A new relativistically covariant approach is discussed for the derivation of local conservation theorems for homogeneous anisotropic and, in particular, dispersive media. We start from a three-dimensional operator equation for the electric field and obtain mainly by coordinate-invariant methods the results basically expressed by the slowly varying amplitudes of the electric field. Apart from local energy and momentum conservation formulated by the energy-momentum conservation, we find a local conservation theorem for the action which is more general and which is the only one which remains also true for inhomogeneous media.
KeywordsSpatial and Frequency DispersionPermittivity TensorGroup VelocityCold PlasmaCoordinate-Invariant MethodsLorentz GroupLorentz Invariants
Einstein, A. (1913) Zeitschrift für Angewandte Mathematik und Physik, 62, 225-261.
Tamm, I.E., Smorodinsky, J.A. and Kuznetsov, B.G. (1965) Albert Einstein, Sobranyje nauchnykh truduv, I (Collection of Scientific Works, Part I). Nauka, Moskva.
Landau, L.D. and Lifshitz, E.M. (1962) The Classical Theory of Fields (4th Revised Edition). Nauka, Moskva.
Rashevski, P.K. (1964) Riemann Geometry and Tensor Analysis. 2nd Edition, Nauka, Moskva. (In Russian)
Pauli, W. (1921) Relativitatstheorie. Enzyklopadie der mathematischen Wissenschaften, Band V, Heft IV, Art. 19, B.G. Teubner, Leipzig.
Lorentz, H.A., Weyl, H. and Minkowski, H. (1922) Das Relativitatsprinzip, B.G. Teubner, Leipzig. https://doi.org/10.1007/978-3-663-16170-7
Landau, L.D. and Lifshitz, E.M. (1982) Electrodynamics of Continuous Media. 2nd Edition, Nauka, Moskva.
Tamm, I.Ye. (1976) Osnovy teorii elektrichestva (Foundations of the Theory of Elctricity). 9th Edition, Nauka, Moskva.
Møller, C. (1972) The Theory of Relativity. Clarendon Press, Oxford.
Jackson, J.D. (1962) Classical Electrodynamics. John Wiley, New York. https://doi.org/10.1063/1.3057859
Felsen, L.B. and Marcuvitz, N. (1973) Radiation and Scattering of Waves. Prentice-Hall, Englewood Cleffs.
Yariv, A. and Yeh, P. (1984) Optical Waves in Crystals. Wiley-Interscience, New York.
Melrose, D.B. and McPhedran, R.C. (1991) Electromagnetic Processes in Dispersive Media. Cambridge University Press, Cambridge. https://doi.org/10.1017/CBO9780511600036
Jimenez, J.L. and Campos, I. (1995) The Balance Equations of Energy and Momentum in Classical Electrodynamics. In: Barrett, T.W. and Grimes, D.M., Eds., Advanced Electromagnetism, World Scientific, Singapore, 464-495. https://doi.org/10.1142/9789812831323_0016
Wünsche, A. (1970) Annals of Physics (Leipzig), 25, 201-214. https://doi.org/10.1002/andp.19704800207
Wünsche, A. (1978) Annals of Physics (Leipzig), 35, 303-320. https://doi.org/10.1002/andp.19784900407
Leonhardt, U. (2000) Physical Review A, 62, Article ID: 012111. https://doi.org/10.1103/PhysRevA.62.012111
Leonhardt, U. and Piwnicki, P. (2001) Journal of Modern Optics, 48, 977-988. https://doi.org/10.1080/09500340108230969
Wünsche, A. (2005) Acta Physica Hungarica B, 23, 151-162. https://doi.org/10.1556/APH.23.2005.3-4.5
Lorentz, H.A. (1904) van Wetenschappen te Amsterdam, 6, 809.
Tyapkin, A.A. (1973) The Principle of Relativity (with a Preface of D.I. Blokhintsev). Atomisdat, Moscow.15
Lorentz, H.A. (1904) Weiterbildung der Maxwellschen Theorie. Elektronentheorie. In: Enzyklopadie der mathematischen Wissenschaften, Bd. 2, Teubner, Leipzig, Art. 14, 245.
Lorentz, H.A. (1915) Theory of Electrons. 2nd Edition, Teubner, Leipzig.
de Groot, S.R. and Suttorp, L.G. (1972) Foundations of Electrodynamics. North Holland Publ. Comp., Amsterdam.
Bloembergen, N. (1965) Nonlinear Optics. W.A. Benjamin, Inc., New York.
Wünsche, A. (1971) Monatsberichte der Deutschen Akademie der Wissenschaften zu Berlin, 13, 479-492.
Hebenstreit, H. and Suchy, K. (1979) Zeitschrift für Naturforschung A, 34, 1147-1157. https://doi.org/10.1515/zna-1979-1001
Minkowski, H. (1909) Physikalische Zeitschrift, 10, 104-111.
Lorentz, H.A., Einstein, A., Minkowski, H. and Weyl, H. (1952) The Principle of Relativity. Dover, Mineola.
Landau, L.D. and Lifshitz, E.M. (1988) The Classical Theory of Fields. 7nd Edition, Nauka, Moskva.
Szivessy, G. (1928) Kristalloptik. In: Geiger, H. and Scheel, K., Eds., Handbuch der Physik, Band XX, Springer, Berlin, 635-904. https://doi.org/10.1007/978-3-642-90780-7_11
Born, M. and Wolf, E. (1999) Principles of Optics. Seventh Edition, Cambridge University Press, Cambridge.
Ginzburg, V.L. (1974) Kak i kto sozdal teoriyu otnocitel’nosti? (How and Who Created Relativity Theory?) In: Einshteynovski sbornik (Einsteinian Anthology), Nauka, Moscow, 351-384.