Can the Abraham Light Momentum and Energy in a Medium Constitute a Lorentz Four-Vector?

ABSTRACT

By analyzing the Einstein-box thought experiment with the principle
of relativity, it is shown that Abraham’s light momentum and energy in a medium
cannot constitute a Lorentz four-vector, and they consequentially break global
momentum and energy conservation laws. In
contrast, Minkowski’s momentum and energy always constitute a Lorentz
four-vector no matter whether in a medium or in vacuum, and the Minkowski’s
momentum is the unique correct light momentum. A momentum-associated
photon mass in a medium is exposed, which explains why only the Abraham’s
momentum is derived in the traditional “center-of-mass-energy” approach. The EM
boundary-condition matching approach, combined with Einstein light-quantum
hypothesis, is proposed to analyze this thought experiment, and it is found for
the first time that only from Maxwell equations without resort to the
relativity, the correctness of light momentum definitions cannot be identified.
Optical pulling effect is studied as well.

Cite this paper

C. Wang, "Can the Abraham Light Momentum and Energy in a Medium Constitute a Lorentz Four-Vector?,"*Journal of Modern Physics*, Vol. 4 No. 8, 2013, pp. 1123-1132. doi: 10.4236/jmp.2013.48151.

C. Wang, "Can the Abraham Light Momentum and Energy in a Medium Constitute a Lorentz Four-Vector?,"

References

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[2] I. Brevik, Physics Reports, Vol. 52, 1979, pp. 133-201. doi:10.1016/0370-1573(79)90074-7

[3] M. Mansuripur, Optics Express, Vol. 12, 2004, pp. 5375-5401. doi:10.1364/OPEX.12.005375

[4] R. N. C. Pfeifer, T. A. Nieminen, N. R. Heckenberg and H. Rubinsztein-Dunlop, Reviews of Modern Physics, Vol. 79, 2007, pp. 1197-1216. doi:10.1103/RevModPhys.79.1197

[5] W. She, J. Yu and R. Feng, Physical Review Letters, Vol. 101, 2008, p. 243601. doi:10.1103/PhysRevLett.101.243601

[6] C. Baxter and R. Loudon, Journal of Modern Optics, Vol. 57, 2010, pp. 830-842. doi:10.1080/09500340.2010.487948

[7] S. M. Barnett, Physical Review Letters, Vol. 104, 2010, p. 070401. doi:10.1103/PhysRevLett.104.070401

[8] V. G. Veselago and V. V. Shchavlev, Physics-Uspekhi, Vol. 53, 2010, pp. 317-318. doi:10.3367/UFNe.0180.201003k.0331

[9] I. Brevik and S. A. Ellingsen, Physical Review A, Vol. 81, 2010, p. 011806. doi:10.1103/PhysRevA.81.011806

[10] M. Mansuripur, Optics Communications, Vol. 283, 2010, pp. 1997-2005. doi:10.1016/j.optcom.2010.01.010

[11] T. Ramos, G. F. Rubilar and Y. N. Obukhov, Physical Review A, Vol. 375, 2011, pp. 1703-1709. doi:10.1016/j.physleta.2011.03.015

[12] M. Mansuripur, Optics Communications, Vol. 284, 2011, pp. 594-602. doi:10.1016/j.optcom.2010.08.079

[13] B. A. Kemp, Journal of Applied Physics, Vol. 109, 2011, p. 111101. doi:10.1063/1.3582151

[14] I. Brevik and S. A. Ellingsen, Physical Review A, Vol. 86, 2012, p. 025801. doi:10.1103/PhysRevA.86.025801

[15] C. Wang, “Plane Wave in a Moving Medium and Resolution of the Abraham-Minkowski Debate by the Special Principle of Relativity,” arXiv:1106.1163. http://arxiv.org/abs/1106.1163

[16] A. Einstein, Annals of Physics, Vol. 20, 1906, pp. 627-633.

[17] A. P. French, “Special Relativity,” W. W. Norton & Co., New York, 1968, p. 16.

[18] According to the principle of relativity, all inertial frames are equivalent for descriptions of physical laws, and Maxwell equations have the same form in all inertial frames. Accordingly, the Abraham EM momentum density vector and energy density must have the same definitions in all inertial frames (although observed in the medium-rest frame the medium is stationary and the refractive index is isotropic while observed in the lab frame the medium is moving and the index is anisotropic). Consequently, the Abraham photon momentum and energy, given by Equation (A-3), must have the same form in all inertial frames. In addition, keep in mind (the basic mathematical result of Lorentz transformation) that the scalar product of any two of four-vectors is a Lorentz invariant. Thus if Equation (A-3) were a four-vector, then would be a Lorentz invariant because the Planck constant h must be a Lorentz invariant. But is a wave four-vector, and must be a Lorentz invariant. From this it follows that both and are Lorentz invariants; thus leading to an incorrect mathematical (physical) result: both the photon’s frequency and the medium refractive index are Lorentz invariants.

[19] A. Sfarti, Europhysics Letters, Vol. 84, 2008, p. 10001. doi:10.1209/0295-5075/84/10001

[20] U. Leonhardt, Nature, Vol. 444, 2006, pp. 823-824. doi:10.1038/444823a

[21] W. Rindler, “Relativity: Special, General, and Cosmological,” 2nd Edition, Oxford, 2006, p. 77.

[1] doi:10.1103/PhysRevA.8.14

[2] I. Brevik, Physics Reports, Vol. 52, 1979, pp. 133-201. doi:10.1016/0370-1573(79)90074-7

[3] M. Mansuripur, Optics Express, Vol. 12, 2004, pp. 5375-5401. doi:10.1364/OPEX.12.005375

[4] R. N. C. Pfeifer, T. A. Nieminen, N. R. Heckenberg and H. Rubinsztein-Dunlop, Reviews of Modern Physics, Vol. 79, 2007, pp. 1197-1216. doi:10.1103/RevModPhys.79.1197

[5] W. She, J. Yu and R. Feng, Physical Review Letters, Vol. 101, 2008, p. 243601. doi:10.1103/PhysRevLett.101.243601

[6] C. Baxter and R. Loudon, Journal of Modern Optics, Vol. 57, 2010, pp. 830-842. doi:10.1080/09500340.2010.487948

[7] S. M. Barnett, Physical Review Letters, Vol. 104, 2010, p. 070401. doi:10.1103/PhysRevLett.104.070401

[8] V. G. Veselago and V. V. Shchavlev, Physics-Uspekhi, Vol. 53, 2010, pp. 317-318. doi:10.3367/UFNe.0180.201003k.0331

[9] I. Brevik and S. A. Ellingsen, Physical Review A, Vol. 81, 2010, p. 011806. doi:10.1103/PhysRevA.81.011806

[10] M. Mansuripur, Optics Communications, Vol. 283, 2010, pp. 1997-2005. doi:10.1016/j.optcom.2010.01.010

[11] T. Ramos, G. F. Rubilar and Y. N. Obukhov, Physical Review A, Vol. 375, 2011, pp. 1703-1709. doi:10.1016/j.physleta.2011.03.015

[12] M. Mansuripur, Optics Communications, Vol. 284, 2011, pp. 594-602. doi:10.1016/j.optcom.2010.08.079

[13] B. A. Kemp, Journal of Applied Physics, Vol. 109, 2011, p. 111101. doi:10.1063/1.3582151

[14] I. Brevik and S. A. Ellingsen, Physical Review A, Vol. 86, 2012, p. 025801. doi:10.1103/PhysRevA.86.025801

[15] C. Wang, “Plane Wave in a Moving Medium and Resolution of the Abraham-Minkowski Debate by the Special Principle of Relativity,” arXiv:1106.1163. http://arxiv.org/abs/1106.1163

[16] A. Einstein, Annals of Physics, Vol. 20, 1906, pp. 627-633.

[17] A. P. French, “Special Relativity,” W. W. Norton & Co., New York, 1968, p. 16.

[18] According to the principle of relativity, all inertial frames are equivalent for descriptions of physical laws, and Maxwell equations have the same form in all inertial frames. Accordingly, the Abraham EM momentum density vector and energy density must have the same definitions in all inertial frames (although observed in the medium-rest frame the medium is stationary and the refractive index is isotropic while observed in the lab frame the medium is moving and the index is anisotropic). Consequently, the Abraham photon momentum and energy, given by Equation (A-3), must have the same form in all inertial frames. In addition, keep in mind (the basic mathematical result of Lorentz transformation) that the scalar product of any two of four-vectors is a Lorentz invariant. Thus if Equation (A-3) were a four-vector, then would be a Lorentz invariant because the Planck constant h must be a Lorentz invariant. But is a wave four-vector, and must be a Lorentz invariant. From this it follows that both and are Lorentz invariants; thus leading to an incorrect mathematical (physical) result: both the photon’s frequency and the medium refractive index are Lorentz invariants.

[19] A. Sfarti, Europhysics Letters, Vol. 84, 2008, p. 10001. doi:10.1209/0295-5075/84/10001

[20] U. Leonhardt, Nature, Vol. 444, 2006, pp. 823-824. doi:10.1038/444823a

[21] W. Rindler, “Relativity: Special, General, and Cosmological,” 2nd Edition, Oxford, 2006, p. 77.