Two equations exist describing momentum transfer between matter and electromagnetic fields. Both seem to be supported by contradicting experimental data. The two existing equations were first suggested by Hermann Minkowski (1908) and Max Abraham (1909),   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, Feigel 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”. More recent work suggests that this characterization is incorrect.
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."
The two equations for the photon momentum in a dielectric with refractive index are:
- The Minkowski version:
- The Abraham version:
A 2010 study 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 contradicting experimental results using this interpretation. However, a recent study shows that in the principle-of-relativity frame the Abraham momentum would break global momentum-energy conservation law in medium Einstein-box thought experiment (also known as “Balazs thought experiment”), and it claims that, the justification of Minkowski momentum as the correct light momentum is completely required by the principle of relativity and the momentum-energy conservation law, which are all the fundamental postulates of physics.
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.
Some scientists 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, 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”.
In addition to the Poynting-vector criterion, Laue and Møller suggested an 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 indicated 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”.
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, and Fermat’s principle still 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?
- Robert N. C. Pfeifer, Timo A. Nieminen, Norman R. Heckenberg, and Halina Rubinsztein-Dunlop (October–December 2007). "Colloquium: Momentum of an electromagnetic wave in dielectric media". Review of Modern Physics 79 (4): 1197. arXiv:0710.0461. Bibcode:2007RvMP...79.1197P. doi:10.1103/RevModPhys.79.1197.
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- Wikisource translation: The Fundamental Equations for Electromagnetic Processes in Moving Bodies
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- Wikisource translation: On the Electrodynamics of Moving Bodies
- Abraham, Max (1910), "Sull'Elletrodinamica di Minkowski", Rendiconti del Circolo Matematico di Palermo 30: 33–46
- Wikisource translation: On the Electrodynamics of Minkowski
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