Relativistic-Covariant Energy-Momentum Tensor for Homogeneous Anisotropic Dispersive Media

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.


Introduction
The four-dimensional energy-momentum tensor was introduced 1913 by Eins-How to cite this paper: Wünsche sults or rederived older results. In [32] [33] were given new arguments in favor of the Abraham tensor.
Apart from Abraham or Minkowski tensor there exist yet other serious problems with the energy-momentum tensor. As it is well known, the energy-momentum tensor is not uniquely determined by the requirement to satisfy a differential conservation theorem, e.g., [3] [20]. Practically, one has to do in every case of a calculated energy-momentum tensor with a whole class of equivalent energy-momentum tensors. One main use of this non-uniqueness is the possibility to remove the highly oscillatory terms in the approximation to waves with slowly varying amplitudes (beams) if we first insert the whole electromagnetic field with a positive and corresponding negative peak in the frequency distribution. We discuss this in Section 14 but mention already here that the energy-momentum tensor for media (only vacuum excluded) is basically non-symmetric and that by using the non-uniqueness it cannot be reduced to a symmetric one. Another difficulty for dispersive media is that their energy-momentum tensor cannot be derived starting from a Lagrange function as consequence of translation invariance of the medium in space and time (Noether theorem) as it is standard for electrodynamics of the vacuum. For inhomogeneous media the energy-momentum tensor does not exist at all in a local conservation law.
In present article we consider first the equations of macroscopic electrodynamics as averaged from microscopic electrodynamics (Section 2) then the constitutive equations in the concept of media with spatial and frequency dispersion and the symmetry of the permittivity tensor for neglect of dissipation (Section 3) and its symmetry under presence of discrete symmetries (space inversion, time inversion and their product, Section 4). Then we derive a three-dimensional operator equation for the electric field (Section 5) which is relativistically covariant but on the first glance it may seem to be paradoxical that this is possible. From this operator equation we derive a local conservation theorem for action (Section 6) and for energy and momentum (Section 7) and discuss the obtained energy-momentum tensor (Section 8). A peculiarity of our approach is that we obtain basically all results expressed by the electric field alone and after limiting transition to plane monochromatic waves we make the transition to more usual representation by the electric and magnetic field (Section 9). Then we consider the role which the group velocity plays (Section 10). After this we discuss the neglect of dispersion and the calculation of the group velocity in this case (Section 11). Next, we consider the special case of a cold plasma (Section 12). In the discussion of controversial opinions to local conservation of angular momentum we show that a complete symmetry of the four-dimensional energy-momentum tensor is not necessary but only symmetry of the stress tensor (Section 13). The non-uniqueness of the energy-momentum tensor is considered under new aspects (Section 14). Connected with the general non-symmetry of the energy-momentum tensor arise some difficulties for the General relativity theory (Section 15) and, finally, we mention some possibilities for generalizations of the discussed material (Section 16). In two Appendices we have separated the deri-

Maxwell Equations of Macroscopic Electrodynamics in Two Concepts
The basis of our derivations is the following Maxwell equations of macroscopic electrodynamics  [36]; see also [37] [38]) in the sense of the transition from microscopic to macroscopic electrodynamics ( [7], de Groot and Suttorp [39], (II. sec- This transition can include different averaging processes, for example, spatial, temporal and statistical ones (denoted by overlining of the corresponding quantity). The averaged microscopic current density micro j and charge density is taken into account. Such an identification is possible in almost all cases with exception of some static cases (e.g., electrostatics, stationary currents, magnetostatics) which have to be considered in this concept as limiting cases. As usual, we define the "electric induction" 3 , . t t t π ≡ + D r E r P r (2.5) From vectorial equations in (2.1) follows then by forming the divergence 2 Lorentz denotes microscopic fields with small letters corresponding to the Capital letters commonly used in macroscopic electrodynamics (but d instead of e for microscopic electric field). Most authors use micro H instead of micro B but since there is no difference between them in microscopic theory this is only of some didactic importance. (2.6) that are the scalar equations in (2.1) differentiated with respect to the time. Therefore with exclusion of the static limiting case, the scalar equations from (2.1) in according to, e.g., de Groot and Suttorp [39] and Bloembergen [40] (2.9) which obey then the following field equations instead of (2.7) The field ( ) , t H r is mostly called magnetic field but it is not the averaged microscopic magnetic field ( ) micro ,t B r and therefore not the "genuine" magnetic field [7] (chap IV, section 29, after Eq. (29.8)) (in our treatment with equations (2.7) we have The Maxwell equations of macroscopic electrodynamics (2.1) or (2.7) form a closed system of equations only together with constitutive equations which depend on the kind of the considered medium. We discuss this now.
The general linear constitutive relation between the electric induction ( ) ,t D r and the electric field ( ) ,t E r for spatially and temporally homogeneous media is (summation convention over equal indices; ρ∇ is displacement operator of arguments of a function of ( ) ,t r to ( ) where the real-valued tensor function ( )ˆ, ij ij ε ε τ = ρ characterizes the material properties and where the integration is written as going over the whole space-time and restrictions of this integration (e.g., to prehistory, causality) are thought to be included by vanishing of this tensor function in certain regions. These restrictions, for example, to the prehistory of the field evolution lead to properties of analyticity and thus to relations between real and imaginary part of the Fourier transform of ( ) , ij ε τ ρ which are called Kramers-Kronig relations which we do not discuss here (e.g., [7]). Relation (3.1) means that the most general linear constitutive relations are also nonlocal in space that describes the spatial dispersion 4 . The homogeneity of the medium is expressed by the property that ( ) according to [9] [10] the constitutive Equation (3.1) can be represented by According to Silin and Rukhadse [11] (p.14) the notion "spatial dispersion" was introduced by Gertsenshteyn. Clearly, the name "spatial dispersion" is not analogous to "frequency dispersion" which then has to be better named "temporal dispersion" or vice versa the "spatial dispersion" then "wave-vector dispersion".
The dispersion of the medium is here expressed by the dependence of the permittivity tensor ( ) As a consequence, the permittivity tensor ij ij Local or differential conservation laws of energy and momentum can only be derived under the condition that the medium is lossless which means that it does not have any dissipation or accumulation or transmission of energy and momentum to other frequencies and wave vectors. As the later derivations show, the condition for this is the following symmetry which after Fourier transformation according to (3.2) and in connection with (3.7) takes on the following form (3.9) and which for dispersive media can only be satisfied approximately for certain regions of wave vector and frequency. Such kind of conditions are closely related to Onsager conditions for quasi-stationary processes [7] (section 21, Ed. 1982) but instead of a rigorous derivation from basic principles we prefer again that one can conclude this from the necessary conditions for the most general possibility of derivation of differential conservation laws of energy and momentum. These conditions should not be confused with the influence of point group symmetries on the medium properties which additionally may be present or may not. The symmetry conditions which are related to different inversion symmetries are discussed in next Section. To treat spatial dispersion it is mostly appropriate to make a Taylor-series expansion of ( ) that means for the expansion (3.10) The approximate transition from one to the inverse tensor in (3.10) is easily to make.
We mention here shortly that in the most common treatment of macroscopic electrodynamics with two constitutive equations This shows that a possible magnetization appears here as effect of spatial dispersion of second order in wave-vector k that is important for the energymomentum tensor and also for the boundary conditions at such medium. Furthermore, we see that the tensor ( ) ijkl α ω in (3.10) is more general and usually contains more non-vanishing terms than this special tensor proportional to k l k k in (3.12) 5 . Only for magnetostatics this concept not used in present article is less appropriate.

Additional Restrictions of Tensor
for Discrete

Symmetries of Spatial and Time Inversion
We now consider the most simple discrete symmetries of order 2. 1) Spatial inversion (presence of symmetry center) The presence of spatial inversion that means invariance of the medium with respect to the transformation → − r r of the coordinates where due to the homogeneity the chosen coordinate origin is arbitrary and due to the property of E and D to be genuine vectors changing their sign under this transformation (in contrast, B is a pseudo-vector) leads to where in the second step (3.7) was used in addition. The first part of this condition means that for media with spatial inversion the components of the permittivity tensor ( ) , ij ε ω k are mutually independent from each other and are even functions of the wave vector k , whereas it does not mean a restriction for its dependence on the frequency ω . In composition with the condition (3.9) for absence of dissipation we have 5 For a deeper understanding we recommend here again the very instructive section 79 in [7] (Ed. 1982).
which relates different components of the permittivity tensor and is not true for regions of k and ω where dissipation is not negligible. If we deal with spatial dispersion by expansion of ( ) , ij ε ω k in powers of the wave vector k then we can use mainly the first part of these symmetry conditions which are true also in case of dissipation.
2) Time inversion (nonmagnetic symmetry classes) The presence of time inversion that means of invariance of the medium with respect to the transformation t t → − taking into account that E and D do not change their sign under this transformation (in contrast, B changes it) leads to where in addition (3.7) is used in last equality. The first part of this condition means that for media with spatial inversion the components of the permittivity tensor ( ) , ij ε ω k are mutually independent from each other even functions of the frequency ω , whereas it does not mean a restriction for its dependence on the wave vector k . In composition with the condition (3.9) for absence of dissipation we have here In expansions of the permittivity tensor ( , ) ij k ε ω in powers of k one may use here mainly the part for simplifications which, however, are true only under neglect of dissipation.
3) Product of spatial inversion with time inversion (nongyrotropic media) The presence of the product of spatial inversion with time inversion (including, evidently, the case of presence of both symmetry elements separately and therefore also of their product) leads to the symmetry where again the condition (3.7) is used in last equality. In composition with the condition for absent dissipation (3.9) we find The most interesting part of this relation sesses time inversion as symmetry element in addition to the spatial symmetry elements of one of the considered 32 crystal classes which are then called non-magnetic crystal classes and for which (4.3) is true. From the 32 crystal classes 11 do not possess a symmetry center and 11 possess it and therefore the spatial inversion as symmetry element and thus are non-gyrotropic and (4.1) and (4.5) are true for them in addition. There are 90 magnetic crystal classes from which 32 are trivial ones and correspond to the usual crystal classes but without time inversion as symmetry element.
Furthermore, there exist 58 magnetic crystal classes which contain time inversion not directly as symmetry element but in the form of the product of time inversion with the elements of a coset to an invariant subgroup of one of the 11 groups with only rotations [7] [41]. The 90 magnetic classes form the basis for the symmetry classification of ferromagnetics and anti-ferromagnetics. This concerns natural absence of time inversion as symmetry element but this absence can be generated also artificially under the influence of the medium by an external magnetic field from a primarily non-magnetic class. In the same way, among the 122 crystal classes (magnetic and non-magnetic ones) there are 32 classes with symmetry center for which (4.1) is true and 90 without symmetry center. This is contained in a compact form in Figure 1 copied from our paper [41].

Elimination of Magnetic Field and Three-Dimensional Operator Equation with Relativistic Covariance for the Electric Field
By differentiation of the second vectorial equation in (2.7) with respect to time and using the first vectorial equation, the magnetic field can be eliminated and using the constitutive Equation This equation for the electric field contains the full information about the electromagnetic field in the medium with exception of some static cases which have to be considered as limiting cases.
Since Equation (5.1) carries the full information about the electromagnetic field the conservation theorems may be derived from it that possesses considerable advantages, in particular, taking into account the dispersion as we will demonstrate this in the following. For such derivations, roughly speaking, we have to multiply this equation from the left with other electric fields where the tensor operator is defined in the following way ( ) The vectorial equation for the electric field (5.2) together with definition (5.3) forms a closed system of equations of macroscopic electrodynamics of homogeneous media and, moreover, is relativistic-covariant (contrary to (5.1)) and are appropriate for the derivation of the energy-momentum tensor. After Fourier transformation of the electric field according to (3.4) we obtain from (5.2) the equation for the Fourier components ( ) ,ω E k of the electric field and then the magnetic field with the tensor operator , , is the general susceptibility tensor. Since the tensor ( ) , ij χ ω k may be a complicated function of the wave-vector k and, in particular, of the frequency ω it is hardly possible to write down a Lagrange function for the system and the usual formalism of derivation of the energy-momentum tensor from such function is almost impossible. Equations (5.4) with operator (5.5) as transformed Equation (5.2) possess also a relativistically covariant form in three-dimensional orthogonal coordinates for arbitrary inertial systems ′ ε ω k by a transformation which we derive in detail in Appendix A. For an inertial system ′  moving with velocity V in the inertial system  according to the special Lorentz transformation (A.14) it with the following relations between ( ) ,ω ′ ′ k and ( ) ,ω k and their inversion by (5.9) and with relativistic invariant ε ω k in the resting system of the medium does not depend on the wave vector k . However, this dependence on the wave vector k in the system moving with velocity V which formally means spatial dispersion of the medium is of some other kind than the natural dependence of the permittivity of a medium on wave vector k in resting system and it is not reasonable to expand it in a Taylor series in k . For the formal derivation of the energy-momentum tensor these differences are not of importance.
The transformation formulae (5.8) for the permittivity tensor from one to another inertial system and for (5.9) simplify essentially in non-relativistic  if we neglect quadratic and higher terms in c V in comparison to linear terms in c V (e.g., 1 γ → ) that we do not write down.
From transformations (5.8) together with (5.9) we see that if we change at the same time the signs of k and ω this also changes at the same time the signs of ′ k and ω′ according to As expected this means that the condition (3.9) for the dissipation-free case transforms into a corresponding condition for the dissipation-free case in an arbitrary inertial system ′  moving with velocity V in inertial system  and is therefore invariant with respect to Lorentz transformations as one could have to expect for such a physical property. It is seen that the condition (3.9) for absent losses can be continued to the following condition for Therefore, if (5.13) is satisfied we can write down in addition to (5.2) the following equation for the electric field The two Equations (5.2) and (5.14) form the basis of our derivations of local conservation laws and it possesses a great advantage that we have only one field function for the electric field in these equations in comparison to the electric and magnetic field in the common derivations.
We now consider quasiplane and quasimonochromatic waves in the form is a slowly varying complex vectorial amplitude and 0 k a mean wave vector and 0 ω a mean frequency. We suppose that 0 k and 0 ω are real and exclude in such way, but only for simplicity, evanescent waves with complex values of these quantities which may exist even in lossless media (for example, waves under total reflection in the lossless optically thinner medium or surface waves). The inclusion of such waves would complicate the following considerations but does not destroy the existence of local conservation theorems. The approximations which we make in the following are that due to slowness of changing of the amplitudes in such way that we may take into account in expansions only a small number of spatial and temporal derivatives of these amplitudes. This means that the wave vectors and frequencies in the Fourier decomposition of the quasiplane and quasimonochromatic wave are concentrated around 0 k and 0 ω (and, clearly, around 0 −k and 0 ω − ) and the two complex conjugated parts in (5.15) are well separated. The supposition and at once approximation in the following is that we can deal with both parts as independent solutions of the wave equation for the electric field. This is apparently equivalent to some averaging procedure over terms with rapidly varying frequencies and wave vectors which then vanish from the equations such as made in [7] and is justified for quasiplane and quasimonochromatic waves. If we insert the first part from the right-hand side of (5.15) as independent solution into Equation (5.2) we obtain the following equation for the slowly varying complex amplitude where index "0" means that the corresponding derivatives have to be taken for In the following, we take the derivatives of the slowly varying amplitudes up to the second order but before this we introduce a shorter relativistic-covariant notation of the equations for Our derivation of the energy-momentum tensor is similar to the derivation of approximate equations for beam solutions with the only difference that in last case the determinant of ij L has to be taken as starting point for the expansion to get the equation for the main component of ( )

Local Action Conservation in Relativistic Covariant Form
The derivation of local (or differential) laws of action conservation and of other local conservation theorems becomes much more concise if we introduce for abbreviation the following four-dimensional notations of special theory of relativity 6 ( ) An advantage of the four-dimensional formalism is that we obtain the results in relativistic covariant form.
We may write the equations for the electric field (5.2) and (5.14) in the concise form 7 ) 6 Modern development mainly for preparing the transition to General relativity theory favors to use only representations by real components for space-time. This makes it necessary to distinguish between contravariant and covariant components of vectors and tensors but this becomes very inconvenient for our purposes. According to Pauli [5] (Part III, p. 71), the historically older notation ( )  which we do not want to change here into that of (6.1). However, by comparison with second line in (1) it seems that this does not cause problems.
and k substituted by i − ∇ . The quasiplane and quasimonochromatic waves (5.15) take on the shorter form In the same approximation as in (5.16) we obtain from the first of Equation and from the second equation We first derive a conservation theorem which is even more fundamental than the theorem for energy-momentum conservation since it may be extended to inhomogeneous media.
If we multiply (6.5) by ( ) The terms are explicitly written down up to first-order derivatives of the slowly varying amplitudes but the higher-order terms on the right-hand side can also be represented as 4-divergence of a 4-vector ( ) In three-dimensional separation according to the definition with the vector field of action flow density. From (6.10) and (6.9) we find in three-dimensional representation up to explicitly given first-order derivatives of the slowly varying amplitudes of the electric field which last take into account the diffraction of beams Before discussing these expressions we derive the local form of energymomentum conservation.

Local Energy and Momentum Conservation in Relativistic Covariant Form
In analogy to (6.7) we consider the following combination which can be represented as the 4-divergence of a second-rank 4-tensor Thus we obtained a local conservation theorem of the form The four-dimensional covariance of ( ) T r κλ with respect to index λ is the same as in the action 4-vector ( ) T r λ and the covariance with respect to index κ is evident from construction (7.1) with 4-wave vector k κ . That this is connected with homogeneity (or translation invariance) in space and time is easily seen since in case of absence of this symmetry it is impossible to have globally constant wave vectors and frequencies as used in the derivation. Thus we have the justification to call ( ) T r κλ the energy-momentum tensor of homogeneous anisotropic dispersive media in the approximation of quasiplane and quasimonochromatic waves. In general, the tensor and is, in general, not equivalent to a symmetric one that means it is intrinsically non-symmetric. We now transform the energy-momentum tensor ( ) T r κλ to another form which is interesting for the physical interpretation. For this purpose we use the identities Inserting this into (7.3) and using the representation (6.9) of ( ) T r λ and the Equations (6.5) and (6.6) for the slowly varying amplitudes we obtain up to first-order derivatives of these amplitudes In the limiting transition from the slowly varying amplitudes to constant ampli- of the energy-momentum tensor. This is in full analogy to a homogeneous particle flow as discussed in e.g. [3] [12] (see also [34] and below) where, however, macroscopic electrodynamics provides a greater variety of possible dependencies A. Wünsche of the momentum of one particle on the group velocity than classical mechanics. Taking seriously this analogy to a homogeneous particle flow this leads in a natural way to a quantization of the electrodynamic flow and to its interpretation as a flow of quasiparticles. The energy-momentum tensor ( ) T r κλ in higher approximations according to (7.6) does not fully factorize into the product ( ) 0, k T r κ λ and the remaining terms are important at such space-time points where the 4-gradient of the slowly varying amplitudes of the electric field components is important. This may be interpreted as the tendency that energy and momentum flow at these points are forced to choose deviating directions in comparison to the homogeneous particle flow and expresses some interaction of the particles within the flow or some (direction-dependent) pressure or stress. This is in rough agreement with the diffraction of beams, for example, of Gaussian beams which cannot remain to be focused over the whole length of the beam.

Three-Dimensional Representation of Energy-Momentum Tensor
We now make the transition to the three-dimensional separation of the terms in the local laws of momentum and of energy conservation. The 4-dimensional energy-momentum tensor can be separated into three-dimensional parts in the following way defining (in common sense) the introduced new quantities on the right-hand side Then from (7.2) we find the following differential law of momentum conservation is the (Maxwell) stress tensor and ( ) , k g t r the momentum density 8 . Furthermore, the following differential law of energy conservation holds According to (7.3) and (7.6) taking into account (8.1) the stress tensor possesses the form 8 The three-dimensional stress tensor kl T is sometimes defined with opposite sign. Our sign of kl T agrees with that in the same notation kl kl T T = in Landau and Lifshits [3] (Ed. 1962) and with kl kl T σ ≡ − in later editions (e.g., [45] from 1988). Apparently, the notation kl σ agrees also with respect to sign to the same notation in [13] and in [10].
and the momentum density is For the energy flow density we find from (7.3) and (7.6) taking into account The terms with spatial and temporal derivatives of the slowly varying amplitudes describe in addition to the stable form of propagation of a wave group its diffraction.
Integral forms of the conservation of momentum and energy in time follow from integration of the conservation theorems within a volume V with surface S and normal unit-vector N directed to the inside of the surface S by (Gauss

Limiting Transition to Plane Monochromatic Waves in Anisotropic Dispersive Media
In the limiting transition from quasiplane and quasimonochromatic waves to plane monochromatic waves the slowly varying amplitudes become constant amplitudes In the three-dimensional separation expressed by the formulae ( where the action flow density l T and the action density s according to (6.12) become The sign of  We will show in the following that these expressions are not in contradiction to known expressions for the energy-momentum tensor (mostly more special or otherwise formulated ones).
If we use the explicit form of ( ) , ij L ω k given in (5.5) we obtain from (9.2) (9.4) and from (9.3) for action flow density l T and action density s As already discussed, in the transition to the factorized form in (9.1) and (9.
c c c c Using these equations, we can transform (9.4) exactly to the following "mixed" forms of representation with the amplitudes of the electric and magnetic field 0 E and 0 B and the electric induction 0 D which dominate in their kind in literature (compare also [7] [9] [10]). 10 We emphasize again that this restriction to real wave vectors and frequencies is not a principal restriction for lossless media but simplifies our derivations considerably since it does not introduce additional difficulties with inhomogeneous (evanescent) waves in lossless media which necessarily are to be discussed without this restriction.
For the action flow density l T and the action density s, we find The appearance of 0 ω in the denominators for action flow density l T and action density s shows that they are formed in nonlocal way by the fields that in the space-time picture is impossible to express by quadratic local field combinations only and which, perhaps, is a reason that they did not find much attention (exception: similar considerations to adiabatic invariance). We see that all parts of the energy-momentum tensor in (9.7) contain a part with origin from the dispersion of the medium. The momentum density k g which possesses the direction of the mean wave vector 0,k k and the energy density w are modified by terms in ′  depends apart from transformed frequency ω′ also on transformed wave vector ′ k and appears there as medium with "unnatural" spatial dispersion (see Appendix A).
The trace of the energy-momentum tensor which is a relativistic invariant is non-vanishing taking into account the dispersion. From the limiting case of If we neglect dispersion the trace of the energy-momentum tensor becomes vanishing as it is seen from this expression. Due to factorization of the stress tensor which is a Lorentz-invariant and thus this relation is true in arbitrary inertial systems. Due to factorization (9.1) of the energy-momentum tensor in considered approximation we find T 0, w − = Sg (9.11) remaining true after Lorentz transformation.
The energy-momentum tensor (9.7) is intrinsically non-symmetric expressed by relation (7.4) also under neglect of dispersion. In general, for anisotropic media the momentum density k g and the energy flow density l S possess different directions and there is no way to remove this but also the stress tensor kl T is non-symmetric for anisotropic media. From the two old proposals for this tensor which are the Minkowski tensor and the Abraham tensor (see, e.g., [5] [10]) the tensor (9.2) is nearer to the Minkowski tensor and makes the transition to it in case of neglected dispersion. However, this problem of the correct tensor did not genuinely exist in our derivations since under the condition (3.9) that the medium is lossless the local form of the conservation laws could be formulated as exact vanishing of a 4-divergence of an energy-momentum tensor. We can subdivide the energy-momentum tensor T κλ in (9.7) in additive way into a pure electromagnetic field tensor ( ) Their explicit forms may be taken from (9.7). It should be emphasized that such a subdivision remains to be formally since each of the two parts does not separately obey a local conservation law.

Group Velocity in Energy-Momentum Tensor for Anisotropic Dispersive Media and Its Calculation
A wave packet in a homogeneous medium propagates in first approximation with shape stability and without diffraction with the group velocity and therefore energy and momentum of this wave packet should propagate also with the group velocity. The introduction of the group velocity into the energy-momentum tensor in the limiting case of plane monochromatic wave reveals a simple basic structure of this tensor (see also, [7] [9] [10]). Plane monochromatic waves with real wave vector and real frequency satisfy Equation The group velocity v is then defined by It is a "regular" velocity also in the relativistic theorem of addition of velocities. Inserting , , with arguments of involved functions of ( )   . Although not difficult to obtain, however, they are long taking into account the dispersion and, therefore, we will not write them down (we give them in next Section under neglect of dispersion). Instead of this we will use last part of (10.8) which reveals interesting relations to the action 4-vector and to the energy-momentum tensor. According to (10.8) where we used the representation (6.12) for action flow density T and action density s in the limiting case of plane monochromatic waves. The energymomentum tensor for this limiting case of plane monochromatic waves can now be represented in the form (see also next Section) 0, 0, 0, 0, 0, 0,  (10.14) which means that the three-dimensional stress tensor T which in considered case is a dyadic product is proportional to the momentum density g and the energy flow density S to the energy density w with the group velocity 0 v as the proportionality factor in analogy to (10.12).
As it is well known [3], the velocity 0 v is not spatial part of a relativistic covariant 4-vector but with following modification by the factor Using it the energy-momentum tensor (10.13) may be represented in the following relativistic covariant form where 0 s is the action density in the inertial system where the wave packet is resting. This is in analogy to a homogeneous particle flow in classical hydrodynamics without interaction of the particles (or without inner pressure) for which the energy-momentum tensor possesses the form is the momentum of one particle, 0 m its rest mass and 0 n the particle density in the inertial system where the particles rest (e.g., [12]).
The analogy of (10.17) to (10.18) for a homogeneous particle flow suggests (with knowledge of quantum theory) to interpret the first as homogeneous flow of quasiparticles and to introduce an abbreviation  according to as action of one particle independently of the considered inertial system (i.e., as a Lorentz invariant and, moreover, even as adiabatic Lorentz invariant as may be shown) and we may write with 0 m the rest mass of one particle. Usually, the relation between 4-vectors p and u in macroscopic electrodynamics is a 4-tensorial one with tensor components depending on components of v separately where this cannot be expressed by only relativistic scalars such as 0 m . A certain exception is formed by transverse waves in a cold isotropic plasma (Section 12). It was mentioned but not explicitly shown that the local action conservation (6.8) or (6.11) is a more general conservation law than the local energy-momentum conservation (7.2) or (8.2) together with (8.3) and holds also for inhomogeneous media (in general, spatially and temporarily inhomogeneous). If we suppose that the action conservation is true for an inhomogeneous medium that means it is informative to see how the energy-momentum conservation is lost for such a medium in case of propagation of almost plane monochromatic waves as here considered. We may assume that in a weakly inhomogeneous medium as main effect the 4-wave vector 0 k becomes dependent on the consi- The right-hand side is non-vanishing that corresponds to local non-conservation of energy-momentum and the 4-divergence of the energy-momentum tensor (if we overtake its formula from the homogeneous medium) becomes a linear combination of the components of the action vector

Neglect of Spatial and Frequency Dispersion and Group Velocity
In the considerations of Section 10 about the group velocity and the representation of the energy-momentum tensor in the limiting case of plane monochromatic waves in analogy to that for a homogeneous particle flow we did not use the explicit form of the determinant . In the following, we will make some explicit calculations of the group velocity under neglect of the dispersion (spatial and frequency one).
Neglecting spatial and temporal dispersion of the medium means that we consider as a constant permittivity tensor 0 ε in the inertial system of the resting medium or, at least, as a good approximation for a neighborhood of the considered mean wave vector 0 k and mean frequency 0 ω . Using the first of the relations for the group velocity in ( where we emphasize that the permittivity tensor 0 ε herein is, in general, not a symmetric tensor that includes gyrotropy of the medium (see also (4.5)). Furthermore, in general, the directions of k and v in anisotropic media are different. From (11.2) follows immediately for the scalar product of wave vector with group velocity , ω = kv (11.3) that proves to be equivalent to vanishing of the trace of the energy-momentum tensor under neglect of dispersion (see (9.9) in connection with (9.7) and (10.12)). According to (15) this also means that the frequency ω′ in the inertial system 0 ′ =   which moves with the group velocity v in the inertial system  of the resting medium (i.e. = V v ) vanishes and due to (A.17) that the wave vector is transformed in the following way (11.4) However, already the presence of frequency dispersion (and, moreover, of spatial dispersion) destroys these relations since we have then additional terms in the denominator of the right-hand side in (11.2) which contain the derivatives The derivations were made for real wave vector k and since S and g are proportional to k in this case the positivity of Sg is also understandable from this side.
In the inertial system ′  where the excitation rests that means which moves with group velocity v in 0  and thus where ′ → = v v 0 we have the energy momentum tensor  (12.18) and the excitation appears in ′  as a pure oscillation of the electric field in time with plasma frequency p ω which due to vanishing wave vector cannot be classified as transverse or longitudinal one but is its unification. The specialized formula (A.20) for the susceptibility in the system 0  which moves with group velocity 0 v of a certain excitation in  is relatively complicated. The transformation to this system makes only one considered wave to a resting excitation, whereas all other ones are not resting. Therefore, in the system corresponding to a rest energy of about ≈ 0.5 MeV. Since the appearance of a rest mass is a collective effect (quasi-particles), we cannot separate different parts of energy and momentum from the pure field and from the moving particles (electrons) on the background of the heavier ions considered as resting and making the medium (plasma) macroscopically neutral.
We mention that an energy momentum tensor of the form (12.17) with only one nonvanishing component 44 T in the energy density part for a point-like resting particle is the starting point for establishing the direct connection of Newton's gravitation law with Einstein's equations of general relativity (see, e.g., [3], section 99, Eq. (99.1)). A warm plasma with spatial inversion as symmetry element possesses a transversal and a longitudinal part of the permittivity tensor proportional to

Angular Momentum Conservation in Resting Isotropic Media
The most prominent supporter of the Abraham tensor in old time was Pauli [5] in his younger years. his death, to the re-edition of his encyclopedic article [5]. In Note 11 with reference to von Laue [14] Pauli praised emphatically the Minkowski tensor as the right one and it seems that he wants to correct his earlier opinion 13 . For anisotropic media which were never explicitly considered by Pauli in this regard it is clear that the energy-momentum tensor cannot be symmetric since the momentum density is in direction of the mean wave vector and the energy-flow density (Poynting vector) in direction of the ray vector which, in general, are not parallel to each other as it is well known from experimental and theoretical crystal optics.
In recent time the Abraham tensor was declared in papers of Leonhardt and coworkers [32] [33] as the correct one. It is easily to conjecture that the same as the young Pauli they want to have a symmetric energy-momentum tensor because the General Relativity theory requires such but they should ask themselves how it can be generalized as such symmetric tensor to general anisotropic media. As mentioned most authors favor the Minkowski tensor as the correct one also for its relativistic covariance but many of them do not consider anisotropic me- 13 The full text of Note 11 in [5] is:"M. v. Laue [see his Relativitätstheorie, Vol. 1 (6th edn., 1955) § 19] has shown that only the unsymmetric energy-momentum tensor of Minkowski is correct for a phenomenological description of moving bodies (just as it is in crystals at rest). His argument also emphasizes the validity of the addition theorem of velocities for the ray-velocity (see Eq. (312) of the text), which is in agreement only with this unsymmetric tensor.". The Editors of the Russ. Transl. V.L. Ginzburg and V.P. Frolov make further remarks to this problem with four additional citations, in particular, [16] [17]. dia where the problem becomes more clear though more difficult. The reason for different views to this tensor is different separations of ponderomotive forces in the conservation theorems which in this case do not possess the exact form of local conservation laws. This is discussed in detail by Ginzburg [10]. The discussion of the relations between the Abraham and the Minkowski tensor is usually restricted to isotropic media and we do not know an explicit more general form of the symmetric Abraham tensor for anisotropic dispersive media. For anisotropic media it is evident that the momentum density should possess the direction of the mean wave vector and the energy flow density should be in direction of the group velocity of quasiplane and quasimonochromatic waves which, in general, are different for anisotropic media or, in other case, essential parts of crystal optics would be wrong. This is provided if the momentum density is , E B as in the Minkowski tensor. Taking into account the dispersion both expression have to be modified as was shown ( [9] and Section 9 of present article). In the case of taking into account the dispersion the constitutive relations bring into play additional derivatives of the electric (and in certain cases of the magnetic) field which have to be taken into account in the derivation of conservation laws. Our strategy is to formulate the differential conservation laws without taking into account absorption (dissipation or absorption or even amplification, open system) as exact vanishing of 4-divergences. With dissipation this is impossible. The condition for the permittivity tensor to describe a dissipation-less medium was discussed in Section 3.
Even in the special case of an isotropic medium and under neglect of dispersion (spatial and temporal ones) that means in case of the constitutive relations that is nonsymmetric for 0 1 ε ≠ . However, the stress tensor kl T is symmetric in this case in the inertial system where the isotropic medium is resting (in moving systems it is no more isotropic) . kl lk Moreover, this symmetry remains to be true also in the general case of taking into account the dispersion as can be seen from (10.13) and (10.8) since for isotropic media the group velocity v and the wave vector k possess in general case the same direction (see below). This partial symmetry of the three-dimensional part kl T (and only this) of the full four-dimensional energy-momentum tensor T κλ is necessary for the existence of a local law of angular-momentum conservation in isotropic media due to invariance with respect to the three-dimensional rotation group in this inertial system where the medium rests.
Multiplying the differential momentum conservation (8.2) by ijk j r ε ( ijk ε is three-dimensional Levi-Civita pseudo-tensor) we may transform this according  . It is not possible to generalize in some simple way the form (14.5) of the energy-momentum tensor containing the full fields to the case of taking into account the dispersion. In particular, the corresponding expressions cannot be local in the fields that means cannot be taken only at the same space-time points ( ) ,t r (see Section 3). Therefore, we have to restrict us in the following discussion of the non-uniqueness concerning the terms with rapidly varying phase factors with T κλ denoting the energy-momentum tensor in (9.1) without terms with rapidly varying phase factors and represented by the slowly varying amplitudes and, in addition, neglecting derivatives which is true for the special form of ij L given in (5.5) only under neglect of dispersion and is related to (11.12) or also to (11.3 and comparison of (14.6) with the noncovariant form in [30] shows that the choice of ( ) r κλµ ψ itself for removing one and the same terms is to certain extent also not fully unique. Evidently, substitutions then one has also to differentiate the slowly varying amplitudes and there remain some new terms with these phase factors which, however, are small compared with the main terms with these phase factors before. In a second step and successively in higher steps one can try to remove also these smaller terms. However, there is no possibility to remove any parts in the energy-momentum tensor (9.7) which do not contain such rapidly varying phase factors without creating new terms with rapidly varying phase factors or terms which grow in space and time in unreasonable way (e.g., linearly). Clearly, we can derive higher-order approximations of the energy-momentum tensor than in (9.7) (see sections 5-7) and can try to ∂ ∂ ∂ and so on (this is not yet done) but they also do not contain such rapidly varying phase factors. Therefore, the nonuniqueness cannot be used to make energy-momentum tensors without rapidly varying phase factors to symmetric ones and nonsymmetric energy-momentum tensors of such kind remain intrinsically nonsymmetric. This suggests also that expressions of the kind (9.7) or (9.4) and of their generalization possess some distinguished position with the possibility of direct physical interpretation among all other equivalent energy-momentum tensors in local conservation theorems. Our derivations provided these expressions directly without the necessity of suppression of terms by the discussed non-uniqueness.
There is another case where the non-uniqueness of the energy-momentum tensor seems to be of great importance. These are evanescent in such waves provides further possibilities to remove terms in the energy-momentum tensor which are difficult to interpret and to get equivalent tensors but this makes the problems of non-uniqueness more complex.
Summarizing, it seems to us that the non-uniqueness of the energy-momentum tensor can mainly be used to remove or to change terms with periodically or exponentially rapidly changing phase factors, whereas the others are hardly to touch. This problem of non-uniqueness has little to do with the discussion of the correctness of the Minkowski or the Abraham tensor which in absence of dispersion was decided in favor of the Minkowski tensor.

Difficulties for General Relativity Theory Connected with General Asymmetry of Energy-Momentum Tensor in Media
The energy-momentum tensor T κλ forms the source term in Einstein's gravitation equations which determine the metric tensor g κλ and thus also the curvature of a Riemannian space-time as a generalization of Minkowski's space-time.
Ricci tensor and thus Einstein tensor in these equations are symmetric ones and, consequently, the energy-momentum tensor has also to be symmetric. Since macroscopic electrodynamics is an averaged microscopic electrodynamics it can be assumed that its energy-momentum tensor provides the source term for a correspondingly averaged gravitation field in the medium and requires boundary conditions in case of transition to vacuum with a sharp boundary. The connection of the classical energy-momentum tensor in Minkowski space as source of a curvature in Einstein's equations is patchwork since it starts from a pseudo-Euclidian space but it cannot be fully wrong concerning its symmetry.
We will shortly discuss some difficulties which result from this for General relativity theory.
If it would be possible to extend the Ricci tensor in Einstein's equations to a possible non-symmetric one that up to now did not be achieved then, nevertheless, there remain some serious problems. The energy-momentum tensor in the local conservation laws is not uniquely defined (see Section 14) and there arises the problem which of these tensors provides the right source term in Einstein's equations of General relativity theory. Moreover, in spatially and (or) temporally inhomogeneous media such a tensor in local conservation laws does not exist at all. In these cases only the local conservation theorem of action remains with a four-dimensional vector of action and action-flow density. One may expect that then this four-vector must be involved in some way as source in generalized Einstein equation but this to our knowledge was also not found up to now.
The unification of the basic laws of physics is a steady desire of physicists. Af- This non-symmetry of the energy-momentum tensor is intrinsic and cannot be removed by considering the current and charge distributions of media on the background of the vacuum. Since in the principal correctness of the existing classical electrodynamics of continuous media cannot be doubt the least which is required is some extension or generalization of the General relativity theory if not a more basically new theory. As it seems to us this problem has to be solved before a successful unification with the other fundamental forces of nature can be accomplished.
The General relativity theory has great success for explanation of astronomical observations and for cosmology. It is a beautiful theory which is considered as experimentally verified. It must not be incorrect due to some of the shown classical difficulties and we hope that they can be overcome in the course of time by generalization or somehow in other way and that it remains true as an approximation.

Possible Additions and Generalizations
The preceding theory of the energy-momentum tensor can be extended in dif-

A. Wünsche
with an operator equation only for the electric field and therefore with results which are basically expressed by the electric field.

Remark
A shorter paper of this theme (in particular, without any statements to difficulties for General relativity theory due to asymmetry of the energy-momentum tensor and to application of a plasma) written in German with nearly all basic formulae as now was made in about 1979 but was rejected from Editor of Annalen der Physik in GDR Professor Gustav Richter with wrong arguments. His main wrong argument was that in my formulae for the limiting case to plane monochromatic waves stands the derivative of the permittivity with respect to the frequency that he declared as wrong "for physical reason". However, the correctness of these formulae was already known from cited monographs and papers, in particular, of Landau and Lifshits and of Agranovich and Ginzburg. When I wanted to explain G. Richter who was also Member of our Institute at that time (in age of a few years below 65) in personal talk why mentioned formulae are correct for beams in limiting case he became very angry. Similar things happened shortly before when I wanted to publish my paper about generalized boundary conditions which, finally, was published after intervention by a prominent physicist of GDR from Editorial Board of "Annalen" and recently I published a continuation of this topic. When I tried to send the mentioned paper about boundary conditions to a Western journal I never got an answer. One could not check whether or not it was really sent since mail to Western countries went before this in an open couvert to the Chief and through Security or was a response withheld. I found now the hand-written comments of G. Richter and will pose them into my Home-page. in space-time representation and separated in three-dimensional form together with the transformation relations is ( lmn ε three-dimensional Levi-Civita pseudo-tensor) 16 Einstein denotes "inertial systems" with letter K, likely, from the German "Koordinatensystem". is the 4-vector of current density in space-time representation. We now make the transition to the Fourier transforms of the field functions (definitions see (3.4)).
From the transformation formula for the 4-vector of current density After transformation of the known formula for the tensor of the electromagnetic field into a corresponding formula for the Fourier components and then after the elimination of the magnetic field by means of the first vectorial equation in (2.7) we arrive after some intermediate calculations to the following transformation formula for the electric field ( ( ) ( ) ( ) ( )