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VI. The energy-momentum tensor of the field 18 I. INTRODUCTION A. The energy-momentum tensor of the field in matter 18 The definitions of the electromagnetic (EM) momen- B. The meaning of Poynting’s energy density tum in matter and of the energy-momentum density ten- and Minkowski’s tensor 19 sor of the electromagnetic field in matter has been dis- C. Orbital angular momentum and spin 20 cussed for over a century in what is known in the lit- erature as the Abraham-Minkowski controversy. It is VII.Theaveragemethod 20 among the most prominent unsettled problems in clas- A. Space-time average 21 sical physics. At the quantum level, resolution of the B.Macroscopicforce 21 Abraham-Minkowski controversy is of paramount inter- C. Densities of dipole moment 23 est since it answers the fundamental question “what is VIII.Testingtheforces 23 the momentum of a photon in matter?” The main con- A.Experimentalresults 23 tenders in the dispute are the non-symmetric tensor con- B. Walker, Ladhoz and Walker experiment 24 structed by Minkowski in 1908 [1] and the symmetric C. Electrostriction 25 one proposed by Abraham in 1909 [2, 3], but other op- D. Magnetostriction 25 tions have been discussed in the literature [4–10]. Each construction provides a definite expression of the force IX. Electromagnetic waves in matter 26 exerted by the field on matter. A. Wave-matter interaction 26 Although hundreds of papers investigate this issue B. Energy and momentum conservation 27 many arguments lack precision in statement of the C.Thecenterofmassandspin 28 hypotheses and definitions or present an inadequate handling of their consequences. Various hypotheses X.Themagnet-chargesystem 29 and mechanisms that depart from the well-established A. Violation of the CMMT 29 Maxwell-Lorentz electromagnetic theory have been ad- B. The motion of the center of mass and spin 30 vanced. For example, proposed forms of the force den- C. Storing momentum in the magnet 30 sity that the field exerts on matter are in general differ- D.Toroidalcurrents 30 ent from the force density imposed by the microscopic Lorentz force. Examples of this are the Minkowski force, XI. Angular momentum of a magnetized cylinder 31 the Abraham force and the Einstein-Laub force [4]. An even more puzzling hypothesis proposed in this context is XII. Conclusion 32 the use of the theoretical quantity known as hidden mo- mentum. Hidden momentum is the caloric of the XXth A. Energy and momentum of dressed dipoles 33 century. It was introduced by Shockley and James in [11] as a resource to force the Center of Mass Motion Theo- B. Integrals of self fields of dipolar systems 34 rem (CMMT) [12, 13] to hold, for systems which appear References 35 to have a contribution of electromagnetic momentum at rest. A weak version of the CMMT, states that the cen- ter of energy of an isolated system moves with a constant velocity in a chosen reference frame. A stronger version claims that such velocity is the total momentum of the system times c2, divided by the energy. As we discuss be- low, these theorems have only limited validity. The true nature of the hidden momentum is never exhibited explic- itly and the hypothesis implies very strange consequences on the mechanical behavior of physical systems when the value of the electromagnetic momentum changes. An unsatisfactory situation exists because none of the sides have been able to convince the other of the validity of their arguments. This has led to an annoying pos- ture well distributed in the literature [14–20] which, re- lying on the fact that the energy-momentum tensor of the matter-field system may be split in many different forms, hypothesizes that all the alternatives are equivalent if ad- equately interpreted. In other words, it postulates that the problem cannot be settled theoretically and is not amenable to experimental elucidation. The same unsatisfactory situation has induced some other people to think that both expressions may be used 3 in the theoretical description with different meanings. is usually not investigated. In agreement with the ex- Often, it is postulated that one of the expressions, usu- perimental evidence, we assume that Maxwell’s macro- ally Abraham’s, is the true mechanical momentum with scopic equations are fulfilled. So, our first remark is that the other being some other kind of momentum, either it is necessary to focus on the dipolar coupling of the pseudo-momentum or quantum mechanical momentum macroscopic electromagnetic field to P and M instead of [21–23]. dwelling too much on the constitutive relations between In this paper we show that fancy interpretations, ex- E and B and D and H. This approach is mandatory otic forces and hidden momentum are unnecessary hy- when dealing with ferromagnetic materials. potheses. We contend that the motivations to introduce Our second remark concerns relativistic invariance. them are wrong and that a totally consistent description The understanding of the implications of relativistic in- of electromagnetic momentum exists within the frame- variance have evolved since the first decade of the past work of Maxwell-Lorentz electromagnetism. We show it century when the controversy began. Putting things sim- is possible to give a definite answer to the question of ply, a relativistic description requires one to work with which is the energy-momentum tensor of the electromag- covariant objects. Cavalier use of the electric and mag- netic field in matter; it is not necessary to take differ- netic permeability constants ǫ and µ obscures the rel- ent expressions to represent the classical and quantum ativistic invariance of the equations. Attempts to re- electromagnetic momentum in order to deal with the re- gain relativistic invariance by introducing the velocity of sults of the experiments. The energy-momentum tensor the material media and switching between the laboratory for the electromagnetic field is neither Minkowski’s nor frame and the co-moving frame of matter without choos- Abraham’s and we explain clearly why. We construct ing convenient covariant objects is a limiting procedure. by two different methods an energy-momentum tensor of Such attempts introduce the well known inconsistency of the electromagnetic field in matter that is consistent with the concept of rigid body and relativity. To guarantee the the microscopic Lorentz force. Our construction finally covariance of Maxwell’s equations in a material medium, resolves the long dated Abraham-Minkowski controversy it is sufficient and necessary that the fields P and M and, and ratifies Minkowski’s expression for the momentum as a consequence, D and H transform in the same way density. It is based on the observation that, in oppo- as the electromagnetic fields E and B. sition to what is claimed in a large part of the recent Our third point concerns the force density that the literature, the matter and field contributions to the total field exerts on matter. The force density and the total energy-momentum tensor can be rigorously distinguished force are observable quantities which can be measured by focusing on the mechanical behavior of matter and in a manifold of situations by observing the mechan- using momentum conservation in the form of Newton’s ical behavior of matter. The energy-momentum ten- Third Law. sor of matter is determined as the tensor whose four- To resolve this important debate, it is essential to de- divergence is the force density. It includes all the en- scribe very clearly the framework in which the discussion ergy parcels which contribute to inertia, even those of will be pursued. So we begin with a few remarks high- electromagnetic origin. Momentum conservation, a con- lighting our suppositions and stating where we disagree sequence of the homogeneity of space, then determines with other authors. the energy-momentum tensor of the field as the tensor Classical electromagnetism in material media is a very whose four-divergence is the negative of the force den- subtle subject with plenty of room for misunderstandings sity. The force density is also subjected to theoretical and confusion. To begin with, the same notation is often constraints. It should be relativistically covariant and used for microscopic and macroscopic electromagnetic should reduce to the Lorentz force when a microscopic fields but they are very different objects. The microscopic description of the electric and magnetic dipoles is re- field, at the classical level, allows for only monopolar cou- introduced. Einstein-Laub’s, Minkowski’s, Abraham’s plings and interacts with all the microscopic charges and and the Lorentz-Ampere force densities do not satisfy µ 1 µ αβ currents ρ and j. From the theoretical point of view, it is these criteria. The force density fd = 2 Dαβ∂ F does, accepted that the description using the microscopic field as discussed below. should be valid up to the scale where quantum effects Our fourth introductory remark addresses the asym- begin to be relevant. But in bulk matter this description metry of the energy-momentum tensor. It is frequently cannot always be experimentally explored making it nec- assumed that the energy-momentum tensor of an iso- essary to introduce the macroscopic field which is inter- lated system should be symmetric for one of the fol- preted as an average field at some scale. The macroscopic lowing three reasons. 1) To guarantee the conservation field has a monopolar coupling with the free charges and of angular momentum. 2) Because there is a theorem currents ρf and jf and also has a dipolar coupling with [24–27] which states that it is always possible to sym- the polarization vector P and the magnetization M. Of metrize the tensor. 3) Because it is imposed by Ein- course ρf , jf , P and M are manifestations of quantum stein’s equations of gravitation. None of these reasons and classical properties of the microscopic quantities ρ is valid. The energy-momentum tensor should be sym- and j. The macroscopic fields may also have quadrupo- metric for conserving the orbital angular momentum, not lar and higher order couplings although this possibility the total angular momentum. There is no symmetrizing 4 theorem for the energy-momentum tensor. Belinfante- pression for the momentum density. This confirms our Rosenfeld’s construction [24, 25] yields a symmetric ten- recent computation [30] where in a simple setup we show sor, made up from the energy-momentum tensor and that if Maxwell’s equations and Ohm’s law are valid, only derivatives of the spin density, which is not the me- Minkowski’s momentum density is consistent with mo- chanical energy-momentum tensor. Einstein’s equations mentum conservation. On the other hand, we recognize impose that their source, not the mechanical energy- Poynting’s energy density and Minkowski’s tensor as the momentum tensor, should be symmetric. The source of objects which describe the whole electromagnetic wave Einstein’s equations is physically a different object and thus satisfying the von Laue-Møller criterion. In section there is no contradiction in relating it to the Belinfante- VII, we take an alternate path and re-obtain our results Rosenfeld tensor. by directly averaging the microscopic field equations and Our next observation is related to the hidden momen- imposing consistency with the results of a macroscopic tum approach and the CMMT. In its usual form the field with an arbitrary dipolar coupling. In the remaining CMMT is not valid in relativistic mechanics. The CMMT sections we discuss some examples and consequences. In was invoked repeatedly in discussions of the electromag- section VIII, we compare the experimental consequences netic momentum to justify introduction of hidden mo- that predict the force density we proposed with the other mentum. It also was used by Balazs to support Abra- alternatives. We propose a method to elucidate which is ham’s momentum arguing on a mechanical example. But the true force density by measuring electro and magne- the CMMT is not valid for systems with spin in general tostriction. In section IX we consider the transmission of and is not valid for the electromagnetic field interacting an electromagnetic wave in a medium, which corresponds with polarizable matter in particular. What is valid for to the setup of Balazs. We compute the force exerted by systems which include spin and for which total angular the fields on matter and show that the standard CMMT momentum is conserved, is a modified version of the the- does not hold, but that instead the center of mass and orem which states that for an isolated system a quantity spin behaves inertially, as we claim it should. In sec- which we call the center of mass and spin behaves in- tion X, we consider an interesting magnet-charge system ertially. This fact has been overlooked in all previous and show that its description is fully consistent with our discussions. equations. We also show that, for this system, the center Our final point is about the distinction between the of mass and spin behave inertially in a non trivial way. In electromagnetic field and the electromagnetic wave when section XI, we consider a situation in which there is ex- such a wave is propagating through a medium. In that change of angular momentum between matter and field, case, traveling together with the electromagnetic fields which can be handled neatly within our approach. Sec- there are disturbances of P, M and the internal stress. tion XII contains our conclusions. The energy and momentum contributions of these dis- turbances should appear in the energy-momentum-stress II. ANGULAR MOMENTUM AND THE CM tensor of matter. Poynting’s equation which is a conse- MOTION THEOREM quence of Maxwell’s equations for homogeneous media is best suited to describe the wave as a whole. In partic- A. Orbital and spin angular momentum ular Poynting’s energy density, as we will argue in this article, is the energy of the wave, not the energy of just the field. Many of the misunderstandings concerning the Let us consider a system with an energy-momentum µν µ energy-momentum tensor of the field may be traced back tensor T upon which a force density f is exerted. A to this observation. This point is also very important to local version of Newton’s Second Law holds understand the meaning of the von Laue-Møller [28, 29] µν µ condition on the velocity of propagation of the energy ∂ν T = f . (1) density. For a localized system with f µ = 0 the total energy U = Our paper is organized as follows. In section II, we T 00dV and the total momentum pi = c−1 T i0dV are show the limitations of CMMT for systems with spin, Rconserved. The current density of the orbitalR angular discuss the concept of center of mass and spin, and de- momentum is defined as Lµνα = xµT να xν T µα and it duce the modified version of the theorem which holds for satisfies identically − those systems. In section III, we discuss Poynting’s equa- µνα νµ µν µ ν ν µ tion and its generalizations. In section IV, a brief discus- ∂αL = T T + x f x f . (2) sion of the Abraham-Minkowski controversy is presented. − − The von Laue-Møller criterion and the Balazs argument The last two terms of the right hand side of this equation are considered. In section V, we obtain an expression of represent the torque of the force density. The asymme- the force which is covariant and reduces to the Lorentz try of the energy-momentum tensor plays the role of a force in the microscopic description. Using this force in torque density exerted on the orbital angular momentum. section VI, we construct the true energy-momentum ten- If the force density is zero and there are neither a torque sor of the field. Although the energy density is not given density τ µν nor a spin density Sµνα, the orbital angu- by Poynting’ s expression one recovers Minkowski’s ex- lar momentum is conserved and the energy-momentum 5 tensor must be symmetric. This is not necessarily true quantity which has the proper Poisson brackets to gen- when these conditions are not fulfilled [31]. In presence of erate this transformation is β g with g = ctp mcx. spin the relevant quantity is the total angular momentum For a system of particles of center· of mass X, total− mass J µνα = Lµνα + Sµνα whose equation of motion M and total momentum P the generator of the Galilean boost is g = ctP McX. Conservation of this quantity µνα µν µ ν ν µ − ∂αJ = τ + x f x f , (3) is equivalent to the CMMT. Note that g corresponds to − the orbital boost momentum L0k of the relativistic case. µν allows for an external torque density τ . Then the right On the other hand, the Galilean boost does not modify hand side is the total torque density acting on the system. the angular velocitiy or the shape of a body, that is, it For isolated systems one assumes that the total angular does not modify the spin of the system (the angular mo- momentum is conserved. mentum with respect to the CM). In Newtonian physics Combining (2) and (3) the spin equation is obtained the spin commutes with the Galilean boost, and there- fore the existence of spin does not impair the CMMT. ∂ Sµνα = T µν T νµ + τ µν . (4) α − This means that spin boost momentum does not have a The internal torque density represented by the asymme- non-relativistic counterpart. Like boost momentum the temporal components of the try of T µν couples spin and orbital angular momenta. If the system has no spin, but there is a distributed torque torque also form a polar vector which is different from the spatial torque. In some particular reference frame a sys- then T µν T νµ = τ µν . Another interesting case is when the− system is isolated;− the total angular momen- tem may in principle have a non-zero boost momentum and a vanishing J. This is also true for the torque. tum is conserved, but there may be an exchange between spin and orbital angular momentum as a result of micro- scopic interactions. That is what happens in a magnet C. The CM motion theorem when the magnetization changes. Since the magnetiza- tion is proportional to the spatial part of the spin density, if a demagnetization occurs, spin becomes orbital angu- Let us consider an isolated, localized system with a non-symmetric energy-momentum tensor upon which no lar momentum. This is the Einstein-de Haas [32] effect µν which is used routinely to measure the gyromagnetic ra- forces are exerted so that, ∂ν T = 0. tio [33] of atoms and molecules and provides an example To discuss the CMMT define the center of mass by where the total energy-momentum tensor cannot be sym- 1 metric. The statement, found some times in the litera- Xi = xiT 00 dV . (5) T U ture [3], that conservation of angular momentum requires Z a symmetric energy-momentum tensor, is not only false Here the index T refers to the fact that the CM depends but in this and other cases the opposite is true: conserva- on T µν . It is worth remarking that this definition of tion of (total) angular momentum requires that the total center of mass depends on the frame of reference. If in energy-momentum tensor should be non-symmetric. some frame of reference at time t the center of mass is X, and the event (t, X) is labeled by (t′, X′) in some other frame, and if Y′ is the CM in the new frame at time t′, B. The boost momentum then in general Y′ = X′. Statements about centers of mass should be done6 with caution. For example, the fact The temporal component of the angular momentum that the CM moves with constant velocity in some frame equation plays an important role in the treatment of of reference does not imply that the CM also moves with the CMMT. The total angular momentum of a system constant velocity in some other frame of reference. J µν = c−1 J µν0 dV is an antisymmetric tensor. The If there is no spin, T µν is symmetric and orbital angu- spatial partR of this tensor is the angular momentum ax- lar momentum Lµν = c−1 Lµν0 dV is conserved. Then, ij k 0k ial vector J, J = ǫijkJ . The temporal components J it is easy to see that the centerR of mass moves with ve- transform as a polar vector, which is a different physical locity c2pi/U. Instead, for non-symmetric T µν it is also quantity, in the same sense that E is different from B. easy to see that This quantity is seldom treated in the literature and its c physical meaning is not well appreciated. There is not X˙ i = T 0i dV . (6) even a generally accepted name for it. A sensible name T U Z could be the boost momentum, since J 0k are the gen- erators of the Lorentz boosts, or the arrow{ of} angular mo- Note that what appears in (6) is the energy current mentum as we proposed elsewhere [34]. To better under- density, not the momentum density. The non-vanishing stand the significance of the boost momentum consider temporal spin of the system jeopardizes the validity of the CMMT. To see this define the spin matrix Sµν = a non-relativistic particle of mass m, position x and mo- −1 µν0 mentum p. In a different frame of reference moving with c S dV and consider the quantity velocity cβ the position is incremented by ∆x = ctβ, R c − i 0i and the momentum is incremented by ∆p = mcβ. The XS = S . (7) − −U 6

From the conservation of total angular momentum it fol- The traditional way of treating polarizable matter is lows that to assume that there are microscopic electric and mag- P M d c d 1 d netic dipoles. The polarization and magnetization Xi = L0i = x0T i0 xiT 00 dV are defined so that an element of material of volume dV dt S U dt U dt Z −   has a total electric dipole moment PdV due to the elec- c2pi d M = Xi . (8) tric dipoles and a total magnetic moment dV due to U − dt T the magnetic dipoles. In the non-relativistic approxima- This shows that the center of mass and spin defined by tion this picture works fine but it is problematic in the general case, since moving electric and magnetic dipoles Xi = Xi + Xi (9) have moments of both kinds. Because of this reason in Θ T S this paper we refrain from using those microscopic enti- ˙ i 2 i ties. In the following we present a phenomenological and moves with constant velocity XΘ = c p /U. i macroscopic definition of P and M. It is worth noting [35] that XΘ corresponds to the cen- ter of mass computed using the symmetric Belinfante- The fields and currents used in macroscopic physics are Rosenfeld’s tensor Θµν , which is a combination of the averages of the microscopic fields and currents over small energy-momentum tensor and the spin density. In the space-time regions. As usual we will assume, that at a literature Belinfante-Rosenfeld’s tensor is frequently con- microscopic scale the equations for the vacuum hold. In sidered as an improved symmetrized energy-momentum a material sample at a macroscopic scale, in addition to µ tensor but we consider that this interpretation is wrong, the averaged current density of free charges jf = (cρf , jf ), at least from the mechanical point of view. there are electric and magnetic dipole moments that cou- As a simple illustration of the ideas presented in this ple directly to the EM tensor F µν . These dipole mo- section, consider an isolated uniformly magnetized sphere ments are due to bound charges. The total bound charge with a device included which heats the sphere respecting in any piece of material is always zero. This is a kind the symmetry. In the rest frame the spin, which is pro- of mechanical constraint. The fields produced by the portional to the magnetization, is purely spatial and both dipoles are described by the current density four-vector µ the center of mass and the center of mass and spin are of bound charges jb . Any piece of material may have located at the center of the sphere. As the sphere heats a bound charge in the surface QbS and a bound charge the magnetization diminishes and finally vanishes and by in the bulk QbV, but the total bound charge should al- conservation of total angular momentum the sphere ro- ways be zero, QbS + QbV = 0. This behavior is described tates. The center of mass and the center of mass and spin in terms of P, defined inside the material, if the charge remain in place. Looking at this system from a reference density of bound charges is ρb = P and the surface −∇ · frame where the sphere is moving in a direction which charge density is σb = P nˆ, where nˆ is the normal uni- · is perpendicular to magnetization, the center of mass is tary vector of the surface. The continuity equation for initially at the center of the sphere. The temporal com- the bound charges is ponents of spin do not vanish and the center of mass ∂ρ and spin is displaced from the center of the sphere in the 0= b + j = (j P˙ ) . (10) direction perpendicular to the magnetization and to the ∂t ∇ · b ∇ · b − velocity. When the magnetization disappears the sphere is rotating about the axis which is in the former direction In general, in addition to the polarization current of the magnetization. The velocity of matter in one side density P˙ , a divergence-less magnetic current density of the sphere is greater than the velocity in the other side, jm = jb P˙ also contributes to the bound current den- and the center of mass is now displaced with respect to sity. This− magnetic current density is not related to any the geometric center. Clearly, it has been accelerated. It macroscopic motion of charge, and is expressed as the may be shown [34] that the center of mass and spin (9) rotational of the magnetization field jm = c M. M which in the final situation coincides with the center of vanishes outside the material. Since there∇× is no total mass behaves inertially along the whole process. magnetic current crossing a piece of material, in the sur- face of any piece of material there is also a magnetic surface current density Σ = cM nˆ. m × III. POYNTING’S ENERGY DENSITY AND An element of material of volume dV has an electrical ENERGY FLUX dipole moment dd = PdV and a magnetic dipole moment dm = MdV due to the surface charges and surface cur- A. The dipolar density tensor rents respectively. The bulk contributions which go like r3 are negligible with respect to the surface ones that go In Gauss units and using the metric tensor ηµν = like r2. diag( 1, 1, 1, 1), the electromagnetic field tensor (either For a covariant description one should define the space- microscopic− or macroscopic) is F = ∂ A ∂ A , time dipolar density D , whose spatial part is the mag- αβ α β − β α αβ and is related to the electric and magnetic fields by netization density Dij = ǫijkMk and whose temporal F 0i = F i0 = E and F ij = ǫ B . part is the polarization D = D = P . The bound − i ijk k 0k − k0 k 7 current density is then The susceptibilities form a four-tensor of four indexes with the following symmetries µ µν jb = c∂ν D . (11) χαβγδ = χγδαβ = χβαγδ . (18) As mentioned before, in order to have relativistic invari- − ance, Dαβ should be a Lorentz tensor. The fact that Dαβ The four-tensor χαβγδ includes the susceptibilities is antisymmetric implies the conservation of the bound three-tensors, the electric-electric χ(e−e), the magnetic- µ µν (m−m) (e−m) current ∂µjb = c∂µ∂ν D = 0. magnetic χ , the electric-magnetic χ and the magnetic-electric χ(m−e). The first two are symmetric tensors, and of the last two one is the transpose of the B. Maxwell’s equations other. In total there are 21 independent components. The three-tensors are defined by

The displacement vector and the magnetizing field vec- (e−e) (e−m) tor are given in our notation by D = E + 4πP and Pi = χij Ej + χij Bj (19) H = B 4πM. In what follows we assume that Maxwell’s macroscopic− equations and (m−m) (m−e) 1 ∂B Mi = χ Bj + χ Ej . (20) D =4πρ , E + =0 , ij ij ∇ · f ∇× c ∂t 1 ∂D 4π Note that χ(m−m) is different from the traditional mag- B =0 , H = j (12) ∇ · ∇× − c ∂t c f netic susceptibility tensorχ ˜, which is defined using H instead of B, χ(m−m) =χ ˜(I +4πχ˜)−1. are valid in any reference frame. To cast this in covariant The relationship between the four-tensor and the notation one defines three-tensors is Hµν = F µν 4πDµν (13) χ(e−e) = χ , (21) − ij 0i0j (m−m) 1 whose components are related to the magnetizing field H χij = ǫilmǫjfgχlmfg , (22) ij 4 and to the electric displacement D by H = ǫijkHk and 0i (e−m) 1 H = Di. χij = ǫjfgχ0ifg , (23) The field equations become 2 (m−e) 1 χij = ǫifgχfg0j . (24) αβ 4π α 2 ∂βH = jf , (14) c The proper way of treating the transformation of the susceptibilities when the frame of reference is changed is ∂ F + ∂ F + ∂ F =0 . (15) by transforming the four-tensor. When matter is at rest µ νλ ν λµ λ µν and the material is centrosymmetric the mixed pseudo- (e−m) (m−e) The second of these equations is the Bianchi’s identity. tensors vanish, χ = χ = 0. This property is The first equation written in terms of the macroscopic lost in other frames of reference. electromagnetic field is Equivalent relations may be written in terms of the permeability tensor. Using (13) one has

µν 4π µ µ ∂ν F = (j + j ) . (16) 1 c f b H = µ F γδ . (25) αβ 2 αβγδ with C. The susceptibility and permeability tensors

µαβγδ = δαγ δβδ δαδδβγ +4πχαβγδ (26) Isotropic linear materials are characterized by the fa- − miliar relations D = ǫE and H =1/µB. These relations The permeability tensor has the same symmetry proper- are meant to be valid in a particular reference frame, ties as the susceptibility tensor. In particular one has, which for homogeneous, isotropic media is the rest frame of matter. But much more complicated situations may Di = ǫij Ej + ξij Bj (27) be contemplated, and in order to obtain a relativistic invariant description it is necessary to introduce gener- and alized relations in terms of a four indexes susceptibility tensor. They take the form, Hi = ζij Bj + ηij Ej . (28) The tensors ǫ and ζ must be symmetric and the 1 γδ ij ij Dαβ = χαβγδF . (17) pseudo-tensors ξ and η are related by η = ξ . 2 ij ij ij − ji 8

These conditions are preserved by Lorentz transforma- In 1884 Poynting [36] proposed his energy conserva- tions. At rest, for a centrosymmetric material, the tion equation using the particular case of these equa- pseudo-tensors vanish, but that is not true in the gen- tions that applies for homogeneous isotropic materials eral case. with time-independent linear susceptibilities D = ǫE and In the particular case of a homogeneous isotropic ma- H =1/µB. He postulates that the energy density is terial with a dielectric constant ǫ and magnetic perme- ability µ, which in the rest frame satisfy D′ = ǫE′ and 1 uP = (E D + H B) . (37) B′ = µH′ one may obtain the permeability tensor by ex- 8π · · pressing the fields in the rest frame in terms of those in With the mentioned restrictions on the medium it follows the moving frame. The familiar relations for the trans- that formed fields are, ∂uP 1 ∂D ∂B D + β H = ǫ(E + β B) (29) = E + H (38) × × ∂t 4π  · ∂t · ∂t  and and equation (34) reduces to 1 H β D = (B β E) , (30) ∂u − × µ − × P + S = E j , (39) ∂t ∇ · − · f where cβ is the velocity of the medium. where the energy current density S is Poynting’s vector, Solving these equations we get, γ =1/ 1 β2, − p c ǫ = γ2[(ǫ β2/µ)δ (ǫ 1/µ)β β ] , (31) S = E H . (40) jk − jk − − j k 4π ×

2 2 This led to the interpretation of S as the energy current ζjk = γ [(1/µ ǫβ )δjk (ǫ 1/µ)βjβk] , (32) density of the electromagnetic disturbance, and E j as − − − · f and the time rate of work done by the field on the free charges. No work is done on the polarizable matter. At this point 2 ξjk = ηjk = γ (ǫ 1/µ)ǫjlkβl . (33) there is place to ask if uP should necessarily be inter- − − preted as the energy density of the field and if the work done on the free charges is all the work done by the field D. Energy flux equation and Poynting’s equation on matter. In the following sections we argue that the answers to these questions are both negative. The flux of energy is determined by the following The condition (38) is valid also for anisotropic lin- relation which is a direct consequence of macroscopic ear susceptibilities. Poynting’s energy density for an Maxwell’s equations (12) anisotropic material in any reference frame is given by 1 u = (ǫ E E + ζ B B ) . (41) 1 ∂D ∂B P 8π jk j k jk j k E + H 4π  · ∂t · ∂t  c From this expression it is readily shown that for time- = (E H) E jf . (34) independent tensors the relationship (38) is fulfilled, and −4π ∇ · × − · that therefore Poynting’s equation (39) is valid. This may be written in the two alternative forms Note that in order for ǫjk and ζjk be time independent 1 ∂ ∂ c in any reference frame, it is required that the medium (B2 + E2)+ (E P)+ (E H) properties are not only time-independent but also ho- 8π ∂t ∂t · 4π ∇ · × mogeneous. Poynting’s equation is valid in any frame ∂E ∂B = E jf + P + M (35) of reference only for material media which are homoge- − · · ∂t · ∂t neous, time-independent, and that are moving in a uni- and form translation. 1 ∂ c (B2 + E2)+ (E B) 8π ∂t 4π ∇ · × E. Electrostatic energy = E (j + j ) . (36) − · f b The different terms in these equations should describe The essential differences between non-dipolar and the energy density and the energy current density of the dipolar matter already appear in electrostatics. In this field and the power supplied to matter by the field. There subsection we discuss the physical significance of various is clearly more that one way to establish these correspon- ways of defining electrostatic energy for dipolar matter. dences, but we will argue below that there is one way that Let us start by considering non-dipolar matter. The en- should be preferred. ergy of a system of charges is the sum of electrostatic 9 interaction energy Ue and UM which represents all other the electrostatic energy P E and the deformation en- kinds of energies. The electrostatic energy is ergy P E/2. The bare matter− · increases its energy by the deformation· energy. Poynting’s density adds the defor- 1 mation energy to the standard energy density u of the Ue = ρφ dV , (42) S 2 Z field. This density is useful for determining the total en- ergy of a system (field plus matter), for example in a ca- which can also be expressed as pacitor, but it cannot be used for determining the energy exchange between matter and field. Poynting density is 1 2 Ue = E dV . (43) 8π Z useless for matter with non-linear polarizabilities. If it is assumed that the matter energy is distributed in the volume there would be a matter energy density uM to IV. ABRAHAM-MINKOWSKI CONTROVERSY be added to the standard field energy density uS which represents the electrostatic interaction A. Splitting the Electromagnetic tensor 1 u = E2 . (44) S 8π To discuss electromagnetic momentum it is necessary to have, not only an energy density and an energy current For dipolar matter where there is a polarization P two density, but a complete energy-momentum tensor. For an facts have to be considered. 1) The energy density of isolated field-matter system the conserved total energy- matter becomes polarization-dependent and should be momentum tensor T µν should be split in two terms, now denoted by ub(P) (here the label b stands for bare). µν µν µν 2) uS still represents the total electrostatic energy, but a T = Tmatter + Tfield (49) part of this energy density, given by corresponding to each part of the system. Contrary to ud = P E (45) what is sometimes stated (see section VIII), the matter − · contribution to the tensor is completely determined by corresponds to the interaction energy of electric dipole its mechanical behavior through the equation moments. The total energy density is given by ub(P)+ u , but u should be considered part of the energy of µν µ S d ∂ν Tmatter = f , (50) matter, since it is attached to it and contributes to its inertia. The energy density of (dressed) matter is then where f µ is the measurable force exerted by the field on matter. Correspondingly, all the portions of the en- uM = ub(P)+ ud . (46) ergy which contribute to inertia should be included in the matter energy-momentum tensor, even those of electro- Correspondingly, the dipolar interaction energy must be magnetic origin. On the other hand, conservation of mo- subtracted from the standard field energy density in order mentum imposes that the field energy-momentum tensor to obtain the actual field energy density µν Tfield should satisfy Newton’s Third Law u = u u . (47) S − d ∂ T µν = f µ . (51) ν field − This modifies the expression of power released to matter. The power density on the bare dipoles is E P˙ , whereas This gives a very stringent criterion for choosing the for the dressed dipoles it is P E˙ . In section· VI, we show energy-momentum tensor of the field, which may be only that the generalization of (47),− · which includes the mag- avoided at the expense of changing the description of the netic field contribution should be regarded as the energy mechanical behavior of matter relaxing (50), or by intro- density of the field. ducing other non standard elements like hidden momen- This field energy density is not only different from tum. Below we show that none of these alternatives is the standard one for non-dipolar matter, it also differs necessary. Note that whereas in the temporal component from the Poynting density for dipolar matter E D/8π. of (51) the energy density and the energy density current · are present, the spatial components are written in terms To compare with this last density consider an isotropic ij medium with linear polarizability. The bare matter den- of the stress tensor T and the momentum density g, sity u should have a deformation term quadratic in P b c∂ gi + ∂ T ij = f i . (52) 0 j − 1 u (P)= u (0) + P 2 . (48) b b 2χ For any covariant energy-momentum tensor introduc- ing Maxwell’s equations in (51), the zero component Here χ is the electric susceptibility. When an electric yields an equation equivalent to (35), with a specific field is present the matter polarizes itself by minimizing choice of the energy density, energy flux and power on uM with the result P = χE. The dressed matter changes matter. These quantities are given by definite expres- its energy by an amount of P E/2 which is the sum of sions of the fields, polarizations, and currents. − · 10

B. Field energy-momentum tensor in vacuum C. Abraham and Minkowski momenta

Let us start from the well-known results of microscopic Many theoretical arguments and some experimental EM field in vacuum, which couples minimally to a mi- evidence have been used to try to elucidate which form µ µ croscopic conserved electric current density j , ∂µj =0 of the energy-momentum tensor of the electromagnetic through the vector potential Aµ. The four-vector Lorentz field in matter should be adopted. Some reviews written force density acting on the currents is through the years from different points of view may be found in [15, 19, 39–53]. µ 1 µν Equations (37) and (39) and the requirement of fL = F jν . (53) c Lorentz invariance were the main ingredients which Minkowski [1] used to propose a non-symmetric energy- The electromagnetic tensor is determined by Maxwell’s µν equations and Newton’s Third Law. Besides Bianchi’s momentum tensor TMin, which he interpreted as the identity (15) the vacuum field equations are energy-momentum tensor of the field. In this tensor the energy density and the energy current density are Poynt-

µν 4π µ ing’s expressions (37), (40), ∂ν F = j . (54) c 00 0i TMin = uP , cTMineˆi = S (59) The standard energy-momentum tensor of the EM field is [37] and the density of linear momentum is,

µν 1 αβ µν 1 µ να −1 i0 1 T = F F η + F F . (55) gMin = c T eˆi = D B . (60) S −16π αβ 4π α Min 4πc × Using (53), (54) and (15) it is straightforward to prove For photons with energy E this implies a momentum the identity nE/c with n the refraction index. Minkowski’s tensor is given by µν 1 µν µ ∂ν TS = F jν = fL , (56) 1 1 − c − T µν = F Hαβηµν + F µ Hνα (61) Min −16π αβ 4π α valid for any solution of Maxwell’s equations. Note that the force density acting on the EM field is the opposite and it is non symmetric . It is written here for the general of the force density that the EM field exerts on the mat- case of arbitrary Dµν , but was originally intended only ter. That is, the action-reaction law holds locally. As for linear materials. In components, besides (59) and a consequence, if there are no other interactions, the to- (60) it defines Maxwell’s tensor by tal energy and momentum of EM fields and matter are conserved. ij 1 ij 1 T = (E D + B H)δ (EiDj + HiBj ) . (62) To compare with our discussion below let us consider Min 8π · · − 4π the standard tensor (55) when it is written in terms of For linear isotropic materials this expression is symmet- the macroscopic field with dipolar coupling. In this case, ric, but in general it is not. Using (51) the corresponding (55) is referred in the literature as the Livens tensor [38]. force density is Maxwell’s equations take in this case the general form µ µ µ (16) with j in (54) replaced by jf + jb . Since (56) is an 1 1 f µ = F µ jν + (D ∂µF αβ F ∂µDαβ) . (63) identity, it is possible to write directly the new identity Min c ν f 4 αβ − αβ 1 µν µ ν µ Minkowski’s force differs from Amp`ere-Lorentz’s force ∂ν TS = F ν (jf + jb ) − c (58). In the rest frame, for linear and homogeneous 1 µ ν µ να materials with no boundaries, with constant suscepti- = F ν jf F ν ∂αD . (57) − c − bility tensor (17), the second term above vanishes, and The negative of the right-hand side of this equation is Minkowski’s force density reduces to Lorentz’s force den- what is known as the Amp`ere-Lorentz force, sity on free charges. The equation for the flux of energy (35) in Minkowski’s approach is written as, µ 1 µ ν µ fAL = F ν (jf + jb ) . (58) ∂u c P + S = E j + (64) ∂t ∇ · − · f µν One should note that TS may not be identified with 1 ∂E ∂B 1 ∂P ∂M P + M E + B . the energy-momentum tensor of the field if the force den- 2  · ∂t · ∂t  − 2  · ∂t · ∂t  sity on the currents of bound charges which contribute to the Amp`ere-Lorentz force is not the same as the true The last two terms cancel each other only for linear ho- force density acting on the dipolar densities. We argue mogeneous materials, allowing Poynting’s (39) equation in the next sections that they are indeed different. to hold. 11

In 1909, Abraham [2], requiring the energy-momentum In the rest frame it reduces to tensor to be symmetric, gave an alternative suggestion 0 0 for the energy-momentum tensor of the field. In the rest fAbrI = fMin (72) frame it agrees on the energy density and the energy current density with Minkowski’s tensor, but the linear ǫµ 1 f i = f i + − ∂ Si . (73) momentum density postulated is AbrI Min c2 t

−1 i0 −2 The second term in this equation, gAbr = c TAbr = c S . (65) 1 f = ∂ [D B E H] (74) This yields a symmetric energy-momentum object for ho- Abraham 4π 0 × − × mogeneous materials in the rest frame of matter, and ǫµ 1 = ∂ [P B + E M]= − ∂ S implies a momentum E/nc for the photons. It also com- 0 × × c2 t plicates the issue of relativistic invariance. The origi- is what is usually called Abraham’s force. The exis- nal Abraham’s discussion was done partially from a pre- tence of this force was apparently supported by some relativistic point of view, and includes a comparison with nice experiments [98, 99] in 1975, but in section VIII we Cohn’s old fashioned electrodynamics. The tensor ex- show that those experimental results are better explained pressed only in terms of the fields (see the components within our approach. below) cannot be written in covariant form (this was For the forthcoming discussion it is convenient to in- shown again recently in [56] by an explicit computation). troduce a second tensor, which may be considered for Invariant versions of the tensor which reduce to the orig- non linear materials, and reduces to Abraham’s proposal inal tensor in the rest frame, may be written [39, 54, 55] for isotropic linear materials. It is given by for a homogeneous isotropic medium at the expense of in- troducing a dependency on the velocity of the medium. 1 T µν = (T µν + T νµ ) + ∆T µν , (75) Pauli [39] gives the expression AbrII 2 Min Min AbrII

µν µν µν TAbrI = TMin + ∆AbrI (66) µν 1 β αµ ν αν µ ∆TAbrII = 2 FαβU (H U + H U ) c 8π h where β αµ ν αν µ HαβU (F U + F U ) . (76) (ǫµ 1) − i ∆µν = − F U αU (U µHβγ U γHβµ)U ν AbrI − 4πc4 βα γ − This tensor is symmetric for arbitrary polarization (67) and magnetization. Usually the symmetrization of with U α the four-velocity of the medium. This tensor is Minkowski’s tensor in (75) is not discussed because as considered only for a homogeneous isotropic medium, in already mentioned the spatial part of this tensor is sym- which case it is symmetric. In the rest frame the only metric for isotropic homogeneous media, but if a symmet- µν non zero components of ∆TAbrI are ric tensor is required for non linear materials this step is necessary. In the rest frame the equations correspond- ǫµ 1 1 ing to (65) and (69) still hold but now Maxwell’s tensor ∆T i0 = − S = (E H D B) . (68) AbrI − c i 4π × − × i appears explicitly symmetric In this reference frame the components of the tensor are ij 1 ij TAbrII = (E D + B H)δ given by (65) and by 8π · · 1 00 0i (EiDj + Ej Di + BiHj + Bj Hi) . (77) TAbrI = uP , cTAbrIeˆi = S (69) −8π Using (51), the corresponding force density is and µ µ µ 1 fAbrII = fMin + ∆fAbrII (78) T ij = (E D + B H)δij AbrI 8π · · where 1 (EiDj + HiBj ) . (70) µ µν 1 µν νµ −4π ∆f = ∂ν ∆T + ∂ν (T T ) . (79) AbrII − AbrII 2 Min − Min We note that the introduction of U α in the formulation For the space time components in the rest frame we have, is a rather artificial procedure, since this velocity is only 0 defined if the medium is taken to be a rigid body, and ∆fAbrII = 0 (80) this enters in contradiction with special relativity. In consequence, there is an intrinsic approximation in the and approach. The force density corresponding to this tensor 1 ∆f i = ∂ [D B E H]i is given by AbrII 4π 0 × − × 1 µ µ µν ij ji f = f ∂ ∆T . (71) + ∂j (TMin TMin) . (81) AbrI Min − ν AbrI 2 − 12

The first term in this equation is again Abraham’s force. for a while to have settled the issue. But soon appar- The second term in (81) ently independent arguments appeared to back Abra- ham’s proposal, notably one based on CMMT due to i 1 ij ji ∆fSymm = ∂j (T T ) (82) Balazs [59]. Let us briefly discuss both lines of reason- 2 Min − Min 1 ing. = ∂ (E D E D + B H B H ) 8π j i j − j i j i − i j vanishes for isotropic linear materials. For non linear D. The von Laue-Møller argument materials it has various unpleasant characteristics. For example, there is a contribution of the form ρ D which b i Around 1950 von Laue [28] and Møller [29] put forward can be hardly justified on physical grounds. a different criterion to pick the correct electromagnetic In the rest frame the energy density and the energy energy-momentum tensor. They basically argue that be- density current for either of the covariant Abraham ten- ing an observable quantity, the velocity of the electro- sors coincide with those of Minkowski’s proposal. Due magnetic disturbance in matter should behave in a way to (72) and (80) the energy flux equation for both of consistent with relativity. Let us first discuss the argu- them is given again by (64) in that reference frame. In ment as stated by von Laue. He treats the case of linearly other reference frames, additional terms depending on polarized sinusoidal plane waves. He observed that there the velocity should be added to the energy density and are two four-vectors that may be related to the wave. One the energy current density, and in consequence the power is the wave-four-vector kµ, k0 = ω/c. The other should released to matter is also modified. be constructed out of the velocity of energy propagation On the theoretical side the reasons used to justify W = (1/u)S. The wave-four-vector is space-like, while Abraham’s requirement of symmetry of the tensor do not the beam four-velocity should be time-like. Using the survive a careful analysis. They are explained for exam- Lorentz transformations (29) and (30) he shows that the ple in [54]. Here is the core of Abraham’s argument (the beam velocity W satisfies Einstein’s addition rule when translation is ours): changing the frame of reference. From this he deduces “In the presentation of the Maxwell-Hertz theory usu- that the energy current density associated to the system ally the preference is given to the Hertz’s symmetrical in any reference frame should be Poynting’s vector. He stress tensor. The asymmetric Maxwell tensor implies also shows that g = (u/ω)k. Abraham’s energy cur- that inside a crystal torques are exerted by neighbor par- Min rent density (here meaning the one obtained from (66) ticles when placed in an electric field. Those torques or (75)) in a moving frame is not Poynting’s vector. von should be equilibrated by elastic torsion forces, unknown Laue took this result as an evidence that Abraham’s ten- in the ordinary theory of elasticity. An asymmetric elec- sor is not adequate to describe this system. To illus- tromagnetic tensor would be allowed only within the trate his point von Laue considered the reference frame framework of an appropriately extended elasticity the- in which W = 0. If the momentum density were c−2S, ory. As the example treated above shows, the fact that the momentum of a wave-packet would be time-like and torsional forces act on a crystal in an electric field in no the energy would be positive. What actually happens in way allows to infer the existence of an internal electrical such a frame is that E = H = u = ω = 0 and g = 0 torque.” Min since neither D nor B vanish . We note that this is a con-6 This argument is incorrect because it relies on the be- sequence of the mechanical difference between a particle lief that in ordinary elasticity theory the stress tensor and a wave-packet. The particle can be stopped in any must always be symmetric. Actually, the stress tensor is reference frame. The wave-packet cannot be stopped; it proven to be symmetric under the hypothesis that there is always moving with respect to the medium. We also is no internal torque or spin density [57, 58]. An asymme- note that U = pv should be expected for a non-dispersive try of the stress tensor may be developed to compensate wave and that pµ kµ is what is consistent with Planck an internal torque. The existence of an internal torque and De Broglie relations.∝ when P is not parallel to E, or M is not parallel to B Møller discusses the same idea in a more abstract way. follows just assuming that the torque on a particle in He also observes that if some energy-momentum ten- the crystal depends only on the local E and B fields, sor T µν accounts for an observable energy density which and is not affected by the state of the other particles. moves in a medium, then the quantity W i The reason also mentioned by some authors that sym- metry of the total energy-momentum tensor is required T 0i Si W i = c = T (83) for the conservation of angular momentum was long ago T 00 T 00 overruled by the discovery of spin. But as discussed in section II, this was not fully appreciated in the literature which corresponds to von Laue beam velocity discussed and many people still use the argument in this context. above should behave as a velocity under Lorentz trans- By the middle of last century, von Laue [28] and Møller formations. He proceed to show that this implies that [29] introduced a different argument which suggested that 00 0i α i cT cT only Minkowski’s ansatz but not Abraham’s is compati- W = (cγw,γww ) = ( , ) (84) 0 0µ 0 0µ ble with Poynting’s equation and relativity. That seemed T µT T µT q− q− 13 with γ−1 = 1 wiwi/c2) is a Lorentz vector although this paper is to show that the limitations of the CMMT w − it is not writtenp covariantly as a vector. For this to be already discussed make of hidden momentum an unnec- true it is necessary that the following relations are satis- essary hypothesis. fied. 00 ik j i T ij ij T STkST F. Alternative approaches gT = ( k )T STj ,T = k (85) STSTk STSTk i 0i i −1 i0 As we mentioned in the introduction, other energy- with ST = cT and gT = c T . In the case of an elec- tromagnetic wave propagating in a medium, Minkowski’s momentum tensors have been considered besides Abra- tensor satisfies both these equations, but Abraham’s does ham’s and Minkowski’s. The early pre-relativistic dis- not. This implies that Minkowski’s tensor describes in cussion on Maxwell’s tensor already has implications fact an object which moves as a whole in a relativistically on the definition of the force. As discussed by Pauli invariant way. In section IX we show that it corresponds [39], Maxwell and Heaviside preferred a non symmet- to the the electromagnetic wave plus the polarizations of ric expression for the stress tensor whereas Hertz opted the medium. for the symmetrized expression (77). In a relativistic setup the force is given by (52) and without specifying the form of the momentum density, the force density E. Balazs argument and hidden momentum is not uniquely defined. This in particular means that Abraham-Minkowski dilemma was already present in the early development of the theory. In 1953, Balazs [59] proposed a thought experiment Einstein and Laub, made a definite choice for Abra- with the aim to expose the inadequacy of Minkowski’s ham’s momentum expression. Based in the form of the momentum density expression. He argued that since force on elementary dipoles, proposed in the rest frame Minkowski’s momentum in matter is greater than in the stress tensor vacuum, photons crossing a dielectric block will pull the block instead of pushing it and in consequence the 1 1 T ij = (E2 + H2)δij (E D + H B ) . (86) CMMT will be violated. As we discussed in section II EL 8π − 4π i j i j the CMMT is not valid in general. In particular, it is never valid in the description of polarizable matter inter- This defines the Einstein Laub force, acting with the electromagnetic field because the polar- f i = ∂ T iν = ∂ T i0 ∂ T ij (87) EL − ν EL − 0 EL − j EL ization of matter induces a variation of the spin density ǫµ 1 1 i i i E P H M current which contributes to (8). This is illustrated in = fMin + −2 ∂tS + ∂ ( + ) . the examples of sections IX and X. Nevertheless Balazs c 2 · · line of reasoning has shown to have a great convincing Since Einstein and Laub gave no expression for the en- power over some people and it has even been accepted ergy density the power released to matter cannot be com- on equal footing with the experimental evidence in some puted in this case. review papers when discussing the status of the contro- In a long series of papers summarized in Ref. [8], de versy [19, 52]. The misunderstandings which introduces Groot and Suttorp approached the problem from the mi- the inadequate use of the CMMT when discussing elec- croscopic point of view. Their construction depends on tromagnetic momentum also have a prominent place in some hypotheses about the matter distribution and the the line of arguments which led to the introduction (and average procedure. For the electromagnetic field in a for the last fifty years the consideration) of the so called fluid (see Ref. [8], Eq.V-92) they propose the energy- hidden momentum. When hints appeared [11, 60, 61], momentum tensor, that in the framework of Maxwell-Lorentz classical elec- µν 1 αβ µν 1 µ να 1 µγ α trodynamics the CMMT may not hold for some systems, T = FαβF η + F H + [F DγαV GS −16π 4π α c2 they were dismissed in favor of this very speculative hy- µ α β 1 µ γ α β ν pothesis [11], which requires radical modifications of the D αF βV 2 V V DγαF βV ]V . (88) mechanical behavior of matter. To illustrate how hidden − − c momentum is introduced consider a charged plane capac- where V µ is the local bulk velocity of the fluid. In the itor at rest in a transverse magnetic field B. The electro- local rest frame this tensor takes the form (see Ref. [8], magnetic momentum of this system neglecting boundary Eq.V-148) effects is pem = EBV and points in the direction per- 1 1 pendicular to B and E. Here, V is the volume between T 00 = (E2 + B2) ,T i0 = T 0i = Si (89) GS 8π GS GS c the shells of the capacitor, and E is the electric field. ij 1 2 2 ij 1 Hidden momentum supporters argued that there is an TGS = [ (E + B ) B M]δ (EiDj + HiBj ) . opposite hidden momentum of equal magnitude so that 8π − · − 4π the total momentum vanishes and the CMMT holds. For The dependence of the last term in (88) on the local discussions of the electromagnetic momentum using this velocity affects the form of the force density which can- concept see for example [2, 11, 62]. One of the results of not be computed directly from (89). In absence of free 14 charges it is given by (see Ref. [8], Eq.V-156), There is another approach to the problem which con- vinces some authors that the controversy has been re- µ 1 µ αβ 1 µγ α solved. Those authors consider that both options for the fGS = Dαβ∂ F ∂ν ([F DγαV 2 − c2 momentum of the electromagnetic field are correct, but µ α β 1 µ γ α β ν that they are applicable to different situations. For ex- D F V V V DγαF V ]V ) . (90) − α β − c2 β ample, Gordon and Nelson try to identify Abraham’s ex- The first term in this equation is very interesting, and we pression with the mechanical momentum of light, and will turn back to it later but, even for constant V µ, the Minkowski’s expression with the pseudo-momentum or second term spoils the good properties of the first term. crystal momentum. Novak [49] talks about bare and In 1976, Peirls proposed yet another tensor based in an dressed radiation quanta, distinguishing between the analysis of the microscopic momentum in matter for non photon which would carry Minkowski’s momentum, and magnetic media, and concluded that within his approx- the wave packet which would carry Abraham’s. Simi- imation scheme the energy-momentum tensor should be larly, in [23, 51], Minkowski’s momentum is character- Minkowski’s symmetrized tensor. The force density for ized as the canonical momentum to be identified with this tensor is given by the quantum mechanical momentum, and it is proposed to identify Abraham’s momentum as the mechanical mo- ǫµ 1 mentum. We think that without obtaining a consistent f i = f i + − ∂ S + ∆f i (91) Symm Min 2c2 t Symm and physically reasonable description at the classical level such approaches which introduce the additional compli- i with ∆fSymm given by (82). The contribution to the cations of quantum mechanics, including those related power is to the correspondence principle, cannot settle the ques- tion. Moreover, as we show in the rest of this article, 0 1 i Minkowski’s expression is the one which gives a consis- f = ∂i[D B E H] . (92) Symm 2 × − × tent description of the electromagnetic momentum at the classical level. This implies from our point of view that As already discussed above, this incorporates to the force the mentioned distinction proposed in [23], and reviewed density terms which are of difficult interpretation. For in [50, 51], should be basically unnecessary. the symmetrized tensor the energy current density clearly is not Poynting’s vector. Besides those discussed above, other alternative ap- proaches have been proposed to discuss the energy- G. Comparison of the tensors momentum tensor of the electromagnetic field. These include the tensors proposed by Beck [6] and Marx and Before entering in the discussion of the adequacy of Gy¨orgyi [7], the works of Haus and Penfield[64, 65], Gor- the force density expressions discussed in the previous don [21] and Nelson [22], which have been influential in section to describe the outcome of the experiments, let the discussions on the subject. There are also construc- us compare Minkowski’s, Abraham’s and the other ten- tions introduced for specific situations, as for example sors at the light of our remarks in the introduction. First the work in [66] for dielectric fluids, and reference [68] we note that most of the tensors with the exception of on dispersive media, and many others. There is no space de Groot and Suttorp proposal focus on the constitutive in this paper to discuss properly the many details and relations between E and B and D and H, and use Poynt- different points of view included in all these articles and ing’s energy, which is adequate only to describe a time- many others that we are not mentioning. We only point independent homogeneous medium. This is a general fail- out that in our opinion none of those approaches has been ure because none of these tensors could be expected to widely accepted as a solution of the controversy. describe materials with permanent polarization. For some authors [14, 15, 17–20], the original From the point of view of relativistic invariance, we 00 0k −1 Abraham-Minkowski controversy has been considered note that TMin = uP and TMin = c Sk in any frame. conceptually solved on the ground that the division of the Minkowski’s tensor is written in terms of the fields and by total energy-momentum tensor into electromagnetic and construction transforms properly under Lorentz transfor- material components is arbitrary. Minkowski’s, Abra- mations. In consequence, the symmetrized Minkowski’s ham’s, or any other electromagnetic energy-momentum tensor related to Peirls approach also has these two prop- tensor has a material counterpart (with or without hid- erties. All the other tensors depend on the velocity of den momentum contributions), which adds to the same matter. If this velocity is taken to be constant and the total energy-momentum tensor and responds for any un- same for all points of the medium as is usually done, one wanted effect not accounted by the chosen tensor. This has to raise the question of the compatibility with rel- point of view denies the fact that the force and torque ativity. If the velocity is taken to be the local velocity densities are, within the experimental limitations, ob- of matter, the correspondence with the intended expres- servable quantities and that the field and matter con- sions of the energy density, the energy current density, tributions are constrained by (51) and (50) as discussed and the momentum density are valid only in the local rest in section IV. frame, and unwanted terms appear in the force density. 15

Even for constant velocity the components of the energy- V. FORCE AND TORQUE ON DIPOLAR momentum tensor and the force density gain additional SYSTEMS terms which depend on the velocity. In particular, the energy current density for any of Abraham’s covariant A. Conditions on the force tensors is Poynting’s vector only in the rest frame. In this section we discuss the general conditions for In our opinion the key point in the controversy is the the splitting (49) of the energy-momentum tensor and proper identification of the true expression of the force the force on dipolar. density consistent with the experiments. In the next sec- First, the separation between matter and field should tion we present our approach to this problem, and in be independent of the frame of reference. This implies section (VIII) we discuss some of the experimental ev- that each part of the energy-momentum tensor must be µ idence on the force. We have already mentioned some a four-tensor and the force density f a four-vector. problems associated to the force density obtained from Second, the energy and momentum conservation Minkowski’s, Abraham’s and the other tensors but we means that equations (50) and (51) must be identities postpone a detailed comparison up to these sections. for any solution of the dynamical equations. On the field sector, this implies that (51) must be an identity for any solution of of Maxwell equations and that therefore the As explained in section II, we do not regard symmetry µν force density is completely determined by Tfield and vice as a necessary condition on the energy-momentum tensor versa. For example, if the energy-momentum tensor of of the electromagnetic field but on the contrary we ex- the field has the form of the standard one in the vacuum pect this tensor to be in some situations non symmetric. TS (Livens tensor), as it is some times suggested [69], the Nevertheless, we note that the only truly symmetric ten- force density should be the Amp`ere-Lorentz force on the sors of the ones discussed are the symmetrized Minkowski total current density; see Eq. (57). µν tensor and the one we called TAbrII. The usual Abraham Third, for macroscopic matter the microscopic dynam- tensor is only symmetric for isotropic homogeneous me- ical equations are unknown, or in any case their solu- dia, and de Groot and Suttorp tensor is non symmetric. tions are unknown. So, (50) becomes a true dynamical These last authors correctly addressed in their book the equation useful for determining the motion. The split- lack of justification for the symmetry constraint. ting should be such that the energy-momentum tensor of matter keeps the standard form for the different kinds of materials (for example for a continuum), excluding any The detailed balance of the momentum exchange of the electromagnetic field with matter in Balzs thought exper- kind of exotic dynamics. This requirement indicates that there is only one sensible splitting. iment cannot be discussed based only in the expressions Fourth, since the macroscopic electromagnetic fields of the momentum density and the currents of momentum µν µ density of the fields. More input of the matter configu- F are generated by the free current density jf and ration is needed. In sections IX and X we discuss two the polarizations Dµν , a tensor that deserves the name of energy-momentum tensor of the field should not de- specific examples for which it is possible to compare the µ different approaches. pend on matter variables, besides jf and Dµν . Livens’ and Minkowski’s tensors fulfill this requirement. Abra- ham’s fulfills the requirement only in the reference frame The importance of distinguishing between the electro- in which the matter is at rest. Velocity dependent terms magnetic wave and the electromagnetic field acquires rel- have to be added for obtaining a tensor that transforms evance in the light of the argument used by von Laue and properly [3, 55]. Abraham’s tensor makes sense only for Møller for rejecting Abraham’s tensor [28, 29]. When ap- non-rotating rigid matter. Maxwell’s equations are valid plied to a wave in a linear medium, Abraham tensor does in any reference frame, but Abraham’s tensor introduces not describe an isolated relativistic moving object. On a particular frame in which matter is at rest. the contrary, Minkowski’s tensor does. Our interpreta- Fifth, the problem is determined by the force density tion of this fact is not that Minkowski’s tensor is the cor- which is restricted by theoretical and experimental re- rect energy-momentum tensor of the field. In our opin- quirements. For example, the force density should be ion, Poynting’s energy as advanced in subsection III E is consistent with the fact that the fields that produce the not purely electromagnetic but also includes the change effects on charges and currents are E and B, not D or of the energy of matter due to polarization. Correspond- H. Terms in which the free current density, the polariza- ingly the object described by Minkowski’s tensor is the tion or the magnetization are coupled to D or H are not whole wave with the disturbances on matter produced by acceptable. For the electric sector there have never been the wave included. This point of view is confirmed by our any doubts on this. For the magnetic sector there was a results of section VI. Møller-von Laue criterion although long dispute during XIX century starting with the molec- physically relevant does not allow by itself to identify the ular currents proposed by Amp`ere, but nowadays this is- energy-momentum tensor that should be ascribed to the sue has been settled [70–72]. Minkowski’s symmetrized electromagnetic field. tensor, which approximates one proposed by Peirls [9], 16 results in a power term (E + D) j /2 which is of course d = γǫ m and d = d = γd , where γ is the − · f ij ijk k 0i − i0 i unacceptable. The Einstein-Laub force has terms that usual relativistic factor γ−1 = 1 v2/c2. This object − couple M and H that are also unacceptable. transforms as a four-tensor in thep limit R 0. In fact → 1 dαβ = γ (xαjβ xβjα)dV + (R2) . (101) B. Dipolar systems c Z − O

The second order term is due to je. To gain insight, let us discuss first the microscopic de- Let us study now what happens when the system is im- scription of a localized system with dipolar moments. We mersed in external electric E and magnetic B fields. The consider a localized system of charges, whose total charge force on the system is given by the Lorentz expression is zero, observed from a distance much bigger than its size R. The properties of the system, like forces felt, can be 1 F = (ρE + j B) dV , (102) expanded in powers of R, that correspond to the mul- Z c × tipole expansion. To fix ideas let us assume that the system is localized near the origin, the power is

r > R = ρ(r,t) = 0 and j(r,t)=0 , (93) W˙ = j E dV , (103) ⇒ Z · and the torque is ρ dV =0 . (94) Z 1 t = x (ρE + j B) dV . (104) The electrical dipole moment of the system is Z × c × If the fields change slowly over distances of the order of d = xρ(x,t) dV . (95) R the system behaves as a dipole. We need also the Z multipole expansions obtained from the Taylor series of The current density can be split in an electrical current the fields that vanishes when ρ vanishes and a magnetic one that E = E(0) + x E + ..., (105) is independent of the charge density, j = je + jm. The · ∇ corresponding continuity equations are B = B(0) + x B + .... (106) · ∇ ∂ρ + j = 0 (96) ∂t ∇ · e C. Forces and torques on dipoles and The magnetic effects are those depending on jm. This field is divergence-less localized and with closed field jm =0 . (97) ∇ · lines. Since the whole space can be subdivided into in- From the definition (95) it follows that finitesimal closed current tubes, it is enough to treat the case of current loops. The general case can be then ob- ˙ tained making the usual substitution jdV idl. For a d = je dV . (98) ↔ Z closed curve an area pseudo vector is defined by C A On the other hand xi dxj = ǫijk k (107) IC A jm dV =0 . (99) Z or equivalently by The magnetic dipole moment is 1 = x dl . (108) 1 A 2 IC × m = x jm dV . (100) 2c Z × The magnetic moment of a loop of current i is m = i/c . The torque on the loop is then A Conditions (94) and (99) imply that the dipole moments C are independent of the choice of the origin. Equation (98) i tm = x [dl (B(0) + x B)] implies that je is of first order in R. c IC × × · ∇ The definitions of the dipolar moments depend on the i = B(0) (109) reference frame. It is important to determine how they cA× change under Lorentz transformations. For each refer- = m B(0) . ence frame the system as a whole moves with some ve- × locity v. Let us define the antisymmetric object dαβ, The same expression is valid in the general case. 17

Analogously the force on the loop is then the power on the dressed dipole is C i dUdressed ∂E ∂B Fm = dl (B(0) + x B) = d m . (117) c IC × · ∇ dt − ∂t · − ∂t · i = [ (B ) ( B) ] The momentum can be treated in an analogous way. c ∇ · A − ∇ · A Recognizing that the troublesome term in (113) repre- = ( B) m . (110) ∇ · sents a potential momentum that has to be added to the Also in this case the same expression is valid in the gen- bare momentum to get the momentum of the dressed eral case. dipole The power on the current loop is 1 p = p + p = p d B , (118) dressed bare i bare − c × W˙ m = i E dl = i E dS I · Z ∇× · C S then the force on the dressed dipole is i ∂B i ∂B(0) S = d = + ... dp −c ZS ∂t · −c ∂t · A dressed = E d + B m . (119) ∂B dt ∇ · ∇ · = m . (111) − ∂t · Equations (115) and (117) suggest a reinterpretation As before this result is valid in the general case. of equation (35), in which the second term on the left The torque on the electric dipole is hand side is subtracted from the field energy because it 1 pertains to matter, and the last two terms on the right te = x (ρE + je B) dV hand side are incorporated to the power transfered from Z × c × the field to matter. This agrees with the discussion we = d E(0) . (112) × have anticipated in section III E. In section VI B, we show that such an interpretation emerges naturally after the Note that the term with je is quadrupolar. The force on the electric dipole is covariant force density in matter is identified. In appendix A, we show that these troublesome terms 1 indeed may be seen as the energy and momentum of Fe = [ρ(E(0) + x E)+ je (B(0) + ... )] dV Z · ∇ c × the electromagnetic fields produced by the dipoles them- 1 selves. = d E + d˙ B + ... · ∇ c × Before turning our attention to the continuum, let us 1 stress that here we are considering only the electromag- = d ( E)+ E d + d˙ B − × ∇× ∇ · c × netic properties of the dressed dipole. To construct a 1 ∂ mechanical model, the internal stresses which also con- = ( E) d + (d B) . (113) ∇ · c ∂t × tribute to inertia should be included. For the treatment of related problem of a point charge and the resolution The power on the electric dipole is readily obtained of the 4/3 factor problem see reference [73].

W˙ e = je (E(0) + ... ) dV = d˙ E Z · · ∂E ∂ D. Torque, force and power densities in matter = d + (d E) . (114) − · ∂t ∂t · The force and torque acting on the dipole moments of Comparing (110) with (113), and (111) with (114) we a microscopic piece of polarizable matter give place to a note that the electric results have additional terms that torque density and a force density acting locally on each are time derivatives. Such terms would have the unde- element of the material. The fields E and B discussed sirable consequence of spoiling the relativistic covariance in the previous section do not include the self fields pro- of the four-force on the dipole. This one would be ex- µ −1 µ αβ duced by the dipolar system itself. For an element of pected to be K = 2 ∂ F dαβ. Lorentz covariance material the self fields are negligible because they are is secured if one recognizes that those terms are actually proportional to the volume of the element. So the EM contributions to the momentum and energy of the dipo- fields in the following discussion may be taken as the total lar system. Let us consider first the potential energy, macroscopic fields. The torque on an element of polar- φ(x) φ(0) E x, ≈ − · ized material is calculated using (109) and (112). The torque density is obtained after dividing by the volume Ui = φρ dV = E d . (115) P M Z − · of the element. In terms of and it is written as

If one defines the energy of the dressed dipole to include τd = P E + M B . (120) × × the interaction or potential energy This is clearly what should be expected and the only U = U + U = U E d (116) physically sensible result. It is easy to see that this is the dressed bare i bare − · 18 spatial part of the torque density four-tensor E. Energy-momentum tensor of matter τ µν = DµαF ν DναF µ . (121) d α − α Excluding the time-derivative terms which come from The temporal components are given by (113) and (114) in (124) and (123) does not mean that τ 0k = ( P B + M E) . (122) they do not have a mechanical effect. They have to be d − × × k incorporated to the energy and momentum of matter, The temporal part of the torque plays an important role which become dependent on the fields. In the local rest in disentangling the apparent paradox discussed by Bal- frame of matter the potential terms are added to the azs (see our treatment of a wave-packet in section IX ). energy and momentum of bare matter The force and power densities on an element of po- larized material should be computed starting from (119) uM(P, M, E)= ub(P, M) P E (128) − · and (117). Dividing by the volume one gets and fd = Pk Ek + Mk Bk (123) ∇ ∇ 1 g (P, M, E, B)= g (P, M, E) P B . (129) and M b − c × 0 ∂E ∂B cfd = P M . (124) We note that in general the dynamics of matter implies − · ∂t − · ∂t that P and M also depend on the field. This is the case In covariant notation the force density four-vector is for example for a material with linear polarization dis- cussed in subsection III E. The energy-momentum tensor µ 1 µ αβ f = Dαβ∂ F . (125) of matter should have the following form d 2 The Lorentz covariance is spoiled if the force on the µν −2 µ ν µν µν TM = c u˜Mu u + P ∆TFM (130) bare dipoles is used, as has been done in some treatments − of the problem [21, 51]. This shows the importance of im- whereu ˜M is the energy density of matter in the rest posing that the splitting of the energy-momentum tensor frame, uµ is the four-velocity and in the rest frame of in matter and field components should be Lorentz invari- matter, ant. Note that the derivative in (125) is applied on the k0 electromagnetic field. We note that (125) was one of the ∆TFM = (P B)k . (131) × terms in the force density (90) advocated by de Groot and Suttorp [8] for a neutral dielectric fluid. The first term in (130) describes the energy-momentum The total force density on matter is obtained adding flux due to the drift of matter. The last term is the potential momentum density; when it is absent P µν is the force on the free charges with the force on the dipoles µν a four-tensor, but in general only TM is a four-tensor. µ µ µ 1 µ ν 1 µ αβ When there is no other mechanism of energy-momentum f = f + f = F ν jf + Dαβ∂ F . (126) f d c 2 transfer besides the drift of matter, P µν is the stress. In It is easy to verify that (126) is related to Amp`ere- this case, it is purely spatial in the rest frame. In other Lorentz’s (58) and Minkowski’s (63) forces by frames it contributes to the inertia of the system [73, 74]. µν µ µ µα ν It is worth remarking that P is obtained using New- f = fAL + ∂ν (F Dα ) , ton’s second law (1) and the equations that determine µ µ µ 1 αβ the mechanical behavior of the system. If there are other f = f + ∂ ( DαβF ) . (127) Min 4 mechanisms of energy-momentum transfer, for example In consequence, the total force on a piece of material if there are electrical currents, heat flow or waves P µν held in vacuum obtained from any of these expressions may have temporal components even in the rest frame. is the same. Abraham’s total force is different in this In the next section, we use our knowledge of the force situation. Locally, the obtained f µ is not the same as to define the energy-momentum tensor of the electromag- the Amp`ere-Lorentz’s force on the bound charges and netic field which is consistent with Maxwell equations and currents. If one considers a piece of material subdi- momentum conservation. vided into infinitesimal elements, the dipolar moment and the force that each element feels are determined by the surface charges and currents on the element. Only VI. THE ENERGY-MOMENTUM TENSOR OF when considering the total force is that the contributions THE FIELD of adjacent elements cancel out; what remains are the contributions of the bulk and of the external surface of A. The energy-momentum tensor of the field in the piece of material which correspond to the Amp`ere- matter Lorentz total force. To discriminate experimentally be- tween Minkowski’s, Abraham-Lorentz’s and our proposal Now we are in a position to determine the energy- the local force density should be measured, not only the momentum tensor of the electromagnetic field in matter. total force. The matter contribution to the total energy-momentum 19 tensor satisfies (50) with f µ given by (126). The force which differs from Poynting’s conservation equation (39) density on the dipoles is not the same as the force on and also from equation (64) deduced for Minkowski’s ten- the current density of bound charges. The right hand sor for non linear materials. Note that in general there side of (57) is not minus the total force on the matter are contributions to the power different from the Joule (126) and the straightforward identification valid in vac- term on the free charges. Our results are valid for any µν uum, of TS as the energy-momentum of the field does kind of material in any kind of condition: ferromagnets, not hold. The true energy-momentum tensor of the elec- saturated paramagnets, electrets, matter moving or at tromagnetic field is given by rest, solids, fluids, absorbing and dispersive media, etc. In each case the polarization tensor should be identified. µν 1 αβ µν 1 µ να Poynting’s equation (39) may be reconstructed from our T = Fαβ F η + F αH . (132) F −16π 4π result in the particular case of linear polarizabilities, that is, when the polarization tensor D is proportional to because it satisfies αβ the field tensor F µν . µν µ Let us return to Poynting’s energy density. When the ∂ν T = f . (133) F − matter is immersed in electromagnetic fields the energy as can be seen after a simple manipulation of Eq. (57) us- density of matter changes. This energy difference can be ing Bianchi’s identity. Newton’s third law between mat- calculated integrating our power expression (124). For ter and field holds. time-independent linear polarizabilities (19) and (20) the For this tensor, the energy density is work done on matter does not depend on the way the fields change in time, it depends only on the final values 1 of the fields u = T 00 = (E2 + B2)+ E P . (134) F 8π · ∆uM = uM(E, B) uM(0, 0) = dwd 0i − Z The energy current density cTF = S = cE H/4π and the momentum density c−1T i0 = g = D B/×4πc coincide F × = (dE P + dB M) with Minkowski’s expressions (59) and (60). Maxwell’s − Z · · stress tensor is given by, 1 = (E P + B M) . (137) 1 −2 · · T ij = (E2 + B2)δ B Mδ F 8π ij − · ij This includes the electrical potential energy density E 1 P − · (EiDj + HiBj ) . (135) − 4π 1 ∆u = E P + (E P B M) . (138) and also differs from Minkowski’s expression. The differ- M − · 2 · − · ence between our energy density and the energy density for the vacuum, u u = P E, is the negative of the elec- Poynting’s energy density is − S · trostatic energy density of the polarization. As discussed 1 in the previous subsection, this energy is subtracted from uP = u + ∆uM = uS + (E P B M) . (139) the energy of the microscopic field and should be consid- 2 · − · ered part of the energy of matter. This makes physical Therefore it results that Poynting’s energy density cor- sense because it contributes to the inertia of matter in responds to a mixed entity with contributions of “fields exactly the same way as nuclear interaction energy con- plus polarizations”. On the other hand Minkowski’s is 00 tributes to nuclei mass. Note that there is no similar the only relativistic tensor for which T = uP and magnetic term since no potential magnetic energy exists. cT 0k = Sk in any frame of reference. For an electro- The difference between the momentum density in mat- magnetic wave propagating in a medium it makes sense ter and vacuum g gS = P B/c is the opposite of the to include the polarization energy of matter as part − × interaction momentum of the dressed electrical dipoles. of the wave energy. So as we suggested in subsection IVG Minkowski’s tensor properly represents the energy- momentum tensor of the EM wave in a non-dispersive B. The meaning of Poynting’s energy density and medium. It is useful for calculating the theoretical force Minkowski’s tensor on the wave, but it cannot be used for determining the force on matter, since the wave energy has a component The energy current density is given in our approach by which belongs to matter. One has to use (132) for doing Poynting’s vector S = cE H/4π. The energy density that. Note that in general, it is the force on matter the × (134) is different from Poynting’s expression (37), uP = one that can be measured. (E D + B H)/8π. From (35) the energy conservation Minkowski’s energy-momentum tensor may be written · · equation in our formulation reads as, ∂u ∂E ∂B 1 + S = E j + P + M (136) T µν = T µν + D F αβηµν . (140) ∂t ∇ · − · f · ∂t · ∂t Min F 4 αβ 20

In this view the force density on the wave is The orbital angular momentum equation (2) for the EM field is therefore µν 1 µ α 1 µ αβ 1 µ αβ ∂ν TMin = F αjf ∂ F Dαβ + ∂ (F Dαβ) µνα µν µ ν ν µ − c − 2 4 ∂αLF = τd x f + x f . (143) 1 1 − − µ α µ αβ γδ µν = F αjf + ∂ χαβγδF F . (141) − c 8 The dipolar torque density is canceled out by TF ’s asym- metry in the spin equation (4) of the field which reduces The second term is due to refraction. In particular, the to EM wave propagating in a time-independent homoge- µνα µν neous medium conserves its energy and momentum. No ∂αS = τ . (144) F − c magnetostriction or electrostriction effects are described by Minkowski’s tensor [45], but they appear when using where the tensor τc represents an additional torque den- (126) and (132) (see section VIII). sity acting on the currents that could exist. The spin den- Poynting’s equation does not hold when the polariz- sity is not coupled to the torque density on the dipoles. abilities are not linear as for example in the well known There are phenomena, like the absorption of circularly case of optical dispersive media [67, 68, 70]. Our result polarized light, in which the electromagnetic spin is rel- µν should be a good starting point for a fresh approach to evant. In those systems, the form of τc depends on the study such cases. nature of the microscopic coupling responsible for the Further evidence of the inadequacy of Poynting’s en- effect. As a result optical angular momentum may be ergy and Minkowski’s tensor for non-linear systems is transferred to matter [75–78]. As already said, the argument used in favor of Abra- obtained calculating the force on a piece of dipolar ma- µν terial in external fields as minus the gradient of the total ham’s tensor that the field T should be symmetric for energy (field and matter) when the piece is displaced. If angular momentum conservation is incorrect. The field the material is linear this energy can be calculated using exerts a torque density (120) on polarized matter, and Poynting’s energy density, since the polarization terms in the corresponding reaction acts on the field. What is Poynting’s energy density are equal to the polarization- conserved is the total angular momentum of matter and dependent terms in the energy density of bare matter. field (including spin), not the orbital angular momentum This relation does not hold for non-linear systems; for of the field alone. It is precisely the asymmetry of the example for an electret or a permanent magnet this pro- energy-momentum tensor what couples in (2) the orbital cedure using Poynting’s energy yields a force that is one angular momentum with the torque density. Our tensor half of the correct result. On the other hand Minkowski’s has the right asymmetry to ensure angular momentum force, obtained from the first equation of the set (141), conservation. gives the correct result even for this case. Therefore there is an inconsistence if one pretends to calculate the VII. THE AVERAGE METHOD total field-dependent energy of non-linear systems using Minkowski’s (Poynting’s) energy density. Our force den- sity (126) matches the result obtained by evaluating the The traditional method to obtain the phenomenolog- energy using our field energy density and the energy den- ical fields, P and M which appear in the macroscopic sity of the dressed matter which includes the dipolar po- Maxwell equations is by making averages on the micro- tential energy density E P. scopic electric and magnetic dipoles. In this approach, − · the covariant treatment presents some problems in how to define consistently the dipole moments from microscopic charges and currents, particularly when the dipoles are C. Orbital angular momentum and spin moving or changing in time [8, 70]. To understand bet- ter the nature of the energy-momentum tensor obtained The identification of the spin density of the electromag- in the previous section, it is convenient to look at its netic field, and in general of gauge fields, is an interest- relation with this approach. In the present section, we ing problem, which even in the vacuum has not yet been re-obtain our result in a form which is independent of any resolved satisfactorily. The canonical spin density which microscopic model of the material and does not require emerges from Noether’s theorem [24] is gauge-dependent, to use the force on the dipoles as an input [79]. We sup- and up to now there is no consistent and gauge-invariant pose that the macroscopic quantities may be obtained definition of the kinetic spin of the electromagnetic field. by taking the average of microscopic fields and forces Nevertheless, from the kinematic point of view it is clear over small regions of space-time, and that the macro- that whatever expression may it have, equations (3) and scopic force density may be expanded in the macroscopic (4) for the field quantities should be satisfied. fields and their derivatives. In the dipolar approximation µν The non-symmetric part of TF happens to be equal only the first derivatives are relevant. We define the free to the torque density on the dipoles, charge current density as the four-vector that couples to the average field and the polarization density tensor as T µν T νµ = τ µν . (142) the quantities that couple to the first derivatives in the F − F d 21 expansion. Then we obtain self-consistently the expres- emerges that they are equal to the corresponding macro- sions for the free and bound (dipolar) charge and current scopic quantities used in the previous sections. In partic- densities and for the energy-momentum tensor. Finally, ular we have, F¯µν = F µν . Using (148) one immediately we show that the polarization tensor that was defined obtains that Maxwell’sh equationsi (54) are valid for the as the coupling parameters of the force should indeed be averaged macroscopic fields and the averaged currents interpreted as the polarization-magnetization tensor. 4π ∂ F¯µν = jµ (149) ν c h i A. Space-time average and that Bianchi’s identity is valid for F¯µν and also for δF µν = F µν F¯µν . At a microscopic scale, we assume that for the electro- − magnetic field F µν and the current density jµ, Maxwell’s equations (54) and Bianchi’s identity (15) hold and that B. Macroscopic force the force density is given by Lorentz expression (53). Cor- respondingly, the energy-momentum tensor for the mi- We avoid any controversy about the macroscopic force croscopic electromagnetic field is the standard tensor TS by making the sole hypothesis that the microscopic force (55). which satisfy identically ∂ T µν = f µ as a conse- ν S − L is given by the Lorentz expression. To obtain the average quence of (54), (15) and (53). The force density can alter- current density and the macroscopic energy-momentum natively be obtained from the energy-momentum tensor µ µν tensors of fields and matter we compute first the average of matter by fL = ∂ν TM if it is available. of the microscopic forces. Since the force (53) is quadratic The observable macroscopic quantities correspond to in the microscopic quantities, the result in this case is averages of the microscopic quantities over small regions not straightforward. We assume that the macroscopic of space-time. Similar treatments are usually done av- force density f¯µ on matter shares with the microscopic eraging only over space [8]. We average also over time force density the following four properties: 1) It is a local for two reasons: 1) Lorentz covariance is preserved natu- functional of the average field. 2) It is linear in the field. rally, 2) average over time is required to handle very fast 3) The direction of the force is determined by the field. processes as for example electrons moving around nuclei. 4) It transforms as a four-vector. Therefore the force The average is done the following way density can be expanded as a sum of terms in which the derivatives of the field tensor are contracted with matter- A(x) = A(x′)W (x x′) dx′ , (145) dependent tensors, and the free index belongs to the field h i Z − derivative tensors. In the dipolar approximation, we ne- where W (x) is a smooth function that is essentially con- glect the coupling with second and higher order deriva- stant inside a region of size R and vanishes outside tives which correspond to quadrupolar and higher order couplings. Taking into account Bianchi’s identity, we W (x) 0 , xµ > R = W (x)=0 , (146) prove that the most general expression for the macro- ≥ | | ⇒ scopic force density fulfilling the four conditions stated above takes the form W (x) dx =1 . (147) Z 1 1 f¯µ = F¯µ ¯jα + ∂µF¯αβD¯ . (150) c α f 2 αβ Inside the region W (x) W . At the scale R the microscopic≈ h i fluctuations are washed It is written in terms of two phenomenological quantities ¯α out, and therefore all the products of averages and fluc- related to matter, a four-vector jf and an antisymmetric ¯ tuations are negligible. That is, if δA = A A then four-tensor Dαβ to be identified later with the current B δA 0 and also B B . − h i density of free charges and the dipolar tensor density. hh i i≈ hh ii ≈ h i ¯ With these conditions it follows that Note that the temporal component of Dαβ is then iden- tified with the polarization vector P¯ by P¯k = D¯ 0k, and ′ ′ ′ ∂ν A = A(x )∂ν W (x x ) dx the spatial components with the magnetization vector M¯ h i Z − by D¯ ij = ǫijkM¯ k. ′ ′ ′ ′ With the use Bianchi’s identity of and some simple = A(x )∂ν W (x x ) dx − Z − algebra the force expression takes the alternative form = ∂′ [A(x′)W (x x′)] dx′ 1 − Z ν − f¯µ = F¯µ (¯jα + c∂ D¯ αβ)+ ∂ (F¯µ D¯ βα) . (151) c α f β β α + ∂′ A(x′)W (x x′) dx′ Z ν − In the microscopic formulation, the total energy- momentum tensor T µν splits in a contribution of the field = ∂ν A . (148) µν h i given by (55) and a contribution TM of the matter. The In this section we denote the macroscopic fields ob- average of the total microscopic energy-momentum ten- tained by averaging with a bar. From the discussion, it sor is the macroscopic energy-momentum tensor. How it 22

¯µν splits in a macroscopic matter term TM , and a macro- This last equation is satisfied identically by setting scopic field term T¯µν , should be determined by Newton’s F α α ¯ αβ third law. We write, j = ¯j + c∂βD (162) h i f T¯µν + T¯µν = T µν + T µν . (152) and F M h S M i and impose, ∆T µν = F¯µ D¯ να . (163) F − α ¯µβ ¯µ ¯µβ ∂βTM = f = ∂βTF . (153) These solutions are unique. Since (161) must be satisfied − for any field tensor, equation (162) follows. A tensor µν µν The average of the microscopic energy-momentum tensor X whose divergence vanishes identically (∂ν X 0) of electromagnetic fields is not the macroscopic energy- µν ¯µν ≡ could be added to ∆TF , but since TF can be at most momentum of the field. It may be expressed as, quadratic in the fields, and should vanish when the fields vanish such a tensor also vanishes. T µν = T¯µν + δT µν , (154) h S i S h S i Defining the current ¯µν where TS is the standard tensor built with the macro- ¯α ¯ αβ µν jb = c∂βD , (164) scopic fields and δTS is the standard tensor built with µν the fluctuation fields δF . For the actual macroscopic Maxwell’s equations for the macroscopic fields (149) are energy-momentum tensor of fields inside matter we write, written in the familiar form, T¯µν = T¯µν + ∆T µν , (155) F S F µν 4π µ µ ∂ν F¯ = (¯j + ¯j ) . (165) µν c f b where ∆TF is a possible polarization-dependent correc- tion to be determined self-consistently. The terms linear ¯α From here the identification of jf as the current of free in the fluctuations were neglected as explained above. To ¯α charges and jb as the bound current density is evident. obtain the macroscopic energy-momentum tensor of mat- Noting also that in terms of P¯ and M¯ defined above, the ter, the average fluctuation tensor of the field has to be familiar expressions ρ = P¯ and j = P¯˙ + c M¯ included and the dipolar correction must be subtracted, b b are obtained, the identification−∇ · of D¯ να with the∇× dipolar T¯µν = T µν + δT µν ∆T µν . (156) density is almost completed. M h M i h S i− F The dipolar or bound charge is conserved identically, α ¯ αβ Note that the contribution of the fluctuation fields δF ∂αjb = c∂α∂βD = 0. Since the average charge is con- should be considered part of the macroscopic matter. served, equation (162) implies that the free charge is also ¯µ µ ¯α This is the reason why f = f . conserved, ∂αjf = 0. Using (153), the macroscopic6 h forcei density is It is worth noting that this whole scheme leaves out processes, like ionization or capture of carriers, in ¯µ ¯µβ µ µβ µβ f = ∂βTM = f + ∂β[ δTS ∆TF ] . (157) which there is exchange between free charges and bound h i h i− charges. In those cases (150) does not hold. Using (54) and (53) the averaged microscopic force den- Now we proceed to discuss the energy-momentum ten- sity is sor. It is convenient to define the tensor 1 f µ = F µ ∂ F αβ . (158) H¯ αβ = F¯αβ 4πD¯ αβ , (166) h i 4π h α β i − ¯ ¯ ¯ αβ ¯αβ αβ which is related to the electric displacement D = E+4πP Making the substitution F = F + δF one gets 0k and the magnetizing field H¯ = B¯ 4πM¯ , H¯ = D¯ k and ij − 1 1 H¯ = ǫijkH¯k. With this notation, Maxwell’s equations f µ = F¯µ ∂ F¯αβ + δF µ ∂ δF αβ . (159) h i 4π α β 4π h α β i (165) become Using (149) in the first term of this last equation and 4π ∂ H¯ µν = ¯jµ (167) Bianchi’s identity in the second, the average microscopic ν c f force density is then and the macroscopic energy-momentum tensor of fields µ 1 ¯µ α µβ given by (155) and (163) is f = F α j ∂βδTS . (160) h i c h i − h i ¯µν ¯µν µν TF = TS + ∆TF Substituting this result in (157) and equating with (151) 1 µν ¯αβ ¯ 1 ¯µ ¯ να one finally gets = η F Fαβ + F αH . (168) −16π 4π 1 ¯µ ¯α ¯ αβ α This is exactly the result we obtained in the previous F α(jf + c∂βD j )= c − h i section, with the notational difference of the bar used in ∂ (F¯µ D¯ βα + ∆T µβ) . (161) this section to identify the macroscopic fields. − β α F 23

C. Densities of dipole moment The expression that fulfills this equation for any ˆt is ¯ ¯ Σm = cM nˆ . (173) To wholly recover the usual picture let us show that P × and M¯ are the densities of electric dipole moment and The usual computation of the magnetic dipole moment of magnetic dipole moment respectively without relying of the magnetic currents, in the model of microscopic dipoles (This is basically the reverse of the usual textbook computation). 1 1 m = x Σm dS + x ¯jm dV First we find the charges at the surface of a piece of 2c I × 2c Z × material. Near the surface P¯ , goes smoothly to zero in a distance of the order of R. The total dipolar charge is = M¯ dV , (174) Z obtained by integrating ρ = P¯ over the volume of b −∇ · the material. Because of Gauss’ theorem such integral is completes the interpretation of M¯ as the density of mag- zero since P¯ = 0 at the surface. Then the total dipolar netic dipole moment. charge is always zero. It is a bound charge that cannot leave the material. At a macroscopic scale (much bigger than R) there is a discontinuity at the surface. The dipo- VIII. TESTING THE FORCES lar charge at the surface is always opposite to the charge in the bulk. If σb is the surface density of polarization, A. Experimental results then The experimental researches deal with two related but 0= σb dS P¯ dV = [σb P¯ nˆ ] dS . (169) I − Z ∇ · I − · different aspects of the controversy. One is the momen- tum of light in matter and the other is the force density This can happen for any surface if that the electromagnetic fields exert on matter. It is the general view of most the authors which σ = P¯ nˆ , (170) b · worked on the subject that the reported experiments [80– 84] where the momentum of light is observed, support which of course is the usual expression given by the model Minkowski’s expression for the momentum of the pho- of microscopic dipoles. The electrical dipole moment of tons. The analysis done of Doppler and Cherenkov effects a piece of material is computed as always giving [85, 86] choose Minkowski’s momentum too. This view is shared even by some works which hold alternative posi- d = xσb dS + xρb dV = P¯ dV . (171) I Z Z tions on the subject as for example reference [87] which reports evidence for Abraham’s pressure of light, or ref- This completes the interpretation of P¯ as the density of erence [52] which argues in favor of Barnett’s proposal of dipole moment. different classical and quantum momenta. Since our mo- ¯ The magnetic current density in the bulk is jm = mentum density expression coincides with Minkowski’s, c M¯ . In addition, there is also a surface current those experimental results also support our approach. ∇ × density Σm that can be obtained in a similar way to that In relation with the force measurements the opinions used for σb. Let us consider a closed curve that is out- are much more divided. The interpretation of the exper- side the piece of material and is the border of a surface iments depends on the measurement of very tiny quanti- that cuts the material. The total magnetic current that ties and it is not a surprise that it has been difficult to ¯ crosses the surface is the flux of jm. Because of Stokes’ discard that other effects related to the collective behav- theorem such flux is the circulation of cM¯ in the curve, ior of the samples may be responsible for the outcome of which is zero. Therefore for any surface S that cuts the the experiments. material, the total magnetic current that crosses the sur- Also the theoretical analysis of the force density face is zero. At a macroscopic scale the bulk current is is confusing. In a recent paper [88], Jazayeri and opposite to the surface current. Let be the intersec- Mehrany presented a comparative study of the most pop- C tion of S with the surface of the piece of material, ˆt the ular force densities that have been proposed, Amp`ere- unitary vector tangent to the curve and nˆ the unitary Lorentz on the bound charges and currents, Einstein- C vector orthogonal to the surface of the piece. The uni- Laub, Minkowski and Abraham. Although we do not tary vector which is orthogonal to , and tangent to the agree with their whole analysis, it is interesting to con- C surface of the piece is nˆ ˆt. Then, sider their conclusions as a point of comparison. The ar- × ticle is very critical about Abraham’s formula character- 0= Σm nˆ ˆt dl + c M¯ dS ized as fanciful and devoid of physical meaning, Einstein- I · × Z ∇× · C S Laub tensor force which is found to be incompatible with

= [Σm nˆ ˆt + cM¯ ˆt ]dl special relativity, and Minkowski’s expression for exclud- IC · × · ing the interaction on the bound charges and currents inside homogeneous media. They spare Amp`ere-Lorentz = [Σm nˆ + cM¯ ] ˆt dl . (172) IC × · force stressing that it corresponds to the assumption that 24 the EM fields act on the total charges and currents. The B. Walker, Ladhoz and Walker experiment expression for the force density that we are proposing is different from all of those. It considers the local force on The experiment reported by Walker, Ladhoz and the dipoles and is compatible with relativity. Our force, Walker [98, 99] was performed by suspending a non mag- Amp`ere-Lorentz and Minkowski’s (when it is valid) each netic disk of high dielectric constant as a torsional pen- predicts the same total force on a piece of material iso- dulum. The suspension was a tungsten fiber and the disk lated in vacuum, but the local distributions are different. was located between the pole pieces of an electromagnet. Therefore they correspond to different stresses inside the The disk had a central hole and the outer and inner sur- material. Measuring the stress, that is magnetostriction faces were coated with a layer of aluminum. The outer and electrostriction, gives an empirical method for deter- surface was connected to ground by means of a thin gold mining which formula is the correct one. fiber and the inner one was connected to the supporting Most of the experiments that may be devised to mea- fiber. A potential V (t)= V0 cos(ωt + φ) was applied be- sure the force which the electromagnetic field exert on tween them. The movement of the disk was observed by matter will measure the total force on a polar or po- reflecting a laser beam from a small mirror attached to larizable sample. As mentioned in section V, such ex- the disk. periments do not allow to distinguish between Amp`ere- The electric field in the disk is Lorentz’s, Minkowski’s or our force density because in V (t) the three cases the total force is the same. Abraham’s E(r)= eˆr (175) and Einstein-Laub’s total forces are different. On the r ln(r2/r1) other hand a determination of the force density depends more on the details of thermal and mechanical properties where V (t) is the applied potential and r1 and r2 are the of the sample. inner and outer radii, respectively. Abraham’s force term (74), would produce a torque in the vertical direction For the specific case of an electromagnetic wave enter- given by ing a medium with a higher refraction index the question of weather the force is in the inward or outward direction ǫµ 1 ∂ tA = − r (E H) dV has a long history. Thomson and Poynting [89] suggested 4πc Z × ∂t × that the force is in the outward direction, a result which ǫµ 1 dV (t) was verified experimentally by Barlow [90] for oblique in- = − B dV . (176) −4πcµ ln(r2/r1) dt Z cidence. This is consistent with Minkowski’s momentum expression while Abraham’s momentum corresponds to Due to the symmetry of the system, the other contribu- an inward force. The experiment of Ashkin and Dziedzic tions to Minkowski’s and Abraham’s forces do not pro- [91] tried to elucidate this point by illuminating a liquid duce any torque. Oscillations of the disk were observed with a laser beam and determining the shape of the sur- with the frequency of the potential adjusted to the res- face. Their observations were consistent with a raising onance value of the mechanical system. The authors of of the liquid level and in a first analysis with an out- Ref. [98, 99] report consistency of the outcome of the ex- ward force and Minkowski’s momentum, but issues re- periment with the results predicted by (176) within the lated to the thermal and hydrodynamic response of the overall experimental error limit of about 10%. medium were invoked to object this conclusion [21, 92]. We can give an alternative explanation of this result, The same effect was observed in [93–95] but the opposite which makes no use of Abraham’s force term, and is fully result has been recently reported in [87]. Related exper- consistent with our approach. As observed in subsec- iments which also support Minkowski’s momentum are tion V E in a macroscopic description, the momentum of discussed in [96, 97]. matter acquires a term which depends on the field (see In relation with Abraham’s force term, there have been equation (129)). Correspondingly, the total angular mo- a few works which claimed experimental evidence for mentum is given in our approach by it. These include the work in references [98–102]. Of 1 them, Walker, Ladhoz and Walker’s experiment is the L = r gb dV r (P B) dV . (177) one which influenced most the opinion of the researchers Z × − c Z × × in favor of Abraham’s point of view. In the next subsec- Assuming that the only mechanism of energy and mo- tion we discuss the experiment and show that its result mentum transport in the disk is the motion of mass, the is also completely consistent with our construction. matrix P µν that appears in (130) is the stress which is As a further theoretical insight on the problem we re- purely spatial in the rest frame, P 0µ = P µ0 = 0. In the cently show [30] with a very simple computation that non-relativistic limit the bare momentum density is just −2 for an electromagnetic wave interacting with a conduc- gb =u ˜Mc v and the bare angular momentum Lb is the tor sheet in a dielectric medium if one supposes that usual classical angular momentum. The time derivative Maxwell’s equations and Ohm’s law are valid, only of the total orbital angular momentum L is equal to the Minkowski’s momentum density is consistent with mo- total torque tt applied to the disk. The opposite of time mentum conservation. derivative of the potential angular momentum acts as an 25 additional torque on the bare angular momentum parallel to the plates cancel. Our force is parallel to the plates, while in the other two cases there is also an inward dLb tension perpendicular to the plates. = tt + r ∂0(P B) dV . (178) dt Z × × It is interesting to consider a variant of this example where the dielectric is a liquid. The system is then made For the conditions of the experiment of two communicating vessels, where one is a capacitor

dLb ǫ 1 dV of parallel plates. In this case [45], the surface of the tt = − B dV . (179) liquid inside the capacitor rises above the level of the dt − −4πc ln(r /r ) dt Z 2 1 liquid outside. The level difference may be computed by This is only slightly different from the expected effect an elementary method using the energy of the capacitor of the Abraham term in the general case and exactly and corresponds to a pressure difference (ǫ = 1 for the the same in the conditions of the experiment of Walker, gas) Ladholz and Walker in which the disk was made of a ǫ 1 nonmagnetic material (µ = 1). ∆p = − E2 . (183) It is not easy, using linear materials, to take advantage 8π of the differences between (179) and (176) to distinguish Brevik [45] cites the work of Hakim and Higham [103] as experimentally between the two alternatives because, to supporting a different experimental result consistent with the effect to be measurable ǫ should be large. But we a force density proposed by Helmholtz [104]. The work can see from (74) that the difference between the Abra- in Ref. [103] uses a different geometry and a very intense ham term and the effective force in our approach is the electric field which may introduce nonlinear phenomena. presence in the former of a term that depends on the The rise of the liquid in this experiment is enough magnetization. Using a ferromagnet (or a paramagnet to exclude Amp`ere-Lorentz’s force since it cannot be close to the Curie point) with small electric permittivity achieved by applying a force density at the lateral sur- it could be possible to rule out Abraham’s force. faces. Both (180) and (182) agree with (183), the differ- ence being that Minkowski’s force predicts that the pres- sure difference appears at the liquid-gas interface inside C. Electrostriction the capacitor (the liquid is pulled from the top), while for our force it appears in the submerged border of the Consider a material with an isotropic dielectric con- capacitor (the liquid is pushed from the bottom). There- stant ǫ. Our force density is fore by measuring the local pressure inside the liquid it would be possible to discriminate between the two forces. (ǫ 1) (ǫ 1) 2 f = Pk Ek = − Ek Ek = − E , (180) Another experiment which may be used to establish ∇ 4π ∇ 8π ∇ the correct force density is to consider a cylindrical elec- Amp`ere-Lorentz’s is tret with polarization along its axis. Suppose the electret is immersed in a uniform electric field parallel to its polar- f = ( P) E , (181) ization. For ours (180), Minkowski’s (182) and Amp`ere- − ∇ · Lorentz’s (181) force densities the total force on the elec- and Minkowski’s is tret vanishes. Our force density itself vanishes and pro- 1 duces no stress on the electret. The bound charges for f = ( ǫ)E2 . (182) this system are located at the circular surfaces of the −8π ∇ electret. Amp`ere-Lorentz’s force predicts a stretching in In the textbook example of a capacitor of parallel the direction of the axis. Minkowski’s force is applied at plates partially filled with a slab of dielectric material, all the pieces of the material’s surface because there the all three forces give the same total force on the slab. dielectric constant varies discontinuously. Correspond- This force can also be obtained starting from the energy ingly, besides a longitudinal stretching it also predicts a of the capacitor as a function of the penetration of the radial stretching. It is possible to discriminate between slab. What is different in each case is the interpreta- the alternatives if the electret is sufficiently elastic or if tion. Our force (180) is exerted on the dipoles where the the internal stress is detected by optical means. magnitude of the electric field is varying. This happens at the border of the capacitor that is filled by the slab. Amp`ere-Lorentz’s (181) force would appear due to the D. Magnetostriction interaction between the free charges in the plates and the polarization charges on the surface of the slab. For a The experiment discussed above has a magnetic anal- homogeneous material Minkowski’s force (182) vanishes ogous. Consider a cylindrical permanent magnet magne- in the volume, but there is a surface force on the slab tized along its axis in an intense magnetic field oriented in because of the discontinuity of ǫ; the total force is de- the same direction. Our force density vanishes again and termined by the force on the surface of the slab that is predicts no additional stress on the magnet. Amp`ere- inside the capacitor since the forces on the two surfaces Lorentz’s (181) force density acts on the magnetization 26 current in the cylindrical surface and produces a radial though it is only a few percent of the total volume change stress. Minkowski’s force density acts again at the whole it should not be completely neglected. boundary and produces a longitudinal and a radial stress. It is possible to discriminate between the alternatives if the magnet is sufficiently elastic or if the internal stress IX. ELECTROMAGNETIC WAVES IN MATTER is detected by optical means. Consider now magnetostriction in a ferromagnet in the A. Wave-matter interaction case where there is no magnetizing field, H = 0. Then, as in any case with permanent magnets, Minkowski’s tensor ij A good test for the energy-momentum tensor and the is useless; it predicts no effect since TMin = 0. Instead force density presented in this paper is to compute the Maxwell’s tensor for our formulation reduces to momentum and energy exchange between a packet of 1 electromagnetic waves and a dielectric medium [109]. T ij = M B δ . (184) F −2 · ij Here the solution of Maxwell’s equations is an indepen- dent input which depends only on the properties of the Outside the material both matter and field tensors van- ij ij medium which is implicitly supposed to be homogeneous, ish. Inside it must be TM + TF = 0. Therefore there is isotropic and rigid. Suppose that the region x> 0 is filled an isotropic pressure inside the material p = 1/2M B. · by a non-dispersive medium with dielectric constant ǫ This pressure can be calculated directly from the forces and magnetic permeability µ. A packet of linearly polar- between magnetic dipoles. For example, if the magnet ized plane waves approaches the yz surface traveling in is a long rod uniformly magnetized along its axis, the the x direction. stress along the axis corresponds to the force of attrac- As the wave enters into matter the electromagnetic tion between the two pieces of the rod divided by a plane field exerts a force on the medium sharing its energy and perpendicular to the axis. momentum with it. The medium becomes polarized and The stress along directions perpendicular to the axis correspondingly changes its energy and momentum. The may also be calculated. The force density (123) vanishes dipolar energy in (128) goes along with the electromag- in the bulk. It is felt only in the narrow superficial region netic fields at the light velocity of the medium. For the in which the magnetization decreases from the bulk value idealized model of a semi-infinite medium used in this ex- to zero. This force density is perpendicular to the exter- ample, the momentum exchange comprises two different nal surface and directed inward; when it is integrated processes. First, the pulling (see below) that the field ex- over the thickness of the surface region the pressure is erts on the medium sets in movement a mechanical wave obtained. which travels at the sound velocity, absorbs the impulse The pressure contributes to the spontaneous volume transferred to the matter and does not interact with the magnetostriction of the ferromagnet. It results in a con- field anymore. Second, as discussed in subsection V E, traction of the ferromagnet during the magnetization part of the electromagnetic momentum (129) is trans- process. Besides the purely magnetic effect there are ferred to the medium. As we show below the medium also contributions due to the Heisenberg interactions that response is to adjust the internal stress components of align the spins. P µν to compensate this gain of momentum. The result- The purely magnetic volume magnetostriction can be ing perturbation which has zero total momentum also calculated dividing the pressure by the bulk elastic mod- travels with the transmitted wave packet at the light ve- ulus K locity. δV 1 Let us turn to the solution of Maxwell’s equations. The ω = = M B . (185) electric field is V −2K · In the following table there is a comparison of the mea- E (x,y,z,t)= E g(t x/c)θ( x)yˆ . (186) 1 1 − − sured volume magnetization ωexp of various ferromagnets [105, 106] with the values calculated with (185) using the where E1 is the amplitude, θ is the Heaviside step func- saturation magnetization. tion and g(t) is a dimension-less well-behaved function (otherwise arbitrary) that vanishes for t < 0 and t>T . At the surface of the material (x = 0) the packet is re- M (MA/m) K (GPa) ω 106 ω 106 × exp × flected and transmitted. The reflected and transmitted Fe 1.75 170 -11.3 400 packets are Co 1.43 180 -7.1 -250 E (x,y,z,t)= E g(t + x/c)θ( x)yˆ , (187) Ni 0.51 180 -0.9 -270 2 2 − E (x,y,z,t)= E g(t x/v)θ(x)yˆ , (188) Gd 2.14 38 -75.8 -5000 3 3 − where the speed of light in the material is v = c/n with n = √ǫµ. For t < 0 only the incident packet is present, The purely magnetic effect is usually not taken into ac- for t>T the reflected one is in the region x< 0 and the count in the theoretical papers on the topic [107, 108]; al- transmitted one is in the region x > 0. For 0 T ) the energy and momen- The surface component of the force at x = 0 is equal to tum are the momentum flux exiting the vacuum side minus the momentum flux entering the matter side. That is ¯ AcT 2 U2 11 11 U2 = E , p2 = g(2) dV = xˆ . (197) F = A(T ( ) T (+))xˆ . (206) 4π 2 Z − c S S − − F Using (55) and (132), The energy and momentum transferred to the x> 0 side T 11( ) T 11(+) = of the space are S − − F 1 ¯ ¯ [B2( ) B2(+) + 2(1 1/µ)B2(+)] AcT 2 2 AcT 2 8π − − − U1 U2 = (E1 E2 )= E3 ǫ/µ (198) − 4π − 4π ǫE2 ¯ ¯ p = 3 (1/µ + µ 2)g(t)2 . (207) AT 2 2 AT 2 8π − p1 p2 = (E1 + E2 )xˆ = E3 (1 + ǫ/µ)xˆ .(199) − 4π 8π Therefore the total force is The EM energy and momentum of the transmitted 2 AE3 2 packet are F = FV + FS = (1 + ǫ/µ 2ǫ)g(t) xˆ . (208) 8π − ¯ 00 AcT 2 We note that if diamagnetism does not prevail the wave U3 = T (3) dV = E (ǫµ +2ǫ 1) , (200) Z F 8π√ǫµ 3 − packet pulls the dielectric. The impulse is ¯ ¯ 2 AT v 2 AT E3 p3 = g(3) dV = E ǫ√ǫµxˆ . (201) I = F dt = (1 + ǫ/µ 2ǫ)xˆ . (209) Z 4πc 3 Z 8π − 28

2 The total momentum transferred to the region x> 0 for (W + Mc ) being the total energy of matter and XM the t>T is center of mass of matter. The polarization energy travels with speed v along with the electromagnetic wave. Then ¯ 2 AT E3 I + p3 = (1 + ǫ/µ)xˆ = p1 p2 (210) 8π − W X (t)+ Mc2X (t) X F3 b M(t)= 2 (215) in a consistent way with momentum conservation. W + Mc

with Xb(t) the center of mass of the bare matter (the C. The center of mass and spin medium perturbed by the mechanical wave). The energy- momentum of matter has two components, one which is symmetric, that is related to the mechanical wave, and We now treat the motion of the center of mass.Consider the other related to the polarization energy. For the me- first the transmitted electromagnetic wave. For t>T , chanical wave the CMMT is valid, so that MX˙ b = I, the position of the center of mass of the field wave and where I is the impulse computed in (209). Assembling of the whole wave including the polarization energy of the parts we have, matter is the same. This is expressed by the following equation: U c + (U + W )v + Ic2 X˙ 1 = − 2 3 1 T U XF3(t)= xu dV xˆ (211) Ac2T¯ E2 U3 Z = 3 [1 + ǫ/µ +2 ǫ/µ +4/µ 4ǫ] (216) 1 16πU − 2 x p = ¯ xg(t x/v) dx ˆ vT Z − which is not equal to the right hand side of (213). The 1 CMMT does not hold in this case. = ′ xuP dV xˆ = XW3 , U3 Z Let us consider now the spin contribution. The total spin density of system which has contributions from mat- where X is the center of mass of the field and X that F3 W3 ter and field should be used. The identification of these of the whole wave. It immediately follows that X (t)= F3 separate contributions in the general case is a difficult X (T ) + (t T )vxˆ, and that X˙ = vxˆ = X˙ . This F3 − F3 W3 problem which we will not discuss here. Since the wave velocity is indeed constant. It can be written also as is linearly polarized, we do not expect any field spin con- tribution in this case, but this supposition is unnecessary 1 1 0i X˙ W3 = S dV = T (3)eˆi dV . (212) to reach our conclusion in what follows. The matter-field U ′ U ′ F 3 Z 3 Z system is isolated, suppose for a while that the energy- For the strong CMMT to hold for this object it would be momentum tensor of matter is symmetric. The symme- −2 ′ ˙ try question arises only for the contribution related to necessary to take Abraham’s value c U3X as its momen- tum. But then overall momentum conservation would be the transmitted packet. Using (142) we have lost. µνα µν Let us turn to the movement of the center of mass of ∂αS = τd . (217) the whole system. Suppose that the medium is initially µν at rest, has a total mass M and is free to move. To with τd given by (120) and (122). Focusing in the tem- be consistent with the assumption of a rigid medium at poral components, which are the ones that contribute to rest used in solving Maxwell’s equations, let us consider the center of mass and spin definition (7), we have, a situation in which uM uP and the sound speed is 0kα ≫ ∂αS = ( P B + M E)k much smaller than light speed, so that the energy of the − × × µǫ 1 mechanical wave that appears as the wave-packet enters = − (E B)k (218) the medium would be negligible in comparison with U1. − 4πµ × Initially the total energy of the system is U = U + 1 where we have used the constitutive equations Mc2. When the wave is moving towards the dielectric there is no spin contribution to the center of mass and ǫ 1 µ 1 spin and we may write P = − E , M = − B . (219) 4π 4πµ U c2p ˙ ˙ 1 ˙ 1 0ki XΘ = XT = XW1 = xˆ , t< 0 . (213) Spin transport in this system is due to the drift, S = U U 0k0 i i S vm with vm the matter velocity. In this case it van- After the wave has penetrated the dielectric, it is con- ishes because the mechanical wave is much slower. Then 0ki venient to separate the field and matter contributions to ∂iS = 0 and using that the right hand side of (218) the center of mass. For t>T one has, points in the x direction we have

UXT(t)= U2XF2(t)+ U3XF3(t) 010 (µǫ 1)√ǫµ 2 2 2 ∂0S = − g (t x/v)E3 (220) + (W + Mc )XM(t) , (214) − 4πµ − 29

Integrating in space the spin term which appears in equa- z tion (8) for t>T is

∂ AcE2T¯(µǫ 1) U S01 = 3 − = X˙ 1 . (221) ∂t − 4πµ − c S

Taking all together we verify that for t>T , q Ac2T¯ E2 c2p X˙ 1 = X˙ 1 + X˙ 1 = 1 = 1 (222) Θ T S 4πU U as requested by the improved theorem (8). This justifies M our supposition that the matter energy-momentum ten- sor is symmetric. We treat this point with more detail in FIG. 1. Magnetized ring the following paragraphs. The electromagnetic field modifies the matter when it X. THE MAGNET-CHARGE SYSTEM penetrates. According to (128) and (129) part of the energy and momentum of the microscopic field are kid- napped by the matter during the polarization process. A. Violation of the CMMT This is accompanied by a mechanical response which tunes the internal stress to the value necessary to guar- As a second illustration of our discussion [110], con- antee the right properties of the medium and relativistic sider a ring with the shape of a washer made of a fer- invariance. To explore this view consider the transport romagnetic insulator (see Fig. 1). The axis of the ring of the energy of matter in the packet. The energy den- is along the z-axis and the center is at the origin. For sity of the matter, which is still at rest when the wave is simplicity we take ǫ = 1. The ring is magnetized around passing, is its axis with magnetization M of uniform magnitude. In this situation B = 4πM inside the ring whereas H = 0 2π 2πµ everywhere. Suppose that at the center of the ring there u = u (0, 0)+ P 2 M 2 P E (223) M b ǫ 1 − µ 1 − · is a particle of charge q. Since the ring is an insula- − − tor the electrostatic field of the charge penetrates the µν whereas the equation of transport of energy in the matter magnet. The total momentum defined by TF does not is vanish but points in the z direction. Using cylindrical coordinates (ρ,θ,z), M = Mθˆ and r = ρρˆ + zˆz we have 0k ∂0uM + ∂kP = P ∂0E M ∂0B . (224) 1 qM − · − · g = D B = ( zρˆ + ρˆz) . (226) 4πc × cr3 − By taking the time derivative in (223), it is obtained that 0k and integrating over the volume ∂kP = 0 and, given the symmetry of the problem, it follows that P 0k = 0 everywhere since P µν vanishes out- qM ρ p z side the packet. That is, Poynting’s vector is the whole = dV 3 ˆ . (227) µν c Z r energy current density of the wave-packet. If TM is sym- metric then P k0 compensates the potential momentum The center of mass of the system (including the elec- density appearing in (129) and vice versa, tromagnetic contribution to the energy) is at rest at the center of the ring. Then the usual CMMT is violated. 0k k0 k0 Note that Abraham’s momentum density vanishes since 0= TM = TM = P (P B)k . (225) − × H = 0, so that if one takes it to define the momentum The spatial stress P jk is symmetric because the spatial of the electromagnetic field, the total momentum of the torque on matter vanishes. system including the electromagnetic field is zero. On the other hand, Møller-von Laue criterion allows to Let us now examine what happens when the fields recognize u and g as the total energy and momen- change by considering the scenario in which the washer P Min demagnetizes (e. g. by the action of a device included tum traveling with the wave. Since gMin is the momen- tum of the electromagnetic field no additional momentum in the magnet which allows its temperature to rise above B attached to the matter is following the electromagnetic the Curie point). The magnetic induction field changes wave. The energy-momentum tensor of matter is sym- and therefore an induced electric field appears. It is al- metric and the conditions for the validity of the improved ways possible to split the electric field in a static field, E E E E theorem (8) are met. In our picture the electromagnetic S, S =4πρ, S = 0 and an induced field I, E∇= · 0, E ∇×= c−1B˙ . It is easily shown that field exchanges energy with matter and exerts locally a ∇ · I ∇× I − force on it, although the total force vanishes once the 1 ∂B ˆr EI = dV . (228) wave-packet is completely into the matter. −4πc Z ∂t × r2 30

Neglecting the radiation field B = 4πM. Then at the C. Storing momentum in the magnet origin the induced field is To complete the picture, let us investigate the origin of M˙ ρ the momentum that in ours and Minkowski’s formulation EI(0) = dV zˆ . (229) − c Z r3 is stored in the electromagnetic field. Let us start with the charge far away and analyze how, as it is brought If the demagnetization process is fast enough, the force to the center of the ring, the field momentum builds up. on the charge is F = qEI(0), and the impulse on the The magnet does not produce an electromagnetic field (dressed) particle may also be computed outside itself. Hence there is no force acting upon the charge. There is, though, a force acting on the mag- q ρ net produced by the magnetic field B generated as the ICharge = dt F = dt M˙ dV zˆ . (230) Z − c Z Z r3 charge moves. To keep the magnet in place an opposite force must be applied to it. The impulse generated by Note that no B field is produced during the demagnetiza- this force is precisely the stored momentum in the elec- tion process by the charge and there is no force acting on tromagnetic field. For a charge moving along the z-axis the magnet which therefore remains at rest. The mechan- the calculation can be done easily. ical momentum of the system is now different from zero. Let us call x the position of the charge, x = ζzˆ. The If one uses Abraham’s momentum density, the unavoid- velocity is v = ζ˙zˆ. For v c the magnetic field produced ≪ able conclusion is that momentum is not conserved. But by the moving charge is on the other hand, (230) is exactly the momentum of the q v r′ qζρ˙ θˆ initial configuration calculated (227) using Minkowski’s B = × = , (235) momentum density, which is then shown to be consistent c r′3 c[ρ2 + (z ζ)2]3/2 with momentum conservation. Since the center of mass − where r′ = r x = ρρˆ + (z ζ)ˆz. Using (126), the force of the magnet does not move, the center of mass of the − − system moves in the direction in which the particle moves density on the magnet is with velocity q ∂ ∂ ρ f = M B = ζM˙ ρˆ + zˆ . (236) k k c ∂ρ ∂z ′3 c2 ∇   r X˙ i = pi (231) T U The force is obtained by integrating over the volume q ∂ ρ where U is the total rest energy of the system. The F = dV f = ζM˙ dV zˆ (237) CMMT is again violated. Z c Z ∂z r′3 q d ρ = M dV ˆz . (238) − c dt Z r′3 B. The motion of the center of mass and spin We have used ∂z = ∂ζ. The impulse on the magnet is then − The result obtained is nevertheless consistent with our q ρ I F z center of mass and spin motion theorem (8). To see this Magnet = dt = M dV 3 ˆ . (239) we note that in the initial situation the total spin density Z − c Z r satisfies again (217). For the temporal components we The momentum stored by the field is the opposite of have, this quantity and again agrees with (227). Also in this situation CMMT does not hold. If no external force is 0iα 0i i ∂αS = τ = (M E) , (232) applied to keep the magnet in place, the charge would × move with constant velocity but the magnet would be and using again that matter is at rest we are left with accelerated in the negative direction of the z axis, as the center of mass would be too. S0i0 = c(M E)it + C (233) × 1 D. Toroidal currents with C1 a constant. Integrating over space and using B =4πM we obtain Taylor [61] has treated a related example, with toroidal d c c2 currents instead of a magnet. It is interesting to have a X˙ i = X˙ i = S0i = pi , (234) θ S −dt U U view on the differences with our discussion in the start- ing situation with the charge at rest in the center. In in agreement with (8) and (231). There is a short period this case the proper energy-momentum tensor is known when the magnetization varies for which both X˙ T and to be the standard tensor defined in the vacuum, which is X˙ are variable. symmetric, S = c2g = cE B/4π, and therefore CMMT S × 31

E . J >0 σ´ E E S q B

E .J <0

FIG. 2. Toroidal currents M σ h holds. If one considers a free current distribution of value j = c M, with M the magnetization of our previous example,∇× the magnetic induction field and the induced electric field are the same as in that case and the electro- magnetic momentum density is equal to the one obtained using Minkowski’s expression in that situation. But now g unlike in the magnet configuration the Poynting vector a does not vanish. The force exerted on the currents by E is still zero, but the power density E j is not, see Fig. 2. b In the equivalent situation to the one· discussed above, inside the structure that supports the currents, the en- FIG. 3. Magnetized hollow cylinder ergy is increasing in the top, and decreasing in the base. The total power does vanish. Since the force is zero, the velocity of the supporting structure maintains its null and E = 0 otherwise. The momentum density is value; nevertheless, its center of mass moves because the 1 4πMσa internal energy is transported by Poynting’s vector from g = D B = θˆ , a < r < b . (241) 4πc × − cr the bottom to the top. The CMMT imposes that the velocity of the center of mass should equal the stored Since g circulates around the axis there is a net orbital momentum divided by the total energy. This makes this angular momentum L stored in the electromagnetic field. configuration ephemeral. Even disregarding Joule’s effect For a section of cylinder of height h the angular momen- it could last only until the energy storage in the base is tum is depleted. Regretfully, the analysis presented in Taylor’s (2π)2 paper is not correct because it does not take into account L = r g dV = Mσa(b2 a2)hˆz . (242) × − c − adequately the electromagnetic momentum. The energy Z transport mechanism just described is in contrast to what Note that the EM fields store angular momentum al- occurs with the magnet where the power density and the though macroscopically there is nothing rotating. Poynting vector are zero. For this reason, the center of If the cylinder demagnetizes an induced electric field mass of the magnet may remain at rest. is generated. The field produces a torque on the charged shell and the angular momentum is transferred to the shell1. For a < r b the induced electric field is XI. ANGULAR MOMENTUM OF A ≤ 2π MAGNETIZED CYLINDER E = (r2 a2)M˙ θˆ (243) in − cr − In our last illustration we consider the implications of and the torque on the shell is the momentum density expression to orbital angular mo- 2 ′ (2π) 2 2 mentum equilibrium. Let us consider an infinite hollow τ = r Einσ dS = σa(b a )hM˙ ˆz . (244) cylinder made of a ferromagnetic insulating material with Z × c − ǫ = 1 (see figure 3). The radius of the concentric hole is a The angular impulse is indeed equal to the angular mo- and the external radius is b. The cylinder is magnetized mentum of the electromagnetic fields, along its axis with a uniform magnetization M. In the surface of the hole there is a uniform surface charge den- τ dt = L . (245) sity σ and in the external surface of the cylinder there Z is a cylindrical shell of insulator charged with a surface density σ′ = σa/b. In this condition inside the wall of − the cylinder there is a uniform magnetic field B =4πM 1 In this example there is also a spin proportional to the magneti- and a radial electrostatic field zation which will also be transferred to the angular momentum a of the magnet (Einstein-de Haas effect). The coupling is due to E =4πσ ˆr for a

In this situation only Minkowski’s momentum den- on polarizable matter. This expression is similar but dif- sity is consistent with angular momentum conservation. ferent to some of the expressions discussed in the past. Abraham’s momentum density yields a null angular mo- The total force that the electromagnetic field exerts on an mentum in the initial situation since H = 0. isolated sample of matter computed with our expression is the same as the total Lorentz force computed on the bound charges and currents of the sample. It is also equal XII. CONCLUSION to the total force predicted by the use of Minkowski’s tensor but it is different to the one consistent with Abra- In this paper we attain a complete resolution of ham’s tensor. The term known as Abraham’s force is the Abraham-Minkowski controversy on the energy- not present in our formulation. Our analysis of the force momentum tensor of the electromagnetic field in matter density uncovers also the portion of the energy momen- by introducing a new energy-momentum tensor which tum tensor of matter which depends on the field. This is different from those previously proposed. Our ap- includes the electrostatic potential energy of the dipoles, proach emphasizes the local character of the force and a momentum contribution required for the polarization the torque exerted by the field on matter and requires of matter and, depending on the external conditions on that the energy-momentum tensor be locally well deter- matter, a field dependent mechanical response of matter mined. It cannot be arbitrarily modified by the addition which modifies its stress tensor. of a divergence-less term, even if by doing so the total The energy-momentum tensor of the field is fixed by energy and momentum do not change. the form of the force density and we are able to get its Our discussion departs from the main lines of argu- correct expression. The energy density of this tensor is mentation prevailing in the recent literature on the topic. not Poynting’s classic formula. It is shown to be the en- It is based on four main points: First, the force den- ergy density of the electromagnetic field in vacuum minus sity on matter is in principle a measurable quantity and the electrostatic energy of the dipoles. This term repre- should be computable as the divergence of the matter sents a electromagnetic contribution to inertia. Poynt- tensor. Momentum conservation determines in an unam- ing’s vector maintains its interpretation as the energy biguous way the energy-momentum tensor of the electro- density current once the correct expression of the power magnetic field. Second, all terms which contribute to in- transmitted by the field to matter is taken into account. ertia, including those of electromagnetic origin, should be The momentum density is shown to be Minkowski’s ex- included in the matter tensor and consequently excluded pression. It is shown to be the momentum in vacuum from the electromagnetic one. Third, the force density minus the field dependent momentum of matter. This should be compatible with the microscopic Lorentz force. puts an end to the long standing controversy. Fourth, magnetization is a manifestation of spin and EM The energy-momentum tensor of the field is asymmet- fields produce torques on polarized matter, hence a com- ric. This has important consequences. This asymmetry plete understanding of the mechanical behavior of po- is shown to be necessary for the coupling of the total spin larizable matter in an electromagnetic field requires the density (magnetization) and the orbital angular momen- dynamics of spin too. tum of matter. The asymmetry cancels exactly the effect In this work the necessity of a relativistic invariant of the torque density in the dynamical equation of the description of the matter-field interaction and the under- field spin density with the general result that the spin of standing of the true consequences of such a description the field is decoupled from the polarization of matter. play a key role. Throughout the discussion we insisted We presented a second approach to deduce the form on the idea that the polarizations tensor which includes of the electromagnetic energy-momentum tensor in mat- the electric polarization and the magnetization in a rel- ter. For this we perform a space-time average starting ativistic covariant object, should be used to describe the from the microscopic equations and postulate a monopo- polarization of matter. We avoid arguments that rely on lar and a dipolar coupling of matter with the macroscopic the properties of the electric and magnetic permeabili- field. By imposing consistency with Maxwell’s equations ties which are defined only in the rest frame of matter we identify the monopolar current with the free charges, (which not always exist). As an element which has more and show that the dipolar coupling is described by the general consequences we discuss the serious limitations polarization tensor in such a way that we recover exactly of the CMMT, which does not take into account the con- the formulation of the first approach. Although it may tribution of spin to the dynamics. The uncritical use of be foreseeable, this result has a deep meaning: Relativis- the CMMT in this context has been the source of many tic invariance and the physical hypotheses discussed in confusions and even mistakes in previous papers on this section V uniquely determine the equations of the macro- subject. scopic field up to the dipolar approximation. It will be The conditions imposed on the form of the force den- interesting to investigate if the quadrupolar approxima- sity on matter by the relativity principle, the validity tion maintains such robustness. of Maxwell’s microscopic and macroscopic equations and In our presentation, we also note that although in a the consistency with Lorentz microscopic force allow us general situation Minkowski’s tensor is not particularly to determine uniquely an expression of the force density useful, for a material with non-dispersive linear polariz- 33

−1 µν abilities (Dαβ =2 χαβµν F ) it may be interpreted as of the consequences of the main ideas behind our formu- the energy-momentum tensor of the electromagnetic field lation. In the same spirit, for the magnetized cylinder plus the fraction of the energy of matter arising from the analyzed in section XI we show how our approach gives polarization. Working with care it can be used in this an appealing and consistent description of a system where case. Nevertheless its divergence is not the reaction of the orbital angular momentum stored in the EM fields is the force acting on matter. This explains the absence transferred to matter. of magnetostriction and electrostriction in Minkowski’s As a by-product of the analysis presented in this paper approach. When an electromagnetic wave propagates on the weakness of the CMMT was exposed in specific phys- such a medium Minkowski’s tensor describes the whole ical situations, and the validity of the improved theorem wave and hence satisfies von Laue-Møller criterion. which includes the spin contributions was tested. These The arguments presented in this paper are of theoret- results also show that the hidden momentum hypothesis ical nature. Of course, the ultimate test of the validity is unnecessary. Although we have not made a detailed of our approach should be the direct measurement of the check in each case (which would take a disproportionate force density in different situations. For some of the ex- effort), we think that the many analysis found in the lit- periments that have been done in the past, it is not com- erature which are based in this hypothesis should prove pletely straightforward to conciliate the measurements wrong or at least superfluous. obtained with a definite choice of the force density or Our results open a range of theoretical and experi- momentum expressions. In consequence there have been mental possibilities of research in the study of the in- confronting opinions on how to interpret correctly the re- teraction of the electromagnetic fields with polarizable sults of those experiments. Nevertheless there are some matter. Our construction of the force density suggests to situations in which in our opinion our analysis proves to make a methodological shift in the approach to the many be clearly better. The applications and examples pre- applications in which it plays a role and concentrate in sented in the text were devised with this in mind. the determination of the physically acceptable polariza- In section V we discuss some of the implications of our tion tensor of the system under study, preferably in the approach for the analysis of the experiments which have laboratory reference frame. This may be applied to a been performed to determine the electromagnetic force variety of systems, which include ionized gases, dielectric density in matter. In particular we show that the result liquids, and non linear magnetic matter, to give a few ex- of the Walker, Ladhoz and Walker experiment, which amples. In the past these were either ignored or studied is usually presented as an evidence for the existence of in terms of the permeabilities. We note that the polar- Abraham’s force term, may be better understood as a ization tensor of some of these systems may depend on manifestation of the field dependent terms of the matter the local velocity of matter, so that velocity dependent energy-momentum tensor. Our discussion of the electro- effects are not banned from our formulation. It is only magnetic forces on solid and liquid dielectrics strongly Abraham’s like terms which are found unnecessary. On supports the validity of the force density that we are the experimental side the determination of force density proposing. instead of the total force advocated by some authors and We use our force density, the energy and momentum in particular by Brevik [45] in the past acquires new im- µν definitions obtained from TF and Maxwell’s equations portance. A critical reanalysis an even a remaking of the to verify energy and momentum conservation in the in- experiments which perform a direct measurement of the teraction of a packet of electromagnetic waves with a di- momentum of photon should be useful for establishing electric medium. In opposition to the statement of the without further doubts Minkowski’s momentum as the influential argument first advanced by Balazs [59], that classical and quantum momentum of light. Experiments for sixty years has been considered a strong support in exploring the inertiality of the center of mass and spin in favor of Abraham’s point of view, we show that for n> 1 this context which may include the real versions of the the wave packet effectively pulls the material when it en- idealized situations discussed in sections IX, X and XI ters a medium (See Eq.(208)). This in fact is a most will also contribute to the complete experimental clarifi- comfortable result after observing that it has a simple cation of this subject. physical explanation. Dielectric and paramagnetic ma- terials are attracted while diamagnetic materials are re- pelled in the direction to high field regions. When the wave packet is entering the medium it pulls the material unless diamagnetism prevails. For the same reason when Appendix A: Energy and momentum of dressed the wave leaves, it drags the block forward. We complete dipoles our picture of this process by computing the evolution of the spin density. This calculation shows that the boost The time derivative terms that spoil the relativistic co- momentum gives the precise contribution necessary for variance in (113) and (114) actually correspond to part our center of mass and spin to behave inertially. of the momentum and energy of the microscopic electro- The example of the charge-magnet system presented magnetic interaction. In addition to the external fields in section X, gives in a very simple setup an illustration we considered in section V C there are also the electro- 34 magnetic fields produced by the dipolar system itself we argue that the physically consistent values of the in- tegrals are Et = E + Ed , (A1) B = B + B . (A2) t d Bd dV = 0 (A10) Z The energy density of the total electromagnetic field is and 1 ut = u + ud + (E Ed + B Bd) (A3) 4π · · Ed dV = 4πd . (A11) Z − and the momentum density is Therefore 1 gt = Et Bt U = d E (A12) 4πc × i − · 1 = g + g + (E B + E B) . (A4) and d 4πc × d d × 1 The integrals of the energy density ud and of the mo- p = d B . (A13) i − c × mentum density gd that only depend on the dipolar fields Ed and Bd are part of the energy and momentum of the As we have shown in subsection V C, these contribu- dressed dipole. tions to the interaction, (A13) and (A12), exactly cancel The interaction momentum and energy are the inte- the additional terms in (113) and (114). In other words grals over the whole space of products of an external (113) and (114) correspond to the force and power on field and a dipolar field. We assume that the the external the bare dipole while for the dressed dipole the offending fields are essentially constant where the dipolar system is terms are not present. located and that the fields of the dipolar system are well- approximated by the fields of dipolar moments where the external charges and currents are located (dipolar ap- Appendix B: Integrals of self fields of dipolar proximation). The integrals can be split in a local term systems corresponding to the region around the dipolar system and an external term outside this region, In this section we prove that the whole-space volume integrals of the dipolar fields that appear in appendix A . . . dV = . . . dV + . . . dV . are non-convergent improper integrals. We show that the Zwhole−space Zlocal Zexternal physically consistent values of these integrals are those (A5) given by (A10) and (A11). We assume that the external The local electromagnetic interaction energy and inter- fields are E = φ and B = A with the potentials action momentum also contribute to the dressed dipole vanishing at infinity−∇ as 1/r.∇× Near the dipolar system energy and momentum. The local interaction energy is the fields are constant, then φ φ(0) E(0) x and A A(0) + B(0) x/2. ≈ − · 1 ≈ × Ui = (E Ed + B Bd) dV (A6) The integral of the self magnetic field on a finite volume 4π Zlocal · · V is and the local interaction momentum is Bd dV = Bd x dV Z Z · ∇ 1 V V pi = (E Bd + Ed B) dV . (A7) 4πc Zlocal × × = [ ( Bdx) Bd x] dV ZV ∇ · − ∇ · To evaluate the contribution of the local interaction = xBd dS Bd x dV . (B1) terms we use the fact that the external fields are essen- I∂V · − ZV ∇ · tially constant. Then The surface integral is not convergent as the surface 1 ∂V goes to infinity. For example, for a sphere the surface Ui E Ed dV + B Bd dV , (A8) ≈ 4π  · Z · Z  integral is (8π/3)m regardless of the radius. Instead if the volume is taken as the doughnut-shaped figure whose and surface is made of field lines then the integral vanishes. Let us assume that there is some way of defining the 1 limit that gives the “correct” physical value. Given the pi E Bd dV + Ed dV B . (A9) ≈ 4πc  × Z Z ×  symmetry of the dipolar field the limit of the surface integral must be proportional to the dipole moment These approximations have a pitfall because the whole- space integrals of the dipolar fields are non-convergent xBd dS λm . (B2) improper integrals. Nevertheless, in the next appendix I∂V · → 35

Since B = 0 the total integral is whole interaction electric energy ∇ · d 1 1 E Ed dV = φ Ed dV 4π Z · −4π Z ∇ · 1 = [ (φEd) φ Ed] dV Bd dV = λm . (B3) −4π ∇ · − ∇ · Z Z = φρ dV Z = E(0) d , (B5) − · which of course is the electrostatic energy of the dipole The electrical case is quite similar. Since the magnetic and should be considered part of the energy of the dressed and electric fields have the same form far away from the dipolar system. Therefore, it must be that λ = 0. The dipole, the surface integral must be the same for both integral of the magnetic field is then zero and there are fields. not magnetic contributions to the momentum or energy of the dressed dipole. Nevertheless, the whole interaction magnetic energy is not zero, 1 1 B Bd dV = A Bd dV 4π Z · 4π Z ∇× · E dV = λd E x dV d − ∇ · d 1 Z Z = A Bd dV 4π Z ·∇× = λd 4π ρx dV − 1 Z = A jm dV = λd 4πd . (B4) c Z · − = B(0) m . (B6) · This energy must be assigned to the external system. If an external field B is turned on in a region where there is a constant magnetic dipole, a work B m is done on the bare dipole. The opposite is the− work· done on the For determining the value of λ let us calculate the external system.

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the kinetic energy-momentum tensor of electromagnetic ter of mass theorem for systems with electromagnetic fields in matter arXiv:1404.5250 interaction arXiv:1404.5251 [110] Rodrigo Medina and J. Stephany, Violation of the cen-