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The Abraham-Minkowski Controversy

The Minkowski-Abraham controversy takes its name from two investigators, who, early in the last century, formulated particular varieties of stress tensor for the electromagnetic field in a material medium [1] - [4]. Unfortunately, the tensors give different values of electromagnetic momentum density. In the Minkowski case, the momentum density

   \bm{g}_{\textrm{M}}=\bm{D\times B}

is proportional to the cross-product of the displacement field D and the magnetic flux density field B, while the Abraham momentum density

   \bm{g}_{\textrm{A}}=\bm{E\times H}/c^{2}

is proportional to the cross-product of the electric E and magnetic H fields. The two momenta are the same only in vacuo.

The debate as to which is the correct momentum, which has raged over many years, is complicated by the fact that both the momenta associated with the Minkowski and Abraham tensors can in principle be observed [5].

From time to time, other electromagnetic tensors, such as those of Einstein and Laub [6] - [7] and Peierls [8.] - [9] have also been constructed. All these tensors exist against the background of standard electromagnetic theory and the well-known symmetric, Maxwell and canonical tensors [10].

A review [11] of radiation momenta and photon pressure in dielectric media shows that the conservation equations involving the spatial components of the electromagnetic (Maxwell), Abraham, Minkowski and Einstein-Laub tensors are equivalent statements of Maxwell's equations applied to a homogeneous, linear and dispersionless material medium. Indeed, any linear combination of Abraham, Minkowski and Einstein-Laub tensors satisfy Maxwell's equations. There is therefore no one correct tensor.

Research published in 1993 suggested that the difference between Abraham and Minkowski momenta lay in the distinction between canonical and kinetic momenta [12]. New work [13] has shown this suggestion to be significant.

References

  1. H. Minkowski, Nachr. Ges. Wiss. Göttingen p. 53 (1908).
  2. H. Minkowski, Math. Ann. 68, 472 (1910).
  3. M. Abraham, Rend. Circ. Matem. Palermo 28, 1 (1909).
  4. M. Abraham, Rend. Circ. Matem. Palermo 30, 5 (1910).
  5. Rodney Loudon, Stephen M. Barnett and C. Baxter, Phy. Rev. A, 71, 063802 (2005).
  6. A. Einstein and J. Laub, Ann. der Phys. 26, 541 (1908).
  7. A. Einstein and J. Laub, Ann. der Phys. 26, 532 (1908).
  8. R. Peierls, Proc. R. Soc. Lond. A 347, 475 (1976).
  9. R. Peierls, Proc. R. Soc. Lond. A 355, 141 (1977).
  10. J. D. Jackson, “Classical Electrodynamics” Wiley, New York (1999), 3rd ed.
  11. C. Baxter and Rodney Loudon, J. Mod. Opts., 57, 830 (2010).
  12. C. Baxter, M. Babiker and R. Loudon, Phys. Rev.A, 47, 1278 (1993).
  13. Stephen M. Barnett, Phys. Rev. Letts., 104, 070401 (2010).
This article is only a short summary of the problem. The interested reader is encouraged to consult the references, especially [11], for a fuller account.