T^{\mu \nu }=T_{Abr}^{\mu \nu }=T_{Min}^{\mu \nu }}$ in free space. From this one can see that the EM tensor is not sufficient to correctly define the momentum of light in a medium.

The study by Wang[27] 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.[66] Another typical example is the "hyperplane" differential element four-vector in relativistic electrodynamics, which contradicts both change of variables theorem in mathematical analysis and Lorentz contraction effect in Einstein's special relativity, namely which follows neither the principle of classical mathematical analysis nor the principle of relativity.[62]

In regard to why the EM momentum-energy stress tensor is not enough to correctly define light momentum, the study[27] 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."[27]

Landau-Lifshitz, Weinberg's, and Møller's versions of von Laue's theorem are well known in the dynamics of relativity,[61] 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[67] while Jackson and Griffiths use this version of Laue's theorem to construct a Lorentz four-vector;[2][34] Weinberg presented his version of Laue's theorem in his textbook[68] 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;[69] Møller presented his version of Laue's theorem in his textbook[51] 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".[70]

However, Wang 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".[61] In a recent study, Wang further indicates that Møller's version of Laue's theorem is also found to be flawed, because the divergence-less plus a zero-boundary condition is not a sufficient condition.[62]

In a beautiful 1970 original research work,[71] 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 his well-known 1979 review paper, Brevik argues that Minkowski tensor is an attractive alternative for the description of optical phenomena, because "in a homogeneous medium it is divergence-free" so that its time-column space integrals form a four-vector;[55] in the 2012 work,[70] 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 2013 work,[72] 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"; in the 2016 work,[73] 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"; in a recent Brief Review paper of Modern Physics Letters A, Brevik again emphasizes that Minkowski tensor "is divergence-free, … meaning that the corresponding total momentum components and the total energy form a four-vector";[41] and in the most recent Editors' Suggestion, Brevik reiterates that "its [Minkowski tensor] vanishing four-divergence implies that the energy and momentum photon components constitute a four-vector".[24] However, in all those publications,[71][55][70][72][73][41][24] 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?

Brevik's "implicit scientific guesswork" (Minkowski tensor for a pure radiation field satisfies the zero-boundary condition required by Møller's theorem) corresponds to a challenging EM boundary-value problem: For a (non-zero) radiation wave in a closed system without any source, can the EM fields satisfy a zero-boundary condition for any time?[62] Brevik's guesswork has been endorsed by high-profile peer-reviewed scientific journals again and again, such as Physics Reports,[55] Physical Review A,[70][24] and Annals of Physics;[73] raising a serious ethical question for scientists and journal editors: Does an "implicit scientific guesswork" not need to be supported by any scientific proofs or clarifications? Otherwise, ever how much difference is there between the "implicit scientific guesswork" (endorsed by Physical Review A again and again[70][24]) and the "fabrication of data … with the intent to mislead or deceive" (defined by APS Guidelines for Professional Conduct[74])? Is such professional conduct consistent with "the expected norms of scientific conduct"?

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.[7][59][55][8][37][23][3][9][19][42][32][69][2][44][27][13] For example, when there exist dielectric materials in space,

• Is the principle of relativity still valid?[27]
• Why should the definitions of physical quantities be the same in all inertial frames of references?[16]
• What is the definition of Lorentz covariance for a physical quantity or a physical tensor?[16]
• Are the Maxwell equations, momentum–energy conservation law, Einstein light-quantum hypothesis, and Fermat's principle[75] equally valid in all inertial frames of reference?
• Why is the traditional formulation of Fermat's principle not applicable to plane light waves?[75][40]
• Why are the velocity and direction of equiphase planes of motion undetermined, without Fermat's principle employed?[27]
• Why are the geodesic lines defined as light rays in moving media?[37] not consistent with energy conservation law?
• Does the Poynting vector always represent EM power flow in any system of materials?[40]
• Why is the EM momentum–energy stress tensor not enough to correctly define light momentum?[16]
• Why is the principle of relativity needed to identify the justification of the light-momentum definition?[27][16]
• Why must the photon momentum and energy constitute a Lorentz four-vector?[29][27]
• Does the photon have a Lorentz four-velocity like a massive particle?[27]
• Can the Abraham photon momentum and energy constitute a Lorentz four-vector?[29][16]
• Why is the Abraham EM tensor not a real Lorentz four-tensor?[51][56]
• Is the Abraham electromagnetic force physical?[76][24]
• Are the momentum and energy of the EM fields carried by an electron, which uniformly moves in free space, measurable experimentally?[16]
• Why is the momentum and energy of a non-radiation field not measurable experimentally?[16]
• Why are the momentum-energy conservation law and Fermat's principle additional basic postulates in physics, independent of Maxwell EM theory?[16]
• Is it the wave-particle duality of light that results in Abraham–Minkowski controversy?[23][27]
• Why is it controversial to define the momentum of light only in the Maxwell-EM-theory frame?[16]
• Does the divergence-less of a Lorentz four-tensor imply that the time-column space integrals of the tensor form a Lorentz four-vector?[24][55][61][62]
• Does Minkowski tensor for a pure radiation field satisfy the zero-boundary condition required by Møller's theorem?[62]
• Does the EM or global momentum–energy stress tensor have to be symmetric?[60][61]
• Why does the construction of "hyperplane" differential element four-vector in relativistic electrodynamics follow neither the principle of classical mathematical analysis nor the principle of relativity?[62]
• Why is the Gordon-metric dispersion equation ${\displaystyle \Gamma ^{\mu \nu }K_{\mu }K_{\nu }=0}$ equivalent to the Minkowski-metric equation ${\displaystyle g^{\mu \nu }(K_{\mu }K_{\nu }-K'_{\mu }K'_{\nu })=0}$?[69][54]

Even in free space, still there are some basic concepts to be clarified. For example:

• Is there any photon-rest frame in free space?[16]
• Does the photon rest mass in free space have any physical meaning?[16]
• What is the definition of photon's mass in free space?[16][77]
• Why is the Planck constant ${\displaystyle \hbar }$ a Lorentz invariant (so that ${\displaystyle \hbar K^{\mu }}$ is legitimately defined as the photon four-momentum)?[27]
• Why is the photon four-momentum supposed to be the direct result of Einstein light-quantized EM four-momentum?[27][16]
• Why is there a Lorentz contraction effect for a moving volume, just like a moving ruler, in Einstein's special relativity?[62]
• Why is the Lorentz contraction consistent with the change of variables theorem in classical mathematical analysis?[62]
• What is the correct technique for change of variables in space (triple) integrals?[62]
• In developing his theorem, why did Laue use the change of variables theorem to perform space integral transformation, instead of using the "hyperplane" differential element four-vector?[62][78]

The results through the years have been mixed, at best.[79][11] However, a report on a 2012 experiment claims that unidirectional thrust is produced by electromagnetic fields in dielectric materials.[80] 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:[64]

“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:[64]

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,[81] 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.

At least one report from Britol et al. has suggested Minkowski's formulation, if correct, would provide the physical base for a reactionless drive,[17] however a NASA report stated, "The signal levels are not sufficiently above the noise as to be conclusive proof of a propulsive effect."[82]

Other work was conducted by the West Virginia Institute for Scientific Research (ISR) and was independently reviewed by the United States Air Force Academy, which concluded that there would be no expected net propulsive forces.[83][82]

• Reactionless drive
• Spacecraft propulsion
• Stochastic electrodynamics

• Physical Review Focus: Momentum From Nothing
• Physical Review Focus: Light Bends Glass