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1、arXiv:0708.1153v1 gr-qc 8 Aug 2007 Electrodynamics of moving magnetoelectric media: variational approach Yuri N. Obukhov Institute for Theoretical Physics, University of Cologne, 50923 K oln, Germany and Department of Theoretical Physics, Moscow State University, 117234 Moscow, Russia Friedrich W. H
2、ehl Institute for Theoretical Physics, University of Cologne, 50923 K oln, Germany and Dept. of Phys. Astron., University of Missouri-Columbia, Columbia, MO 65211, USA Abstract Recently, Feigel has predicted a new eff ect in magnetoelectric media. The theoretical evaluation of this eff ect requires
3、a careful analysis of a dynamics of the moving magnetoelectric medium and, in particular, the derivation of the energy-momentum of the electromagnetic fi eld in such a medium. Then, one can proceed with the study of the wave propagation in this medium and derive the mechanical quantities such as the
4、 energy, the momentum, and their fl uxes and the corresponding forces. In this paper, we develop a consistent general-relativistic variational approach to the moving dielectric and magnetic medium with and without magnetoelectric properties. The old experiments in which the light pressure was measur
5、ed in fl uids are reanalysed in our new framework. PACS numbers: 03.50.De, 04.20.Fy, 71.15.Rf Keywords: Electrodynamics, magnetoelectric medium, general relativity, variational principle, Feigel eff ect Electronic address: yohtp.uni-koeln.de Electronic address: hehlthp.uni-koeln.de 1 I.INTRODUCTION
6、The discussion of the electrodynamics of moving media has a long history. At present, the general structure of classical electrodynamics appears to be well established. In particular, in the generally covariant pre-metric approach to electrodynamics 1, 2, 3, 4, 5, 6, the electric charge and the magn
7、etic fl ux conservation laws manifest themselves in the Maxwell equations for the excitation H = (D,H) and the fi eld strength F = (E,B), namely dH = J, dF = 0. These equations should be supplemented by a constitutive law H = H(F). The latter rela- tion contains the crucial information about the und
8、erlying physical continuum (i.e., space- time and/or material medium), in particular, about the spacetime metric. Mathematically, this constitutive law arises either from a suitable phenomenological theory of a medium or from the electromagnetic fi eld Lagrangian. It can be a nonlinear or even nonlo
9、cal relation between the electromagnetic excitation and the fi eld strength. The constitutive law is called a spacetime relation if it applies to spacetime (“the vacuum”) itself. Among many physical applications of classical electrodynamics, the problem of the inter- action of the electromagnetic fi
10、 eld with matter occupies a central position. The fundamental question, which arises in this context, is about the defi nition of the energy and momentum in the possibly moving medium. The discussion of the energy-momentum tensor in macro- scopic electrodynamics is quite old. The beginning of this d
11、ispute dates back to Minkowski 7, Einstein and Laub 8, and Abraham 9. Nevertheless, up to now the question was not settled and there is an on-going exchange of confl icting opinions concerning the validity of the Minkowski versus the Abraham energy-momentum tensor, see, e.g., the review 10. Even exp
12、eriments were not quite able to make a defi nite and decisive choice of electromag- netic energy and momentum in material media. A consistent solution of this problem has been proposed in 4, 11 (cf. also the earlier work 12) in the context of a new axiomatic approach to electrodynamics. Recently Fei
13、gel 13 has studied, theoretically, the dynamics of a dielectric magnetoelectric medium in an external electromagnetic fi eld. He predicted that the contributions of the quantum vacuum waves (or “virtual photons”) could transfer a nontrivial momentum to matter. This prediction was made with the help
14、of the non-relativistic formalism. In our opinion, a proper relativistic analysis is needed for a better understanding of the physics and of the viability of this phenomenon. Here we begin to reconsider this problem in a 2 covariant framework as developed earlier in 4, 11. As a fi rst step, we devel
15、op a variational approach to the description of the dynamics of a moving magnetoelectric medium. The corresponding energy-momentum of matter plus electromagnetic fi eld that arises can be derived straightforwardly in this formalism from the variation of the total action with respect to the spacetime
16、 metric. II.PRELIMINARIES: THE ESSENCE OF THE FEIGEL EFFECT The Feigel eff ect 13 can be described in simple terms as follows: Let us consider an isotropic homogeneous medium with the electric and magnetic constants ,. Electromag- netic waves are propagating in such a medium absolutely symmetrically
17、, with the Fresnel equation describing the unique light cone. This is easily derived from the constitutive rela- tions D = 0E and H = (0)1B. However, if a medium is placed in crossed constant external electric and magnetic fi elds, then it acquires magnetoelectric properties.As a result, we have the
18、 anisotropic magnetoelectric medium with , plus the magnetoelectric matrix (determined by the external fi elds) which modifi es the constitutive relations to D = 0E + B and H = (0)1B T E; here T denotes the transposed matrix. Accordingly, the wave propagation in such a medium also becomes anisotropi
19、c and bi- refringent, with the wave covectors now belonging to two light cones.Applying this to vacuum waves (or, perhaps, better to say to the “vacuum fl uctuations” or “virtual photons”) propagating in the magnetoelectric body, Feigel 13 computed the total momentum carried by these waves and concl
20、uded that it is nontrivial. In accordance with this derivation, a body should move with a small but non-negligeable velocity. Earlier the Feigel process was discussed in 14, 15, 16, 17, 18. In order to evaluate the possible Feigel eff ect, it is necessary to substitute the “vacuum waves” into the en
21、ergy-momentum tensor. This paper is devoted to the derivation of the latter in the framework of a variational approach. 3 III.CONSTITUTIVE RELATION Within the axiomatics of the premetric generally covariant framework 4, the projection technique is used to defi ne the electric and magnetic phenomena
22、in an arbitrarily moving medium. As in 4, we assume that the spacetime is foliated into spatial slices with time and transverse vector fi eld n. When applying the projection technique to the 2-forms of the electromagnetic excitation H and and the electromagnetic fi eld strength F, we obtain the thre
23、e-dimensional objects: the magnetic and electric excitations H and D as longitudinal and transversal parts of H and, similarly, electric and magnetic fi elds E and B as longitudinal and transversal parts of F, respectively, namely H = H d + DandF = E d + B.(3.1) This foliation is called the laborato
24、ry foliation, with the coordinate time variable labeling the slices of this foliation. The spacetime metric g introduces the scalar product in the tangent space and defi nes the line element. With respect to the laboratory foliation coframe it reads (a,b,. = 1,2,3) ds2= N2d2+ gabdxadxb= N2d2 (3)gab
25、dxadxb.(3.2) Here N2 = g(n,n) is the length square of the foliation vector fi eld n, and dxa= dxanad is the transversal 3-covector basis, in accordance with the defi nitions above. The 3-metric (3)gab is the positive defi nite Riemannian metric on the spatial 3-dimensional slices corresponding to fi
26、 xed values of the time . This metric defi nes the 3-dimensional Hodge duality operator . The constitutive relation which links the electromagnetic fi eld strength to the electro- magnetic excitation, H = H(F), can be nonlocal and nonlinear, in general. Here we will confi ne our attention to the loc
27、al and linear constitutive relation. Then, if we write the the excitation 2-form in terms of its components in a local coordinate system xi, (H,D) = H = Hijdxi dxj/2 (with i,j, = 0,1,2,3), the local and linear constitutive relation means that the components of the excitation are local linear functio
28、ns of the components of the fi eld strength (E,B) = F = Fijdxi dxj/2: H = (F),Hij= 1 2 ijklFkl.(3.3) 4 Along with the original constitutive -tensor, it is convenient to introduce an alternative representation of the constitutive tensor: ijkl:= 1 2 ijmnmnkl.(3.4) Performing a (1+3)-decomposition of c
29、ovariant electrodynamics, as described above, we can write H and F as column 6-vectors with the components built from the magnetic and electric excitation 3-vectors Ha,Da and the electric and magnetic fi eld strengths Ea,Ba, respectively. Then the linear spacetime relation (3.3) reads: Ha Da = CbaBb
30、a AbaDba Eb Bb . (3.5) Here the constitutive tensor is conveniently represented by the 6 6-matrix IK= CbaBba AbaDba , IK= BabDab CabAab . (3.6) Assuming that the skewon and the axion pieces are absent, we fi nd that the constitutive matrices satisfy Aab= Aba, Bab= Bba, and D a b = Cab, with Caa= 0.
31、The dynamics of a material medium is encoded in the structure of another foliation (,u) which is determined by the four-vector fi eld of the velocity u of matter and the proper time coordinate . Accordingly, we have to formulate the constitutive law with respect to this, so called material foliation
32、. As a fi rst step, we observe that the relation between the two coframe bases, namely those of the laboratory foliation (d,dxa) and of the material foliation (d,dx f a) is as follows 4: d dxa = c/N vb/(cN) vaa b d dx f b . (3.7) Here, for the relative velocity 3-vector, we introduced the notation v
33、a:= c N ? ua u() na ? ,with := 1 q 1 v2 c2 .(3.8) Substituting (3.7) into (3.2), we fi nd for the line element in terms of the new variables ds2= c2d2 b gabdx f a dx f b, whereb gab= (3)gab 1 c2 vavb.(3.9) 5 The metric b gabof the material foliation has the inverse b gab= (3)gab + 2 c2 vavb,(3.10) w
34、here (3)gab is the inverse of (3)gab . For the determinant one fi nds (detb gab) = (det (3)gab)2. We will denote the 3-dimensional Hodge star defi ned by the metric b gabas b . Given the transformation (3.7) between the 1-form bases of the two foliations (laboratory and material), it is straightforw
35、ard to calculate the components of the constitutive tensor with respect to the moving reference frame from the constitutive tensor which describes the matter at rest. The corresponding explicit general transformation formulas for the 3- dimensional matrices A,B,C can be found in 4. A nontrivial outc
36、ome of this formalism is that the isotropic moving medium is described by the constitutive relation based on the so- called optical metric (fi rst introduced by Gordon 19). We will use this fact in the subsequent derivations. IV.RELATIVISTIC FLUID The earlier work includes that for the ideal fl uids
37、 20, 21, 22 and the case of the fl uids in electromagnetic fi elds was discussed in 23, 24, 25, 26. Recently, a new study was reported in 27, 28. An ideal fl uid which consists of structure-less elements (particles) is characterized in the Eulerian approach by the fl uid 4-velocity ui, the internal
38、energy density , the particle density , the entropy density s, and the identity (Lin) coordinate X.Normally, it is assumed that the motion of a fl uid is such that the number of particles is constant and that the entropy and the identity of the elements is conserved. In other words, i(ui) = 0,(4.1)
39、uiis = 0,(4.2) uiiX = 0.(4.3) By the conservation of the entropy only the reversible processes are allowed. In a variational approach to continuous media, these assumptions are considered as constraints imposed on the dynamics of the fl uid by means of Lagrange multipliers. Then the fl uid Lagrangia
40、n 6 20, 21 reads Vmat= Lmat g d b with (b = spatial volume 3-form) Lmat= (,s) c + 0(uiui c2) + 1i(ui) + 2uiis + 3uiiX.(4.4) The Lagrange multipliers 1,2,3impose the constraints (4.1)-(4.3) on the dynamics of the fl uid, whereas 0provides the standard normalization condition for the 4-velocity gijuiu
41、j= c2.(4.5) For the description of the thermodynamical properties of the fl uid, the usual thermodynam- ical law (“Gibbs relation”) is used, T ds = d(/) + pd(1/),(4.6) where T is the temperature and p the pressure. When the medium possesses dielectric and magnetic properties, then we have to add the
42、 Lagrangian of the electromagnetic fi eld Vem= 1 2 H F = 1 2 d (H B D E) = 1 2 d b ?AabE aEb+ BabBaBb 2CabEaBb ? .(4.7) At fi rst, let us consider the moving isotropic dielectric fl uid without magnetoelectric properties, that is, Cab= 0. Then the electromagnetic fi eld Lagrangian reads Vem= Lem g d
43、 b with Lem= 0 4 gij optg kl optFikFjl. (4.8) Here 0is the vacuum admittance, and gij opt= g ij 1 n2 c2 uiuj(4.9) is the optical metric of Gordon 19. A. Equations of motion without a magnetoelectric eff ect At fi rst, we consider the case when the fl uid does not have magnetoelectric properties: Cab
44、 = 0. We also assume that there are no free charges and currents in the fl uid. The dynamics of the system (fl uid+fi eld) is thus described by the total Lagrangian V = Vmat+ Vem. The corresponding equations of motion are then derived from the variational 7 principle for V with the independent varia
45、bles ui,s,X,0,1,2,3, and Ai(the covector of the electromagnetic potential). Variation with respect to the Lagrange multipliers 0,1,2,3yields the constraints (4.1)-(4.3) and (4.5), whereas variation of V with respect to ,s,X yields, respectively, uii1+ + p c = 0,(4.10) i ? 2ui ? + T c = 0,(4.11) i ?
46、3ui ? = 0.(4.12) In order to derive these results, we used the thermodynamical law (4.6), from which one has /s = T and / = ( + p)/. Finally, the variation of V with respect to the 4-velocity yields 20ui i1+ 2is + 3iX + 0(1 n2) c2 FikF jkuj = 0.(4.13) The last term emerges from Vem. Contracting the
47、last equation with uiand making use of (4.1)-(4.3), (4.5), and (4.10), we fi nd 20= 1 c2 ? + p c + 0(1 n2) c2 FikukF ilul ? .(4.14) The variation with respect to Ai gives the Maxwell equations for the moving fl uid j ? gjk optg il optFkl ? = 0.(4.15) When the fl uid has free currents and charges, we
48、 have to add the interaction term AiJito the electromagnetic fi eld Lagrangian Lem, and then the right-hand side of (4.15) picks up a nontrivial current Ji. B.Energy-momentum tensor The above consideration is not only relativistically covariant, but actually generally co- variant. The metric gijabov
49、e describes an arbitrary curved spacetime. The invariance of the total Lagrangian L under general coordinate transformations (using the standard Noether machinery) yields the conservation of the energy-momentum of the system (fl uid+fi eld). We obtain the energy-momentum tensor as usual from the variation of the total Lagrangian with 8 respect to the metric, i.e., Tij:= 2c g(Lmat+ Lem) ggij.(4.16) The computation is straightforward, and it yields for (4.4) and (4.8) Tij= j i p + p + c2 uiuj+ 1 n2 0c4 uiujFmkukF mlul + 1