Electromagnetic energy-momentum tensors in media

📝 Abstract

It is pointed out that the previous energy-momentum tensors of Minkowski and Abraham for the electromagnetic field in continuous media are based on a covariant formulation which does not reflect a symmetry inherent to the system. Instead, taking into account the intrinsic invariance under Lorentz transformations involving the reduced speed of light in such a medium, a compact and fully consistent theory can be formulated without the old problems.

💡 Analysis

It is pointed out that the previous energy-momentum tensors of Minkowski and Abraham for the electromagnetic field in continuous media are based on a covariant formulation which does not reflect a symmetry inherent to the system. Instead, taking into account the intrinsic invariance under Lorentz transformations involving the reduced speed of light in such a medium, a compact and fully consistent theory can be formulated without the old problems.

📄 Content

arXiv:0805.2606v2 [hep-ph] 30 Sep 2008 Electromagnetic energy-momentum tensors in media Finn Ravndal1 Department of Physics, University of Miami, Coral Gables, FL 33124. Abstract It is pointed out that the previous energy-momentum tensors of Minkowski and Abraham for the electromagnetic field in continuous media are based on a covariant formulation which does not reflect a symmetry inherent to the system. Instead, taking into account the intrinsic invariance under Lorentz transformations involving the reduced speed of light in such a medium, a com- pact and fully consistent theory can be formulated without the old problems. 1 Introduction After close to a hundred years, there are still two possible energy-momentum tensors for the electromagnetic field in continuous media being discussed in the literature[1]. The first was derived by Minkowski[2] and the second proposed shortly afterwards by Abaham[3]. Both of them have problems, in particular when they are used in a quantum context. In most textbooks these difficulties are only hinted upon. The book by Panofsky and Phillips[4] endorses the Abraham energy-momentum tensor. In the latest edition of the book by Jackson[5] one is of the same view although one is open for the possibility that there might be an additional co-traveling momentum from the mechanical momentum of the electrons which could add up to the Minkowski electromagnetic momentum density. The same support for the Abraham version is also presented in the book by Landau and Lifshitz[6], based on the usual requirement of relativistic invariance. A clear presentation of the original ideas behind the theories can be found in the book by Møller[7]. Both of them are covariantly formulated, based on Lorentz trans- formations in the vacuum. But this apparent Lorentz invariance is not an intrinsic symmetry of the medium where light moves with a reduced velocity. Taking into ac- count this physical fact, an effective theory has recently been proposed where these problems are avoided[8]. The free theory can be quantized by standard methods and extended with higher-order interaction terms to also describe dispersive and 1On sabbatical leave of absence from Institute of Physics, University of Oslo, N-0316 Oslo, Norway. 1 Kerr effects. It thus becomes a full-fledged effective field theory for electromagnetic phenomena in media and relates many classical and quantum effects in a systematic way. When light moves from the vacuum into an isotropic and transparent medium, it is in general refracted. Its frequency ν remains the same, but the wavelength λ is reduced by the refractive index n > 1. The corresponding phase velocity νλ is therefore lowered to 1/n when we set the light velocity in vacuum to c = 1. But it should not be forgotten that this is an effective description valid on large scales where the discrete atoms in the material can be replaced by a continuous medium. On the atomic scale light is moving with the vacuum velocity c = 1 between interactions with electrons around the atoms. These will scatter the light in such a way that in the forward direction the scattered waves add up to a plane wave. However, it is delayed by a phase shift of π/2 relative to the incoming wave as for instance explained by Feynman[9]. The interference between these two waves will then effectively slow down the propagating wave. As a result of these microscopic processes, the resulting wave is a highly complex object. In spite of that, experience shows that we can describe it by local electromagnetic fields obeying the standard Maxwell’s equations for continuous media. But these are now effective fields, incorporating complicated physics on very short scales. For this reason one would think that we today have a satisfactory and consistent theory for electromagnetic phenomena in media. And to a large extent that is cer- tainly true. But the energetics of these processes are still unclear, caused by the uncertainty about the energy-momentum tensor. The Minkowski tensor is not sym- metric and therefore has problems with angular-momentum conservation[2]. For this reason Abraham proposed a symmetric tensor at the cost of having to intro- duce a new, electromagnetic volume force in the medium[3]. Its existence has still not been verified experimentally. The new tensor is not conserved and results in a momentum density of the field smaller than the Minkowski value by a factor n2. Since then several other energy-momentum tensors have been suggested. Experi- mentally, the different proposals are most directly tested by their predictions for the radiation pressure[10]. This seems to favor the original construction of Minkowski. The different theories and experiments were reviewed in detail some time ago by Brevik[11]. A more recent survey of the whole situation can be found in the more phenomenological approach by Garrison and Chiao[12]. We will in the following sum up the essence of the Minkowski and Abraham theories and comment in more detail upon thei

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