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. It was suggested that Abraham only accounted for the momentum of the electromagnetic fields, and his equation was an attempt to take into account the momentum of the material as well. 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 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.
- 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.
- Minkowski, Hermann (1908), "Die Grundgleichungen für die elektromagnetischen Vorgänge in bewegten Körpern", Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse: 53–111
- Wikisource translation: The Fundamental Equations for Electromagnetic Processes in Moving Bodies
- Abraham, Max (1909), "Zur Elektrodynamik bewegter Körper", Rendiconti del Circolo Matematico di Palermo 28: 1–28
- 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
- James Dacey (9 Jan 2009). "Experiment resolves century-old optics mystery". physicsworld.com. Retrieved 4 Mar 2010.
- Hector Hugo Brito (1999). "Propellantless Propulsion by Electromagnetic Inertia Manipulation: Theory and Experiment". In Space Technology and Applications International Forum – 1999, Mohamed S. El-Genk (editor). American Institute of Physics. ISBN 978-1-56396-846-4.
- Marc G. Millis (2004). "Report on Prospects for Breakthrough Propulsion From Physics". In Proceedings 2004 NASA/DoD Conference on Evolvable Hardware. IEEE Computer Society. ISBN 0-7695-2145-2.
- Barnett, Stephen (2010-02-07). "Resolution of the Abraham-Minkowski Dilemma". Phys. Rev. Lett. 104 (7): 070401. Bibcode:2010PhRvL.104g0401B. doi:10.1103/PhysRevLett.104.070401. PMID 20366861.