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Momentum of an electromagnetic wave in dielectric media

Robert N. C. Pfeifer,∗ Timo A. Nieminen,† Norman R. Heckenberg,‡ and Halina Rubinsztein-Dunlop§

Centre for Biophotonics and Laser Science, School of Physical Sciences, The University of Queensland, Brisbane,

QLD 4072, Australia

(Dated: May 8, 2007)

Almost a hundred years ago, two different expressions were proposed for the energy–momentumtensor of an electromagnetic wave in a dielectric. Minkowski’s tensor predicted an increase in thelinear momentum of the wave on entering a dielectric medium, whereas Abraham’s tensor predictedits decrease. Theoretical arguments were advanced in favour of both sides, and experiments provedincapable of distinguishing between the two. Yet more forms were proposed, each with theiradvocates who considered the form that they were proposing to be the one true tensor. This paperreviews the debate and its eventual conclusion: that no electromagnetic wave energy–momentumtensor is complete on its own. When the appropriate accompanying energy–momentum tensorfor the material medium is also considered, experimental predictions of all the various proposedtensors will always be the same, and the preferred form is therefore effectively a matter of personalchoice.

PACS numbers: 03.50.De, 41.20.Jb

Published as: Reviews of Modern Physics 79(4), 1197-1216 (2007)

Contents

I. Introduction 1

II. A Sound Foundation 2

III. The Early Years 2

IV. Experiments 3A. Jones and Richards 3B. Ashkin and Dziedzic 4C. James; Walker, Lahoz and Walker 5D. Jones and Leslie 6

V. Hypothetical Constraints and Alternate Theories 6

VI. Proofs of Equivalence 7A. Penfield and Haus; de Groot and Suttorp 7B. Gordon’s Analysis 10C. Mikura, Kranys and Maugin 10

VII. Momentum and Pseudomomentum 10

VIII. The Total Energy–Momentum Tensor 11A. A Total Energy–Momentum Tensor 12B. Forces Acting On a Subsytem 14

1. Qualitatively Distinguished 152. Spatially Distinguished 15

∗Electronic address: [email protected];URL: http://www.physics.uq.edu.au/people/pfeifer/†Electronic address: [email protected]‡Electronic address: [email protected]§Electronic address: [email protected]

3. General Subsystem 16C. Examples 16

1. Reflection of Light Off a Mirror 162. Transmission of Light Through a Nonreflective

Dielectric Block 163. Transmission and Reflection Off a Dielectric

Block 17D. Discussion; Simplifying Assumptions 17

IX. Anomalous Effects? 17A. Rigid Bodies and the Optical Tractor Beam 18

X. Recent Work 19A. Extension of Existing Theory and Practical

Applications 19B. Ongoing Controversy 20

XI. Conclusion 21

Acknowledgments 21

References 21

I. INTRODUCTION

The correct form of the energy–momentum tensorof an electromagnetic wave, and hence the mo-mentum of an electromagnetic wave in a dielectricmedium, has been debated for almost a century.Two different forms of the energy–momentum ten-sor were originally proposed by Minkowski (1908,1910) and Abraham (1909, 1910), though more havebeen added in later years (de Groot and Suttorp,

http://arxiv.org/abs/0710.0461v2

mailto:[email protected]

http://www.physics.uq.edu.au/people/pfeifer/




Colloquium: Momentum of an electromagnetic wave in dielectric mediaRNC Pfeifer, TA Nieminen, NR Heckenberg, and H Rubinsztein-Dunlop

Reviews of Modern Physics 79(4), 1197-1216 (2007)

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1967, 1968, 1972; Grot and Eringen, 1966; Livens,1918; Marx and Gyorgyi, 1955; Peierls, 1976;Penfield and Haus, 1967). The question regularlyattracts experimental interest, as on initial inspection itmay appear that some of the different tensors give riseto distinct and potentially useful physical consequences.

We discuss why this is not the case, reviewingthe key experiments of Jones and Richards (1954),Ashkin and Dziedzic (1973), and Walker et al. (1975),and theoretical work including that of Penfield and Haus(1967), de Groot and Suttorp (1972), Gordon (1973),Mikura (1976), Kranys (1979a,b, 1982), and Maugin(1980), the latter four authors demonstrating explicitlyand directly the equivalence of a number of differentenergy–momentum tensors.

More recent work on the subject may largely be dividedinto three camps. First, there are those concerned withpractical applications, focussing primarily on which ex-pression is most useful in a given circumstance. Second,there are those extending the theoretical work into newdomains. Frequently such papers are hampered by thefragmentary nature of existing literature on the topic andtheir conclusions may be weakened as a result. Third,there are those who are unconvinced that the controversyhas been resolved, perhaps unsurprising as certain keypublications remain relatively obscure. It is clear thatbroad interest still exists in the subject, and from thewidespread prevalence of work in the second and thirdcategories, that the field stands to benefit greatly from acoherent and comprehensive review.

We therefore hope this paper will increase awarenessthat the controversy has been resolved, and that pre-dictions regarding measurable behaviours will always beindependent of the electromagnetic energy–momentumtensor chosen, provided the accompanying material ten-sor is also taken into account.

In particular, renewed interest in the controversyhas been stimulated by the advent of optical trap-ping and manipulation of microparticles in fluid media(Ashkin et al., 1986; Lang and Block, 2003). We wish tolay to rest any concerns that predictions of observableeffects in optical tweezers might depend on the divisionof momentum between field and medium, and to providean introduction to the subject of the energy–momentumtensor in a dielectric medium. We hope that by read-ing this paper and some of its references, those new tothe field will gain a comprehensive understanding of thisinteresting subject.

II. A SOUND FOUNDATION

The Abraham–Minkowski controversy debates the cor-rect expression for electromagnetic momentum within amaterial medium, and by implication the form of thefour-dimensional energy–momentum tensor. Before wedebate the relative merits of the different expressions, wemust first satisfy ourselves that it is possible to constructa well-defined energy–momentum tensor. After all, on amicroscopic level the transfer of energy and momentumconsists of the absorption and emission of photons as theyinteract with the atoms of the medium, a process whichis both stochastic and quantum-mechanical in nature.Fortunately, the Abraham–Minkowski controversy is

conventionally formulated wholly within the realms ofclassical continuum electrodynamics (Nelson, 1991), inwhich the fluctuations due to the inherent unpredictabil-ity of the microscopic realm become negligible. Specifi-cally, the properties of the medium must change negligi-bly over unit cells which are nevertheless large in com-parison with the mean volume per atom. Discontin-uous inter-medium boundaries are permitted providedthey are sharp on the scale of these unit cells. Thetransition from stochastic behaviour on the microscopicscale to the Maxwell equations of the continuum, andhence by implication a domain in which both the refrac-tive index and the energy–momentum tensor are well-defined and piecewise smooth is comprehensively detailedby de Groot and Suttorp (1972).A number of authors do extend the problem

into the microscopic domain (Garrison and Chiao,2004; Haugan and Kowalski, 1982; Hensley et al., 2001;Loudon et al., 2005). Such work must of course be evalu-ated on its own merits, and on its consistency with the re-sults of experiment (Campbell et al., 2005; Gibson et al.,1980; Jones and Leslie, 1978).

III. THE EARLY YEARS

The first author to propose an expression for theenergy–momentum tensor of an electromagnetic wave ina dielectric medium was Minkowski (1908). His expres-sion corresponds to an electromagnetic wave momentumdensity of D × B, where D is the electric displacementfield and B is the magnetic flux density. The total mo-mentum of a propagating electromagnetic wave thereforeincreases upon entering a dielectric medium, from p infree space, to np, where n is the refractive index of themedium, and we are neglecting dispersion. An accessiblederivation of the Minkowski energy–momentum tensormay be found in Møller (1972).The electromagnetic energy–momentum tensor of

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Minkowski was not diagonally symmetric, and this drewconsiderable criticism as it was held to be incompatiblewith the conservation of angular momentum. Abraham(1909, 1910) therefore developed an alternative, symmet-ric, tensor, for which the electromagnetic wave momen-tum density was instead (1/c2)(E × H), with E repre-senting the electric field and H the magnetic field. Themomentum of an electromagnetic wave entering a dielec-tric medium then falls to p/n. Under the Abraham ten-sor, a photon therefore carries less momentum within amedium than in the Minkowski case.However, both of these tensors are incomplete in iso-

lation (de Groot and Suttorp, 1972; Penfield and Haus,1967). Momentum may also be carried by interactionsoccurring within the material medium, and between themedium and the electromagnetic field. This is most ob-vious for the Abraham tensor, as the material tensormust be considered in order to obtain the correct rate oftransfer of momentum to a reflective object suspendedwithin a dielectric medium, as in the experiments ofJones and Richards (1954) and Jones and Leslie (1978).However, increasing the photon momentum from p tonp in accordance with the Minkowski tensor yields thecorrect result directly, and hence many authors neglectto observe that it, too, is associated with a correspond-ing material tensor. Any complete, thermodynamicallyclosed system must conserve energy and momentum, andit is only by taking into account the corresponding mate-rial tensor that the Minkowski formulation can be rec-onciled with conservation of total angular momentum(Brevik, 1982). The behaviour of a transparent objectis less straightforward, with the object acquiring a fixedvelocity away from the source while exposed to the elec-tromagnetic field. The existing literature lacks a sin-gle consistent mathematical approach capable of dealingwith wholly or partially reflective and transparent ob-jects immersed within a dielectric medium. We addressthis lack in Sec. VIII.When the force exerted by an electromagnetic wave

in a dielectric medium is evaluated without recourse tothe material counterpart of the Minkowski tensor, thenthe two tensors will predict different results. The forcedensity predicted by the Abraham tensor pair is smallerby

fAbr =εrµr − 1

c2∂S

∂t, (1)

which is known historically as the Abraham force. Here,S is the real instantaneous Poynting vector E×H, and εrand µr are the relative permittivity and permeability ofthe medium respectively. When the material counterpartto the Minkowski electromagnetic energy–momentumtensor is also employed, as described in Sec. VIII.A, then

the Minkowski tensor pair also gives rise to the smallervalue.In the early part of the 20th Century the importance

of the material energy–momentum tensor was not fullyappreciated, and vigorous debate ensued regarding therelative merits of the Abraham and Minkowski electro-magnetic energy–momentum tensors. We shall discussmore fully how the requirement for the material counter-part to the Minkowski tensor arises from conservation oftotal linear and angular momentum in Sec. VI.Meanwhile, it is worth noting that in some sense “the

momentum of a photon in a dielectric medium” is anabstract concept, as it is impossible to construct an ex-periment to directly measure this momentum. Instead,one can only measure the total energy or momentumtransferred to a detecting apparatus in a given set of cir-cumstances. This may include contributions both fromthe electromagnetic momentum and from the associatedmaterial energy–momentum tensor. One may thereforeask whether it is possible or not to devise an experi-ment which distinguishes between the Abraham and theMinkowski tensors.

IV. EXPERIMENTS

A. Jones and Richards

The first experiment to measure the radiation pressuredue to light in a refracting medium was conducted byJones (1951) at the University of Aberdeen. Jones soughtto verify the prediction that when a mirror was immersedin a dielectric medium, the radiation pressure exerted onthe mirror would be proportional to the refractive indexof the medium. Later refinement of this experiment ledto a paper by Jones and Richards (1954) verifying thiseffect to an accuracy of σ = ±1.2%.The experimental set-up (shown in Fig. 1) consisted of

a pair of mirrors strung on a vertical wire and tethered ateach end by a gold alloy torsion fibre. The lower mirror(vane V ) was subjected to an asymmetric illumination,giving rise to a torque about the axis of suspension andcausing both lower (V ) and upper (M ) mirrors to rotateuntil this torque was neutralised by the torsion fibre. Bymeasuring the angle of the upper mirror, the authors de-termined the magnitude of the torque, and hence the rateof momentum transfer to the lower mirror. By immers-ing the lower mirror in a number of liquids of varyingrefractive index and observing the resultant deflections,the momentum transferred was found to scale with therefractive index, and the accuracy obtained was sufficientto show that this dependency was upon the phase refrac-tive index (nφ), and not the group refractive index (ng) of

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FIG. 1 The suspension and its container. A, screws to tautensuspension; B, brass draft cover; C, phosphor bronze strip0.005 cm thick; D, Johnson Matthey gold alloy suspension0.005 × 0.0005 cm; E, brass tube 0.8 cm internal diam.; F,magnet 0.2 cm long; G, electromagnet; H, 48 s.w.g. copperwire; I, liquid levels; J, screwed brass ring and lead wash-ers; K, support for mirrors; L, lead washer; M, silvered glassmirror 0.2 × 0.5 × 0.02 cm; N, brass swivel joint; O, brassnut and washers; P, copper/glass seal; G[sic], glass tube; R,cotton-wool plug; S, triangular cast iron base; V, rhodium-plated silver vane 0.2 × 0.5 × 0.01 cm; W, circular glass win-dows 0.1 cm thick. Reprinted figure with permission, fromJones and Richards (1954), The Pressure of Radiation in aRefracting Medium, Fig. 1, published by and c©(1954) TheRoyal Society.

the medium. This finding agreed with theoretical expec-tations, as the behaviour examined was concerned withvariations in the electric field at a point, rather than therate of propagation through the medium of a modulationin the field.

In the theoretical introduction to the 1954 paper, Jonesdiscussed whether the choice of expression for the mo-mentum density of the electromagnetic wave was depen-dent either upon D×B or E×H, though without direct

reference to the work of Minkowski or Abraham. Heobserved that the former followed naturally from the ex-pression for the Maxwellian stresses on a body within aradiation field, whereas the latter could be suggested onrelativistic grounds (see Pauli, 1921, 1958) but was alsoaccompanied by a corresponding stress in the surround-ing medium of (1/c)(kµ− 1)(E×H) (in Gaussian units),where k and µ stand for the dielectric constant and mag-netic permeability of the medium respectively, and theexpression is equivalent to the Abraham term (1). Hecommented that “it does not seem possible to devise apracticable experiment to distinguish between the twocases”. Brevik (1979) subsequently argued that whenthe linear momentum flux densities due to the Abrahamelectromagnetic tensor and the corresponding materialstress tensor are combined, the result is the same as thatgiven by the Minkowski tensor. He thus interpreted theexperiment of Jones and Richards as providing more di-rect support for the Minkowski formulation.Although in principle Jones and Richards are correct,

and the Abraham electromagnetic momentum is accom-panied by an induced material momentum, they arewrong to identify this momentum flux with the Abrahamterm. We shall examine the Abraham material momen-tum in greater detail in Sec. VIII.A and return to thisexperiment in Sec. VIII.C.1.

B. Ashkin and Dziedzic

The next significant experiment conducted was that ofAshkin and Dziedzic (1973), in response to a theoreticalpaper by Burt and Peierls (1973). In this paper, Burtand Peierls argued that when a laser beam passed fromair into a dielectric fluid of greater refractive index, trans-fer of momentum to the fluid would cause the surfaceto either bulge outwards, if the Minkowski tensor wascorrect, or be depressed, if the Abraham tensor was cor-rect. However, this conclusion was based on the mistakenassumption that the momentum of the electromagneticwave would rapidly move ahead of any momentum associ-ated with the material medium, allowing them to neglectthe effects of the material tensor in the Abraham case. Infact, the field-related material component of the momen-tum constitutes an excitation induced in the medium bythe leading edge of the electromagnetic wave and elimi-nated by the trailing edge. The excitation of the materialmedium therefore propagates at the same velocity as theelectromagnetic wave.Ashkin and Dziedzic performed this experiment, pass-

ing a laser beam vertically through a glass cell containingair and water (Fig. 2). Bulging or depression of the liquidsurface would cause a lensing effect, altering the radial

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FIG. 2 Basic apparatus: A, Beam shapes for low power (solidcurve) and high power (dashed curve) for positive surface lens;B, shapes (for low and high power) for a negative surfacelens. For A and B the beam is incident from above. C, beamshapes for low and high power for a positive surface lens withthe beam incident from below. F is a 2-4 µm glass fibre usedto identify the surface of the water under the microscope.A beam attenuator was placed either in position (1) or (2)depending on the power level desired within the glass cell.Reprinted figure with permission, from Ashkin and Dziedzic(1973). http://link.aps.org/abstract/PRL/v30/p139

profile of the laser beam. By studying the beam profile asit emerged from the cell, they were able to determine thatthe surface of the liquid was caused to bulge outwards. Intheir paper they acknowledged work by Gordon (1973),at that time still unpublished, which showed that thisresult does not invalidate the Abraham tensor, as thepredictions of the two tensors are reconciled when thematerial counterpart to the Abraham tensor is properlytaken into account.

C. James; Walker, Lahoz and Walker

In 1968 James performed an experiment to directlymeasure the Abraham force (James, 1968a,b). Two fer-rite toroids, axially aligned with one another, were con-nected to a piezo-electric transducer. The toroids weresubjected to time-varying electric and magnetic fields.Both Minkowski’s and Abraham’s energy–momentumtensors predicted a net torque on the toroids, but withdiffering magnitudes due to the existence of the Abrahamterm, Eq. (1). With the varying electric fields arrangedin antiphase, the two toroids exerted torques in oppo-site directions upon the piezo-electric transducer, and by

FIG. 3 Diagram of the experimental apparatus. Reprintedfigure with permission, from Walker et al. (1975). c©(1975)National Research Council, Canada.

measuring the magnitude of this torque, James was ableto experimentally demonstrate the existence of the Abra-ham force.This result went largely unnoticed until the experiment

of Walker et al. (1975). This team measured the torqueexerted on a disc of barium titanate which was suspendedon a torsion fibre in a constant axial magnetic field, andsubjected to a time-varying radial electric field (Fig. 3).The disc behaved as a torsion pendulum and its periodof oscillation was measured by reflecting a laser beam offa mirror attached to the disc. To maximise the responseof the system, the time period of the electric field wassynchronised with the oscillations of the disc. Followingan initial displacement, the oscillations of the disc settledexponentially towards a constant period, supporting theexistence of the Abraham force. The experiment was alsolater modified to confirm the effects of a time-varyingmagnetic field (Walker and Walker, 1977).As the conservation arguments leading to demonstra-

tion of the material counterpart of the Minkowski ten-sor were not widely recognised at this time, this exper-iment was taken as providing support for the Abrahamtensor over that of Minkowski (Brevik, 1979). However,

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Israel (1977) and later, tangentially, Obukhov and Hehl(2003) have demonstrated that for a suitable combina-tion of electromagnetic and material energy–momentumtensors the observed results are also compatible with theMinkowski case, and hence these experiments cannot dis-criminate between the two.

D. Jones and Leslie

In 1977, Jones and Leslie repeated the experiment of1954, taking advantage of recent advances in technology.A laser was employed as a light source instead of a tung-sten lamp, allowing the refractive index of the liquid tobe stated with greater precision as only one wavelengthof light was involved; a superior mirror was employed,reducing heating of the dielectric liquids and hence con-vection forces in the fluid; and the torsion apparatus wasrestored to the neutral position by means of an exter-nal electromagnet. The current in this electromagnet re-lated to the radiation pressure in a highly linear manner,whereas in the previous apparatus, the authors estimatethat nonlinearity in the measured deflection of the sec-ondary mirror may have been as large as 1%.Through these modifications to the experiment, Jones

and Leslie obtained final results with a standard devia-tion of only 0.05% (Jones and Leslie, 1978). In the ac-companying theoretical paper (Jones, 1978), Jones dis-cussed whether this corresponded to a photon momen-tum of nφp in accordance with the Minkowski tensor, orp/ng in accordance with the Abraham tensor (where p isthe momentum of the photon in free space, and nφ andng are respectively the phase and group refractive in-dices). Once again, he concluded that both were accept-able provided that in the Abraham case the total momen-tum nφp was divided into an electromagnetic componentp/nφ, and a mechanical component, which was expressedthis time as nφp[1− 1/(nφng)].

V. HYPOTHETICAL CONSTRAINTS AND ALTERNATE

THEORIES

Numerous attempts have been made to find addi-tional constraints by which one or another of the Abra-ham and Minkowski energy momentum tensors might beproven invalid, the earliest being the suggestion that theMinkowski tensor failed to conserve angular momentum.This led first Einstein and Laub (1908, 2005) and thenAbraham (1909, 1910) to attempt to develop symmetricforms of the electromagnetic energy–momentum tensor.The Einstein–Laub tensor enjoyed only limited attentionas it does not purport to be valid outside the rest frame

of the dielectric medium. Its use — indeed, even its valid-ity — in a science increasingly embracing the Principle ofRelativity was therefore considered questionable. It at-tracted considerably less interest than Abraham’s form,which was developed in a fully relativistic manner, work-ing from microscopic considerations.

In contrast, Dallenbach (1919) claimed to demon-strate the derivation of the Minkowski tensor frommicroscopic considerations. However, according tode Groot and Suttorp (1972) he fails to justify his gener-alisation from the electrostatic to the dynamic case, andPauli (1958, p.109) also deemed his argument “not verycogent”.

As we noted in Sec. III, concerns about non-conservation of angular momentum in the Minkowskielectromagnetic energy–momentum tensor arose fromconsideration of an incomplete system. When the ap-propriate material tensor is also taken into account, an-gular momentum is conserved (Sec. VI). Similarly theEinstein–Laub tensor can be saved from the ignonimy ofdepending upon a specific frame of reference.

Later, von Laue (1950) argued that the velocity of en-ergy propagation, which is less than c in a dielectricmedium, should add in accordance with the relativis-tic velocity addition theorem. An equivalent argument,developed by Møller (1952, 1972, pp. 206-211 and 221-225 respectively), is that the energy propagation velocityshould transform as a 4-vector. This is satisfied by theelectromagnetic energy flux under the Minkowski tensor,but not under the Abraham tensor. In the first editionof his work Møller concludes “. . . this is a strong argu-ment in favour of Minkowski’s theory”, but in the sec-ond edition this is softened to “. . . which shows thatMinkowski’s theory gives the most convenient descriptionof the phenomenon in question”. This alteration perhapsreflects a change in philosophy, away from attempting toprove that one tensor is wrong and the other is right,and towards recognising that both have their place in apractical implementation of electrodynamics.

Although it was ultimately dismissed byTang and Meixner (1961) and Skobel’tsyn (1973),von Laue’s argument was nevertheless influential. WhenPauli first wrote his encyclopedia article on relativity(Pauli, 1921), he favoured the Abraham tensor. How-ever, in the revised edition (Pauli, 1958), he inserted asupplementary note reversing his position on account ofvon Laue’s argument.

In addition to rebutting the arguments of von Laueand Møller, Skobel’tsyn also proposed criteria of hisown based on two thought experiments, which favouredthe Abraham tensor. These are described in detail byBrevik (1979), along with his reasons for rejecting Sko-bel’tsyn’s arguments. Brevik then concludes that the

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second thought experiment, which deals with a capaci-tor free to rotate within a magnetic field, would never-theless provide an experimental means to discriminatebetween the Abraham and Minkowski tensors. The be-haviour of the experiment depends upon the existence orotherwise of the Abraham force, as does the experimentof Walker et al. (1975). The rebuttals of Israel (1977)and Obukhov and Hehl (2003) discussed in Sec. IV.C areequally applicable to Skobel’tsyn’s second experiment,and hence both Skobel’tsyn’s arguments and Brevik’s in-terpretation of the second experiment are refuted.

Lai (1980) also published an argument against theMinkowski tensor based on non-conservation of angularmomentum, similar to Skobel’tsyn’s second argument.This was rebutted by Brevik (1982) by invoking the mate-rial counterpart to the Minkowski electromagnetic tensor.We raise this here to discuss Lai’s counter-argument (Lai,1984). In this, he argues that his thought experiment wasnot intended to distinguish between what he calls the“phenomenological” theories derived from global conser-vation of momentum, but rather to determine which ofAbraham’s and Minkowski’s tensors is “physically cor-rect”. How this is to be interpreted is unclear.

If Lai is advocating comparison of the historical formsof these tensors, then as neither incorporates effects suchas electrostriction and magnetostriction, both are incom-plete and demonstrably incorrect. We can only ask whichis least inaccurate in any particular situation, a conclu-sion of little general value.

If Lai permits the material tensor accompanying theAbraham tensor to be amended to incorporate such ef-fects, but does not permit the Minkowski tensor to re-ceive the same treatment, then he is crippling one of thecontenders before the competition, and to conclude thatone then performs better than the other is hardly sur-prising.

Finally, if both tensors are allowed such a counterpart,then we recover the treatment of Brevik, which Lai dis-misses as “phenomenological” and missing the point.

Alternatively, Lai may be implying that it is possi-ble to discriminate between the Minkowski and Abrahamtensors when both are accompanied by the appropriatematerial tensors. This position, however, has been dis-proven by others (Kranys, 1979a,b, 1982; Maugin, 1980;Mikura, 1976; Schwarz, 1992), and hence Lai’s thoughtexperiment appears to serve no useful purpose, in spiteof his defence.

It is of historical note that the Abraham tensor wasalso initially proposed without a material counterpart(Abraham, 1909, 1910; Brevik, 1970), and it was sev-eral decades before the role of this counterpart in de-termining experimental results was properly appreciated(see Sec. IV.B). Any treatment involving such a ma-

terial counterpart is to some extent phenomenological,with the nature of the counterpart being determinedor verified by observable fact (Jones and Leslie, 1978;Jones and Richards, 1954). It can be argued that onlyapproaches which begin from universal conservation lawsmay be exempt from this criticism, if criticism it is, butit is these very results which Lai appears to dismiss outof hand.Other arguments purportedly favouring one tensor or

the other had also been proposed (Gyorgyi, 1960; Tamm,1939; Tolman, 1934), but ultimately it was not to bethese which decided the debate. In the 1960s and 1970s,two developments occurred which changed the face ofthe argument completely. First, a number of further al-ternative energy–momentum tensors were proposed onvarious theoretical grounds (de Groot and Suttorp, 1967,1968, 1972; Grot and Eringen, 1966; Marx and Gyorgyi,1955; Peierls, 1976; Penfield and Haus, 1967). The de-bate was no longer over whether to choose Abraham’s orMinkowski’s formulation, but rather over the principlesupon which a valid electromagnetic energy–momentumtensor should be based. Second, Blount (1971) identifiedthe Abraham electromagnetic energy–momentum tensorwith classical momentum and the Minkowski tensor withcrystal momentum, or pseudomomentum. Although thework was unpublished it had considerable effect, stimu-lating a recognition that both energy–momentum tensorsmight have a valid role in appropriate circumstances.

Although not every proposed means of discriminat-ing between the Abraham and Minkowski tensors hasbeen explicitly refuted in the literature, there is littleto be gained from such a process. Inevitably, such at-tempts stem from inadequate exploration of the materialenergy–momentum tensor, and ultimately all such argu-ments are overturned by the direct proofs of equivalenceof the differing electromagnetic energy–momentum ten-sors discussed in Sec. VI. These proofs arise directly fromconservation of energy, linear and angular momentum forthe system as a whole, and hence it is only in those rarepapers which discard these constraints that differing pre-dictions can be made in a logically consistent manner.

VI. PROOFS OF EQUIVALENCE

A. Penfield and Haus; de Groot and Suttorp

Towards the end of the 1960s, a number of researcherswere demonstrating greater awareness of the role of thematerial energy–momentum tensor in symmetry and inmomentum conservation (de Groot and Suttorp, 1967;Grot and Eringen, 1966), and developing novel pairs ofelectromagnetic and material tensors with this in mind.

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Penfield and Haus (1967) took this further with the for-mulation of their “Principle of Virtual Power”, developedas a tool to test the equivalence of a number of differ-ent formulations of electromagnetic theory. By applyingthis principle they were able to determine the incom-pleteness of both the Minkowski and Abraham tensors,and also of the tensors emerging from the Boffi (1957),Amperian (Fano et al., 1960) and Chu (Chu et al., 1965;Penfield and Haus, 1967) formulations of electromag-netism, and to identify the missing terms, and demon-strate that with properly constructed material counter-parts, all these energy–momentum tensors are in agree-ment. A cogent summary of their arguments and resultsmay be found in a review paper by Robinson (1975),which also covers the approach of Gordon, discussed inSec. VI.B.

Like Penfield and Haus, de Groot and Suttorp (1972)also recognised the important role of the material energy–momentum tensor, and that conservation laws relatingto energy and momentum should only be applied toclosed thermodynamic systems. However, they disagreewith Penfield and Haus on the correct form of the to-

tal energy–momentum tensor to which electromagneticand material energy–momentum tensors must sum. Thisdisagreement does not bear directly upon the Abraham–Minkowski controversy, but does reflect different overallassumptions about the behaviour of matter in the pres-ence of an electromagnetic wave. De Groot and Suttorphave argued for the superiority of their expression as it isderived from considerations of the microscopic propertiesof the medium. As yet, no experimental comparison ofthese two different total energy–momentum tensors hasbeen performed.

For the moment, however, the correct expressionfor the total energy–momentum tensor is unimportant.While eventually it might prove beneficial to our mod-elling of material behaviour to discriminate between thedifferent propositions, for our purposes it suffices to de-note the selected total energy–momentum tensor by Tαβ.This tensor is then constrained by conservation of linearand angular momentum such that

∂αTαβ = 0 (2)

and

Tαβ = T βα, (3)

(Jackson, 1999), where greek indices range over 0-3.

A more familiar expression of the conservation of linearmomentum may be

∂t gi + ∂j T

ij = 0, (4)

where latin indices range over 1-3, ∂t represents thederivative with respect to time, g is the momentum den-sity, and T is the three-dimensional stress tensor. Thestress tensor T is related to the energy–momentum ten-sor T via

Tij = −T ij. (5)

For an electromagnetic wave in free space, T is calledthe Maxwell stress tensor, but in dielectric media twodifferent conventions exist (Stallinga, 2006b) so unlessexplicitly defined, this nomenclature is best avoided.A further constraint

∂t g0 − ∂j T

0j = 0 (6)

corresponds to the conservation of energy, where g0 isthe energy density, and T 0j is the energy flux. For anelectromagnetic wave in free space, T 0j = Sj/c where S

is the Poynting vector. Identifying (g0,g) as a 4-vectorand writing

Tαβ =

(

g0 T 0j

cgi −Tij

)

, (7)

we see that (4) is equivalent to constraint (2) above.Defining g′ = (1/c)T 0j for symmetry, constraint (3) re-quires that g = g′, giving

Tαβ =

(

g0 cgj

cgi −Tij

)

. (8)

Unlike the system described by the total energy–momentum tensor, the one described by the electromag-netic energy–momentum tensor need not be thermody-namically closed. Therefore, one must consider the com-bined system of both the electromagnetic and materialenergy–momentum tensors, such that

Tαβ = TαβEM + Tαβ

mat (9)

and write

∂α

(

TαβEM + Tαβ

mat

)

= 0 (10)(

TαβEM + Tαβ

mat

)

=(

T βαEM + T βα

mat

)

. (11)

As before, we may rewrite (9) and (10) in terms of g andT:

g = gEM + gmat (12)

T = TEM +Tmat (13)

∂t (giEM + gi

mat) + ∂j (TijEM +T

ijmat) = 0 (14)

∂t (g0EM + g0mat)− ∂j (T

0jEM + T 0j

mat) = 0. (15)

8



9

Again we can define g′ = (1/c)T 0j, giving

T = TEM + Tmat (16)

=

(

g0EM + g0mat cg′j

c(giEM + gi

mat) −(TijEM +T

ijmat)

)

. (17)

g′ must also be subdivided,

g′ = g′EM + g′

mat, (18)

and from (11) we see that

g′EM + g′

mat = gEM + gmat, (19)

but no constraint is placed upon the subdivision of g′ intoelectromagnetic and material components. In energy–momentum tensors where the electromagnetic portion issymmetric, such as the Abraham tensor, it follows that

gEM = g′EM (20)

gmat = g′mat, (21)

but this is not obligatory. For the Minkowski tensor,gEM = D × B, whereas g′

EM = (1/c2)E ×H. Similarly,gmat 6= g′

mat.When the Minkowski tensor is used, Tmat is frequently

ignored as the material components of g and T are typ-ically safely negligible (see Sec. VIII.D). However, be-cause gEM 6= g′

EM under the Minkowski tensor, this givesthe illusion of non-conservation of angular momentum via(3). The Minkowski electromagnetic energy–momentumtensor has received much criticism for this reason, due tofailure to recognise the existence of its material counter-part.Historically, neither the Minkowski nor the Abraham

tensors were proposed in the context of a matched pairof electromagnetic and material energy–momentum ten-sors, and although the need for a material counterpart tothe Abraham tensor was proposed by Jones and Richards(1954), the Minkowski tensor only acquired a mate-rial counterpart through the theoretical re-evaluationsof the 1970s (de Groot and Suttorp, 1972; Israel, 1977;Mikura, 1976), explaining the results of James (1968a)and Walker et al. (1975) discussed in Sec. IV.C.We can now explain why experiments continue to be

announced from time to time which purport to dis-criminate between the Minkowski and Abraham energy–momentum tensors, only to be disproven a few yearslater. For brevity we adopt the notation of Eqs. (2)and (3). Any experiment will be sensitive only to certainterms in the complete energy–momentum tensor. An ex-periment involving only angular momentum will be in-sensitive to independently symmetric terms, and one in-volving only linear momentum will be insensitive to terms

which are independently divergence-free. For any givenexperiment, we may therefore define TX as the terms inthe material tensor which have no significant effect onthe result, and Tmat′ as

Tmat′ = Tmat − TX. (22)

Using Tmat′ in lieu of Tmat to make predictions for thebehaviour of one experiment will therefore still yield thecorrect result, but for another experiment it may not.To illustrate this, consider the Minkowski electromag-netic energy–momentum tensor. In general, this tensoris used in isolation, letting Tmat′ = 0. The Minkowskitensor is asymmetric, and hence from Eqs. (3) and (11)we can conclude that TX also contains asymmetric terms.The Minkowski electromagnetic energy–momentum ten-sor largely functions well in linear experiments such asthat of Jones and Richards (Sec. IV.A above), but inisolation fails to correctly describe the rotary experi-ment of Walker et al. (Sec. IV.C). Israel (1977) andObukhov and Hehl (2003) demonstrated that this is dueto neglect of the corresponding material tensor.A given electromagnetic and material tensor pair may

be found to perform well across a broad range of exper-iments before failing. This failure can arise as a resultof an interaction which was not probed in the previousexperiments, and is therefore described by a term whichdoes not influence those previous experiments. A fre-quent reaction is to announce that the experiment in-validates one of the electromagnetic energy–momentumtensors being considered. However, inevitably a suit-able term will be identified and added on to the cor-responding material energy–momentum tensor, or a newexpression developed which is equivalent to the additionof such a term, and the new “correct” material energy–momentum tensor will be announced, rebutting the inval-idation (Brevik, 1982; Israel, 1977; Obukhov and Hehl,2003). As the tensor pair is named by the structure ofthe electromagnetic member, and the necessary term mayalways be added to the material member, the tensor istherefore always able to be “saved”.As an interesting alternative, the necessary term may

be added to the electromagnetic tensor, and accompa-nied by a change of name. Neither the Abraham norMinkowski tensors as they are commonly used incorpo-rate the electrostrictive effect. When the relevant term isadded to the electromagnetic component of the Abrahamtensor, this is known as the Helmholtz tensor (Brevik,1982).Regrettably, the full ramifications of the works of Pen-

field, Haus, de Groot and Suttorp were not immediatelyappreciated by the physics community, perhaps becausecalculating the full expressions for the material energy–momentum tensors was somewhat laborious, and per-

9



10

haps because the experiment of Walker et al. (1975) hadyet to be performed, that of James (1968a,b) was notwidely publicised, and thus no convincing situation hadyet been recognised for which the conventional forms ofthe Minkowski and Abraham tensors failed or disagreed.

In addition, de Groot and Suttorp derived a num-ber of new tensor pairs which they felt were partic-ularly relevant to specific situations, such as a neu-tral plasma, and which are documented in their book(de Groot and Suttorp, 1972).

B. Gordon’s Analysis

A more practically oriented demonstration of theequivalence of the Abraham and Minkowski tensors wasachieved by Gordon (1973). In response to the exper-iment of Ashkin and Dziedzic (see Sec. IV.B above),he demonstrated that the field-related components ofthe material energy–momentum tensor propagated alongwith the electromagnetic component, rather than at thespeed of sound in the medium as was sometimes assumed(for example Burt and Peierls, 1973), and that the op-tical pulse length in Ashkin and Dziedzic’s experimentwas sufficiently long for pressure within the fluid to reachequilibrium, and so purely material effects such as pres-sure also needed to be considered. He then went on toshow that both Abraham’s and Minkowski’s tensors pre-dicted identical behaviour when an arbitrary radiationpulse traversed an air/fluid interface.

We note that all real material bodies are to some extentcapable of deformation, and hence allow the transmissionof tension and pressure in a manner analogous to the ef-fects considered in Gordon’s proof. The proof thereforereadily generalises to arbitrary isotropic dielectric me-dia for deformations small enough that dissipative effectssuch as hysteresis may be ignored, and has the potentialfor extrapolation to larger deformations and anisotropiccases. Inhomogeneous dielectric materials might then bedealt with by breaking them down into infinitesimallysmall homogeneous chunks. However, as pointed outby Robinson (1975), Gordon’s result is only explicitlydemonstrated for materials of dielectric constant suffi-ciently close to unity.

Gordon’s work was subsequently extended by Peierls(1976, 1977) with a number of interesting consequences,such as the generation of phonons by a finite width beamin an elastic solid.

Recognition of the deformable nature of the dielectricmedium is fundamental to Gordon’s approach. We dis-cuss the hazards associated with inappropriately treatingthe dielectric as a rigid body in Sec. IX.

C. Mikura, Kranys and Maugin

Although the work of de Groot and Suttorp (1972)had provided the theoretical framework within which todemonstrate the formal equivalence of different choices ofelectromagnetic energy–momentum tensor, they stoppedshort of explicitly demonstrating this for commonly usedtensors such as those of Abraham and Minkowski. Thisequivalence had been demonstrated by Penfield and Hausin 1967, but it seems to have been little appreciated.Subsequent, far shorter demonstrations of equivalence

were produced by Mikura (1976) and by Kranys (1979a,b,1982), who appears unaware of the work of either Pen-field and Haus or de Groot and Suttorp, as he cites laterpapers by Israel (1977, 1978) as conjecturing a “slid-ing” relationship between electromagnetic and materialenergy–momentum tensors. The paper by Mikura is par-ticularly readable, but both are recommended to any-one seeking greater mathematical detail than is providedin this review. Ginzburg (1973) also demonstrates thatthe behaviours predicted by the classical Abraham andMinkowski tensors are equivalent up to the action of theAbraham force, but does not take the final step of pos-tulating a material counterpart to the Minkowski tensor.He therefore concludes that while the Minkowski tensoris not strictly correct, it is nevertheless useable undermany circumstances.A response to Kranys (1979a) by Maugin (1980) sup-

ports Kranys’s result but criticises him for not in-cluding enough references, and for confining himself toconsidering only the Abraham and Minkowski tensors.Maugin extends Kranys’s work to consider tensors dueto Grot and Eringen (1966) and de Groot and Suttorp(1972) in non-dissipative materials, with remarks aboutextension to more arbitrary cases, and the use of thisapproach to demonstrate the equivalence of any otherenergy–momentum tensor with those already consid-ered. Schwarz (1992) then extends the work of Kranysand Maugin to include media which may be magnetisedand/or electrically polarised.The interested reader may also have inferred the gen-

eral equivalence of all choices of electromagnetic energy–momentum tensor from Eqs. (2), (3) and (9), as valid ex-pressions for the total energy–momentum tensor T mayonly vary by a term which is symmetric and divergencefree, and hence carries no momentum at all.

VII. MOMENTUM AND PSEUDOMOMENTUM

As mentioned in Sec. V, an unpublished memoran-dum by Blount (1971) is credited with first drawing theparallel between the Minkowski electromagnetic momen-

10



11

tum density and pseudomomentum. In doing so, he in-troduced a viewpoint in which both the Abraham andMinkowski expressions had a valid role. This view is fur-ther developed by Gordon (1973), Peierls (1991), Nelson(1991), and Stallinga (2006a), and our review would beincomplete without discussing this important recogni-tion.

Balazs (1953) provides a thought experiment which wemay use to illustrate this. Consider a perfectly trans-parent glass rod, initially at rest, with a perfect anti-reflection coating on each end. A wave packet may passeither alongside the rod, or through it. No external forceacts on the combined system of rod and wave packet,so the centre of mass of the system must have constantvelocity. However, if the wave packet enters the rod, itslows down by a factor of 1/n. This must therefore beaccompanied by a movement of the rod in the directionof propagation of the wave packet so that the velocityof the centre of mass is preserved. As the rod is per-fectly transparent, no momentum is absorbed from thewave packet, and when the wave packet departs the rodit must leave the rod at rest. By comparing the trajecto-ries of the centre of mass with the wave packet passing ei-ther inside or outside the glass rod, we can show that therod experiences a finite displacement during the traversalof the wave packet, which we may interpret as resultingfrom the action of the Lorentz force on the atomic dipoleswhich make up the rod.

While the rod is moving, it clearly has momentum.The Abraham approach identifies this as mechanical mo-mentum. However, the Minkowski approach recognisesthat this momentum propagates with and arises from thepresence of the electromagnetic wave, and hence classifiesit as electromagnetic momentum.

We may well have qualms about describing the mo-tion of a material body as electromagnetic momentum.Therefore, we may call the Abraham momentum thetrue momentum of an electromagnetic wave, and theMinkowski momentum the pseudomomentum, this beingthe physically useful quantity of all momentum associ-ated with the propagation of the wave. (Nelson adoptsa different terminology, in which pseudomomentum ex-cludes electromagnetic momentum, and the total mo-mentum associated with the propagation of a wave istermed “wave momentum”. His chosen expression forelectromagnetic momentum also differs, though this hasno effect on the total momentum. Nelson identifies theMinkowski momentum with wave momentum, less a termdue to dispersion.)

This argument may appear compelling, but before thereader confidently declares the Abraham momentum tobe the only true electromagnetic momentum, they shouldrecall that every expression appears compelling within

the context of the arguments on which it is founded.For example, Mansuripur (2004) presented an alternativechain of argument leading to an electromagnetic momen-tum density of g = 1

2D×B+ 12E×H/c2. This expression

seems unlikely in the context of the above discussion ofmomentum and pseudomomentum, but in the approachtaken by Mansuripur it is the Abraham and Minkowskiexpressions which appear inappropriate, and this hybridelectromagnetic momentum density which seems the in-evitable conclusion. Of course, it is momentum trans-fer which determines experimental results, and this takesplace identically regardless of whether we describe itas electromagnetic or material in nature. We thereforereprise our position stated in the abstract, that the readermay describe as electromagnetic whatever portion of thetotal momentum flux appears most appropriate, in thecontext in which they are working. Provided all momen-tum is ultimately accounted for, this will have no effecton their predicted experimental results.Gordon (1973) has given the conditions under which

the pseudomomentum approach will reliably describemomentum transfer to an object in a fluid dielectricmedium. The factor |E|2∇ε must be negligible in thevicinity of a surface surrounding the object of interest,and sufficient time must be allowed for pressure in thefluid to reach equilibrium. In contrast the total energy–momentum tensor approach described in Sec. VIII maybe employed even when the constituent variables arelargely arbitrary functions of space and time, though thecost is a considerable increase in mathematical complex-ity.Performing calculations using the pseudomomentum

density D×B (in Gordon’s notation and Gaussian units,ǫG) is directly equivalent to choosing the Minkowskielectromagnetic energy–momentum tensor without a ma-terial counterpart. It therefore omits a small numberof effects associated with the material counterpart ten-sor, including the Abraham force, but these are sel-dom significant. Situations suited to the use of theisolated Minkowski electromagnetic tensor, and hencealso the pseudomomentum approach, are discussed inSec. VIII.C.1 and VIII.D.

VIII. THE TOTAL ENERGY–MOMENTUM TENSOR

The obvious question now becomes, what is the totalenergy–momentum tensor? Can we write down a singleexpression that will serve our needs in all circumstances?Unfortunately, the answer is no. This is not a problemin electromagnetism. The limitation is instead our un-derstanding of material science. We may certainly writedown all contributions to the energy–momentum tensor

11



12

which are relevant in a particular perfectly defined idealmedium, but to do so for a generalised real medium liesbeyond the scope of our current understanding.Fortunately, in many circumstances a complete under-

standing of the material properties of the medium is notrequired. The medium may behave sufficiently like a suit-able ideal, or we may be interested solely in the behaviourof the centre of mass, and we may then neglect the ef-fects of internal tensions which merely redistribute ex-isting force and torque densities. Provided our descrip-tion of the interaction of the material object with theelectromagnetic wave is sufficiently accurate, details ofmaterial-material interactions may then be overlooked.There is another important caveat. When material

stresses induced by an electromagnetic wave cause de-formation of the boundary of the dielectric medium, fo-cussing effects may then alter the subsequent path ofthe beam, as in the experiment of Ashkin and Dziedzic(1973). The results calculated for a given material con-formation will then only be instantaneously true. Thiseffect may of course neglected if the deformation of thematerial medium is sufficiently small.

A. A Total Energy–Momentum Tensor

Mikura (1976) derives a total energy–momentum ten-sor for a nonviscous, compressible, nondispersive, polar-isable, magnetisable, isotropic fluid. Electrostrictive ef-fects, magnetostrictive effects, and acoustic waves are in-cluded, though in most circumstances they may be omit-ted for an increase in simplicity, as shown by Mikura inthe latter part of his paper. We have converted fromMikura’s choice of Gaussian units and matrix notationto SI units and tensor notation. Mikura’s approach mayalso be extended to dispersive media, as discussed inSec. VIII.B.1. We will therefore distinguish between thephase and group refractive indices, denoted nφ and ng

respectively.The total energy–momentum tensor is given by

T µν = T µν

(m) + T µν

(f) + T µν

(P ) + T µν

(M) + T µν

(d) (23)

where

T µν

(m) =(

ρ0c2 + ρ0ǫi

)

uµuν + φ (uµuν + δµν)

T µν

(f) = ε0c2

(

FµγF νγ −

1

4F 2γδ δ

µν

)

T µν

(P ) =1

α

(

PµγP νγ −

1

4P 2γδ δ

µν

)

T µν

(M) = −FµγM∗ νγ −

1

4βM2

γδ δµν

T µν

(d) =1

βMµγMν

γ + F ∗µγMνγ .

We have slightly condensed Mikura’s original scheme.T(m) contains terms due to mechanical fluid flows, elec-trostriction, and magnetostriction. T(f) contains fieldterms identical to those in free space. T(P ) contains termsrelating to the polarisation of the medium, and T(M) andT(d) contain terms relating to magnetisation. T(d) is sep-arated from T(M) to facilitate the subsequent derivationof the Abraham and Minkowski electromagnetic energy–momentum tensors.In these expressions, ρ0 is the matter density in the

local rest frame, u is the 4-velocity of an element of thelocal medium, ǫi is the specific internal energy of non-electromagnetic nature and is a function of ρ0 and thespecific elastic entropy s, α and β are related to the elec-tric and magnetic constants of the medium by α = ε−ε0and β = µ−1

0 − µ−1, δµν = 1 for µ = ν and zero other-wise, φ is the total pressure including electrostrictive andmagnetostrictive effects, given by

φ = φh −1

4KaP

2γδ +

1

4KbM

2γδ (24)

for

Ka = ρ0∂(1/α)

∂ρ0

∣

∣

∣

∣

s

, Kb = ρ0∂(1/β)

∂ρ0

∣

∣

∣

∣

s

, (25)

and φh is the total hydrostatic pressure in the fluid,

φh = ρ20∂ǫi∂ρ0

. (26)

F is the usual electromagnetic field tensor

Fµν =

0 −E1

c−E2

c−E3

cE1

c0 −B3 B2

E2

cB3 0 −B1

E3

c−B2 B1 0

, (27)

and the polarisation and magnetisation tensors P andM are related to the three-dimensional polarisation andmagnetisation vectors P and M in a given frame by

Pµν = (1/c)(vµPν − vνPµ) (28)

Mµν = (1/c)(vµMν − vνMµ) (29)

where v is the 3-velocity of an element of the localmedium. We define v0 = c, P0 = M0 = 0. P andM are related in turn to E, B, D, and H via D = εE,B = µH, and the relativistic constitutive relations

D = ε0E+P+1

c

(v

c×M

)

(30)

H =1

µ0B−M + c

(v

c×P

)

. (31)

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13

For F, M, and P, the notation X2γδ abbreviates

XγδXγδ, and X∗ denotes the dual tensor

X∗µν =1

2εµνγδXγδ (32)

where ε is the Levi–Civita symbol.

The total momentum flux and stress tensor under thistotal energy–momentum tensor may be obtained via gi =1cT i0 and Tij = −T ij.

g = [ρ0(c2 + ǫi) + φ]γ2v

c+ ε0E×B+

1

cαP×

(v

c×P

)

+1

cβM×

(v

c×M

)

−1

cB×

(v

c×M

)

+1

c2E×M (33)

T = −ρ0(c2 + ǫi)γ

2v ∧ v

c2− φ

(

γ2v ∧ v

c2+ I)

+ ε0E ∧E+1

µ0B ∧B−

1

2

(

ε0|E|2 +1

µ0|B|2

)

I

+1

α

[

P ∧P+(v

c×P

)

∧(v

c×P

)

−1

2

(

∣

∣

∣

v

c×P

∣

∣

∣

2

− |P|2)

I

]

+1

β

[

M ∧M+(v

c×M

)

∧(v

c×M

)

−1

2

(

∣

∣

∣

v

c×M

∣

∣

∣

2

− |M|2)

I

]

(34)

−B ∧M−M ∧B+E

c∧(v

c×M

)

+(v

c×M

)

∧E

c−

[

E

c·(v

c×M

)

−B ·M

]

I.

x ∧ y denotes the outer product, (x ∧ y)ij = xiyj .Note that the trace over all field-related components

of the total energy–momentum tensor vanishes, and itis therefore compatible with massless photons (Jackson,1999).Following Mikura, we also subdivide this tensor to

yield the Abraham and Minkowski electromagnetic andmaterial energy–momentum tensor pairs. For the Abra-ham tensor, we simply write

T µνmat,Abr = T µν

(m) (35)

T µνEM,Abr = T µν

(f) + T µν

(P ) + T µν

(M) + T µν

(d). (36)

To obtain the Minkowski tensor we define

Cµν =1

αPµγP ν

γ − FµγP νγ (37)

and then we find

T µνmat,Mink = T µν

(m) + T µν

(d) + Cµν (38)

T µνEM,Mink = T µν

(f) + T µν

(P ) + T µν

(M) − Cµν . (39)

In the nonrelativistic limits, these yield the usual expres-sions for the electromagnetic energy–momentum tensors:

T µνEM,Abr =

(

12 (E ·D+H ·B) 1

cE×H

1cE×H −E ∧D−H ∧B+ 1

2 (E ·D+H ·B) I

)

(40)

T µνmat,Abr =

(

ρ0(c2 + ǫi) ρ0cv

ρ0cv ρ0v ∧ v + φI

)

(41)

T µνEM,Mink =

(

12 (E ·D+H ·B) 1

cE×H

cD×B −E ∧D−H ∧B+ 12 (E ·D+H ·B) I

)

(42)

T µνmat,Mink =

(

ρ0(c2 + ǫi) ρ0cv

ρ0cv − cD×B+ 1cE×H ρ0v ∧ v + φI

)

. (43)

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14

In this representation, the terms in the Minkowski ma-terial counterpart giving rise to the Abraham force mayclearly be seen in Eq. (43). In contrast, the field-relatedcomponent of the Abraham material momentum tensorarises from an implicit dependence of v on E, B, D, andH. Also note that although the Minkowski material mo-mentum density is frequently very small or zero, this doesnot reflect the behaviour of the particles of the dielectricmedium, whose velocity is related only to the term ρ0cv.Now consider a wave packet of total momentum p and

volume V incident upon a dielectric slab of group re-fractive index ng. The velocity of the wave packet slowsto c/ng on entering the slab, and its volume reduces toV/ng. The total momentum must remain constant at p,and hence the total momentum density increases fromgf in free space to nggf within the material. However,under the Abraham tensor the electromagnetic momen-tum density within the material is only gf and henceit must be accompanied by a material momentum of(ng − 1)gf . Identifying this with the corresponding termin (41), and substituting for gf the free space momentumdensity E×H/c2, we obtain

gmat,Abr = ρ0v = (ng − 1)E×H

c2. (44)

This expression holds for both dispersive and nondisper-sive media. It is trivial to show that the same expressionfor v is obtained under the Minkowski tensor pair.This result differs from the expression obtained by

Jones and Richards (1954) and Jones (1978), who arguedthat an electromagnetic wave propagating in a dielec-tric medium at velocity c/ng with momentum densitygm will give rise to a momentum flux gm/ng. Based onan observed momentum flux of nφgf , where gf is the mo-mentum density in free space, they concluded the totalmomentum density associated with the wave within thedielectric must be given by gm = ngnφgf . This yieldsa material counterpart to be added to the Abraham mo-mentum density of

gmat,Jones = (nφng − 1)E×H

c2(45)

which simplifies to

D×B−E×H/c2 (46)

in nondispersive media and generates the Abraham force(1). However, this result is at odds with that ofWalker et al. (1975), who demonstrated that the Abra-ham electromagnetic momentum density is correct, re-quiring that Eq. (45) or (46) instead be deducted from theMinkowski momentum density, as seen in our Minkowskimaterial tensor (43).

The error here is in the treatment of momentum flux.While the expression gm/ng will hold for a single wavepropagating in isolation through the dielectric medium,it breaks down for superpositions such as the incidentand reflected beams present here, and ∂j T

ij must beused instead. The incident beam is associated with amedium particle velocity of v and the reflected beam witha particle velocity of −v. While the accompanying fieldsinterfere, the velocities simply add, and the medium ad-jacent to the mirror remains stationary. The momentumflux is therefore dependent only on field-related terms.Unlike gm/ng, ∂j T

ij is insensitive to the choice of Abra-ham or Minkowski momentum density. Taking properheed of the effect on the magnetic field of time-varyingcurrents and charges induced within the medium, weobtain via Eq. (53) a flux of nφgf , which is in accor-dance with experiment. As this result is independentof the choice of Minkowski or Abraham electromagneticenergy–momentum tensor, we are free to assign (46) tothe Minkowski tensor in agreement with Walker et al.(1975).

We also note a further advantage to our expression formomentum density: Consider a wave packet of total mo-mentum p and unit volume entering the medium. Un-der the total energy momentum tensor approach givenhere, the volume of the wave packet falls to 1/ng, itstotal momentum density rises by a factor of ng, and mo-mentum is conserved. Under Jones and Richards’ ap-proach the total momentum density rises by a factorof nφng, giving a momentum associated with the wavepacket of nφp. Conservation of momentum is usually re-stored by positing a transfer of momentum −(nφ−1)p tothe medium boundary (see, for example, Loudon et al.,2005). Nevertheless, in steady state experiments such asJones and Leslie (1978) we might expect this momentumto be conducted to the mirror via pressure effects withinthe medium, reducing momentum transfer to the mirrorat equilibrium from nφp to just p. This reduction is notobserved, and this supports our approach over that ofJones and Richards.

B. Forces Acting On a Subsytem

While it is always possible to return to first principles,in the manner of Gordon (1973), and individually evalu-ate all the forces acting in a situation, this is frequently afar from simple task. However, where appropriate expres-sions for the total energy–momentum tensor are known,force and torque densities for a specific subsystem maybe calculated directly. The means by which they arecalculated will depend upon whether the subsystem isspatially or qualitatively distinguished from the system

14



15

as a whole.We use the covariant generalisation of angular momen-

tum density

Mαβγ = Tαβxγ − Tαγxβ (47)

given in Jackson (1999), where x is the 4-vector positionco-ordinate relative to the point about which torque isdetermined. M0jk is the 3x3 antisymmetric angular mo-mentum tensor whose three independent elements maponto the angular momentum density pseudovector l via

li =1

2εijkM

0jk (48)

where ε is the Levi–Civita symbol. The elements M0j0 =−M00k represent the puzzling yet orthogonally necessarytemporal components of angular momentum, and M iβγ

describes angular momentum flux, analogous to T iβ inthe total energy–momentum tensor. From (47), (2), and(3), it follows that

∂α Mαβγ = 0, (49)

which corresponds to conservation of angular momen-tum.The torque density tensor will be denoted tjk, and

maps onto the torque pseudovector via ti =12εijkt

jk.

1. Qualitatively Distinguished

Let the system be divided into two parts, A and B,which are qualitatively different but occupy the same re-gion of physical space. We wish to calculate the mo-mentum transfer from A to B. For example, B mightbe a material component, such as a dielectric medium,and A the electromagnetic field. We also similarly di-vide the linear momentum density, angular momentumtensor, and stress tensor into parts A and B. Into gB,MB and TB we place all terms relating to the physicalmomentum of the material component, as it is the forceand torque on the material that we wish to evaluate. Theremainder of the terms are placed into gA, MA and TA.The force and torque densities on component B are thenevaluated using

f iB = −∂t giA − ∂j T

ijA (50)

tjkB = −∂tM0jkA + ∂i M

ijk. (51)

These are momentum balance equations, with the rate ofmomentum and torque transfer into system B equallingthe rate of loss from system A.

Integrating over the volume of the subsystem thengives the total force or torque exerted on B by A as a re-sult of coupling between the two subsystems, for exampleabsorption of momentum from an electromagnetic field.

FiB =

∫

dV f iB , τ jkB =

∫

dV tjkB . (52)

F and τ represent the total force and torque respectively.The introduction of a complex refractive index al-

lows the modelling of electromagnetic wave propagationthrough media demonstrating loss or gain, and it is theresulting dependency of the magnitudes of the electricand magnetic fields on position and time that gives riseto a non-zero time average for the force or torque of (50)and (51). Of course, by the Kramers–Kronig relationsany absorptive material medium is also necessarily dis-persive. This may be modelled by describing a sepa-rate energy–momentum tensor with appropriate dielec-tric and magnetic constants for each mode of the electro-magnetic wave, and expressing the refractive index as acomplex function of the wave number, n(k).For near-monochromatic waves in the steady state, dis-

persion may frequently be neglected and the use of a sin-gle electromagnetic energy–momentum tensor with thematerial counterpart tensor may suffice.

2. Spatially Distinguished

Now suppose we are interested in momentum flux intoor out of a particular region of space. We denote thisregion B and the outside region A. By Gauss’s Law, therate of change of momentum within region B is depen-dent upon the rate of flux through the boundary. Trans-forming Eq. (52) into surface integrals, we obtain

FiB = −

∮

dSj TijA,in −

∮

dSj TijB,out, (53)

τ jkB =

∮

dSi MijkA,in +

∮

dSiMijkB,out. (54)

The normal to the boundary is taken as pointing fromB to A. The first integral in each equation evaluatesflux from A to B, and only terms inbound to region Bshould be retained. Similarly, the second integral evalu-ates flux from B to A and only outbound terms shouldbe retained. It is not required that regions A and B bematerially identical. Evaluation of these surface integralsmay frequently be simplified by judicious application ofGauss’s law.Note that the form of the momentum entering area B

— mechanical or electromagnetic — is not specified.

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3. General Subsystem

In more general circumstances, for example momen-tum transfer to a solid dielectric block immersed in adielectric fluid and exposed to an electromagnetic field,both spatial and qualitative considerations may apply.The block may receive momentum through its bound-aries as a result of the external medium (e.g. pressure)and the external electromagnetic field (e.g. reflection),and within its bulk as a result of coupling between mat-ter and the electromagnetic field within the block (e.g.absorption). We denote the region containing the blockregion B, and the external region A.

As in Sec. VIII.B.1, we divide the linear momen-tum density, angular momentum tensor, and stress ten-sor within the block into material components, denotedgB,mat, MB,mat, and TB,mat, and electromagnetic com-ponents, denoted gB,EM, MB,EM, and TB,EM, purely de-pendent upon whether they relate to the outcome wewish to determine — the physical momentum of the di-electric block. We then label the region external to theblock region A, and similarly divide its stress tensor intomaterial and other components, except that here, a termis considered material if it will transfer momentum intothe material component of section B on encountering theboundary. The equations for the force and torque actingon the dielectric block then become

FiB,mat = −

∫

dV(

∂t giB,EM + ∂j T

ijB,EM

)

−

∮

dSj

(

TijA,mat,in +T

ijB,mat,out

)

(55)

τ jkB,mat = −

∫

dV(

∂t M0jkB,EM − ∂i M

ijkB,EM

)

+

∮

dSi

(

M ijkA,mat,in +M ijk

B,mat,out

)

. (56)

Further subdivision of field-related tensors into the differ-ent electromagnetic modes may be required for the mod-elling of dispersive media, as indicated in Sec. VIII.B.1.

The labelling of terms in region A as “material” shouldnot be taken literally. For example, if the block in regionB is perfectly reflective, then all electromagnetic waves inregion A would be classified as “material” as they wouldimpart their momentum to the material component of re-gion B. The designation of “material” is functional andcontext-dependent, and should not be taken as propos-ing a clear and unique subdivision in the manner of theAbraham or Minkowski approaches.

At this point we will be well served by examining someexamples within the literature.

C. Examples

1. Reflection of Light Off a Mirror

Consider the reflection of light off a perfect mir-ror suspended within a dielectric. We wish to eval-uate the momentum transferred to the mirror, as inJones and Richards (1954) or Jones and Leslie (1978).This takes place only at the surface of the mirror, andtherefore we can use the surface integral (53). For TA

we take the total stress tensor within the dielectric. Wemay discard the term in TB as there will be no momen-tum flux from the mirror to the dielectric. As noted inSec. VIII.A, the medium adjacent to the mirror remainsstationary because the velocities induced by the incidentand reflected beams sum to zero. Momentum is thereforetransferred by the electromagnetic fields alone.Historically the Minkowski electromagnetic energy–

momentum tensor in isolation has also been used herewith great success. Although the assumed momentumdensity differs from that yielded by the total energy–momentum tensor, the simultaneous identification of themomentum flux as g/ng instead of ∂j T

ij yields the cor-rect overall result, up to a factor of ng/nφ. This worksbecause the Minkowski material momentum density istypically numerically very close to zero, and hence the to-tal momentum density may be reasonably approximatedby the electromagnetic momentum density D×B. Thatthe total momentum flux is given by g/ng then followsfrom Gauss’s Law. The result is satisfactory when theerror introduced by this approximation is small.

2. Transmission of Light Through a Nonreflective DielectricBlock

Now consider a beam of light incident upon a blockof perfectly nonreflective dielectric of higher refractiveindex than the surrounding medium. Balazs (1953) con-siders the transmission of a wave packet through a glassrod with anti-reflection coatings on both ends. If we areinterested in the behaviour of the surface, then surfaceterms must be taken into account using (55), but if weare only interested in the motion of the centre of mass ofthe block, then we can neglect the surface forces providedthe time average of the integral (53) goes to zero over aperiod much less than the sensitivity of our apparatus.

For a laser beam traversing a dielectric block, we mayeliminate the field-related terms of the integral if the elec-tromagnetic fields where the beam enters and leaves theblock are identical up to a phase. At its simplest this re-quires zero reflection, zero absorption, and that the twosurfaces be parallel. We may eliminate the remainder of

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the terms by choosing the surrounding medium to be vac-uum. The behaviour of the centre of mass of the blockmay then be determined using (52). For nφ > 1, theblock accelerates away from the beam source while it istraversed by the leading edge of the beam, then contin-ues to travel away from the source at constant velocitywhile the beam is turned on. When the beam is turnedoff, traversal of the trailing edge restores the block torest. The use of (52) can also readily be extended toinclude absorption, giving rise to an additional ongoingacceleration.

3. Transmission and Reflection Off a Dielectric Block

The most general case to consider is a partially reflec-tive dielectric block suspended within an arbitrary dielec-tric medium. In the absence of simplifying assumptionswe must employ the full expression (55), with both a sur-face and a volume term. We may compare this with thework of Mansuripur (2004), who calculated the forces ona dielectric using the Lorentz force. Integration of theLorentz force density over the volume of the dielectricgives a volume term, whereas reflection of a portion ofthe incoming radiation gives rise to a surface term. Al-though Mansuripur examined pressure and tension aris-ing within the dielectric block as a result of the incidentradiation, contributions of these effects to the net sur-face force, as in Gordon (1973), are not discussed. Theseforces make no contribution to the motion of the centreof mass, but will affect the behaviour of the surface if theblock is treated as flexible.Mansuripur obtained an expression for the electromag-

netic momentum density in a dielectric medium fromthe rate of transfer of momentum into the medium bothas electromagnetic fields and via the Lorentz force, butexcluded momentum transferred to the surface via re-flection, which he considered non-electromagnetic. How-ever, this merely represents a further scheme by which tosubdivide the total energy–momentum tensor, and whenelectromagnetic radiation passes from one medium intoanother, both bulk and surface forces will be present.Momentum flux through the boundary under this schemeis therefore indistinguishable from the combined elec-tromagnetic and material tensors of either Abraham orMinkowski. How Mansuripur’s definition of the electro-magnetic momentum density extends to moving mediahas yet to be established.In general, momentum transfer to the surface of a di-

electric block arises both from interactions with the ex-ternal environment and from pressure effects within theblock itself. This may cause regions of relative compres-sion or rarefaction close to the surface, depending on

whether the surface force is of the same or opposite signto the Lorentz force which acts on the bulk of the block.For dielectric fluids, a flow may result. When Eq. (55) isused, these effects are taken into account via the materialcomponents of the total energy–momentum tensor. Sucheffects are seen in the experiment of Ashkin and Dziedzic(1973) discussed in Sec. IV.B, where the surface of thewater bulges towards the source of the laser beam, al-though the Lorentz force within the body of the liquidacts in the opposite direction.

D. Discussion; Simplifying Assumptions

It is frequently possible to make simplifying assump-tions. We have already considered a perfectly reflectingmirror, for which no volume term arises, and a perfectlytransmissive dielectric in vacuum, for which the surfaceterm has no effect on the motion of the centre of mass.Also, it is often possible to use the Abraham or

Minkowski energy–momentum tensors in the traditionalmanner. The Abraham tensor pair are well suitedto problems in highly transmissive dielectrics, such asExample VIII.C.2, as momentum transfer to the ma-terial component may be identified with the Lorentzforce (Stallinga, 2006b). The Minkowski electromagneticenergy–momentum density is frequently used to calcu-late momentum transfer across a boundary, as discussedin Sec. VIII.C.1.It is important to note that electrostriction, magne-

tostriction, dispersion, and acoustic waves are rarelytreated in the literature, though with some exceptions(Garrison and Chiao, 2004; Mikura, 1976; Nelson, 1991;Stallinga, 2006a). When these effects are significant, anappropriate material energy–momentum tensor must al-ways be used.Rigorous treatment using a Lorentz force approach

(Loudon, 2002; Mansuripur, 2004, 2005) is also a vi-able option under any circumstances, provided sufficientcare is taken over field/charge interactions at the surface(Mansuripur, 2004), and may frequently be the simplerapproach. Where the behaviour of the surface is impor-tant, for example in the behaviour of dielectric liquids(Ashkin and Dziedzic, 1973), stress- or pressure-relatedeffects must also be considered.

IX. ANOMALOUS EFFECTS?

Over the years, a large number of tensor-dependenteffects have been proposed, by which it has been sug-gested that it might be possible to distinguish betweenthe different tensor formulations. These tend to arise

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either from inadequately considering the role of the ma-terial energy–momentum tensor, or from inappropriatelytreating the dielectric medium as a rigid body.Failure to adequately consider the role of the mate-

rial energy–momentum tensor may lead to claims of ex-perimental discrimination between the different energy–momentum tensors, as in Walker et al. (1975), and hypo-thetical thrusters based on non-conservation of momen-tum (Brito, 2004).Perhaps more important is the error of treating the di-

electric as a rigid body, an error which recurs frequentlywithin the literature (Arnaud, 1973, 1974; Balazs, 1953;Brevik, 1979; Padgett et al., 2003) and also leads to in-appropriate predictions of unusual physical behaviour.

A. Rigid Bodies and the Optical Tractor Beam

Consider a pulsed electromagnetic wave of momentump entering a material body of refractive index n. If we usethe Minkowski electromagnetic energy–momentum ten-sor, on entering the body the momentum of the wave isincreased to np and hence by conservation of linear mo-mentum, the body acquires a momentum of −(n − 1)p,setting it in motion towards the source of the wave. Whenthe pulse exits the body, the momentum of the materialis restored to zero and the body returns to a stationarystate. In contrast, under the Abraham tensor the mo-mentum of the pulse decreases and the body must beginmoving away from the source of the wave. Jones (1978)discussed such behaviour in the context of an Einsteinbox — a rigid box in which photons are emitted fromone end and absorbed by the other, typically used as athought experiment to demonstrate mass–energy equiv-alency.Because the body is treated as rigid and hence unable

to deform, momentum is transferred to the body as awhole, propagating across it instantaneously. In reality,the transfer of momentum must take place at a finite ve-locity which cannot exceed c and will therefore either bemuch less than, or of comparable magnitude to, the speedof light in the medium, c/n. Because of this, significantcompressive and expansive effects must take place, whichare neglected if the body is treated as rigid.This approach also mis-handles momentum associated

with the material body under the Minkowski tensor. Asseen in (43), the Minkowski material momentum den-sity includes a term (1/c)E×H− cD×B, which causesthe material momentum density as a whole to be neg-ative. However, this term is not associated with themovement of material particles described by v, and istherefore not related to the motion of material particleswithin the block. While it is necessary for the material

tensor momentum density to be negative in order to off-set the increase in momentum density in the Minkowskielectromagnetic tensor, not all of this momentum densityis associated with the particles of the block. In fact, thebehaviour of these v-dependent terms is identical to thatunder the Abraham tensor and the block gains positivemomentum, although the material momentum density asa whole becomes negative under the Minkowski scheme.

When these effects are treated correctly, we recoveran approach equivalent to that of Gordon (1973), asdiscussed in Sec. VI.B, and the traversal of the lightpulse now generates a distortion in the boundary of themedium. When the leading edge of the pulse enters themedium, the boundary bulges outwards, as demonstratedby the experiments of Ashkin and Dziedzic (1973), and asimilar outward bulge is produced when the leading edgeexits the medium. The traversal of the trailing edge ofthe pulse gives rise to a restoring impulse in each case.Meanwhile, the unopposed action of the Lorentz forcewithin the bulk of the medium gives rise to the motion ofthe body as a whole. The resulting behaviour is identicalunder both the Abraham and Minkowski tensors.

As discussed in Sec. VII, Balazs (1953) demonstratedthat a perfectly transmitting dielectric of higher refrac-tive index than its surroundings will advance in the di-rection of propagation of the wave packet, using argu-ments relating to the movement of the centre of massof the system. However he then erroneously applies therigid body argument given above, and concludes againstthe Minkowski electromagnetic energy–momentum ten-sor. In reality, as we have seen in Sec. VIII.C.2, bothchoices when used correctly will yield the same Lorentzforces acting on the body of the rod, accompanied byany surface effects due to pressures within the rod as perGordon (1973) and Peierls (1976), because regardless ofwhether we choose the Minkowski or Abraham represen-tation, the total energy–momentum tensor remains un-changed.

We note that in a real medium, coupling between ad-jacent surface elements (for example, surface tension ina liquid or lattice forces in a solid) may cause distur-bances generated by surface forces to propagate radiallyoutwards. Thus in an absorptive medium, traversal of aradiation pulse may give rise to acoustic (matter) waveswithin the substance of the medium, but once again thepredicted behaviour will be independent of the choice ofenergy–momentum tensors used. Peierls (1976) showedthat such photon/phonon coupling may also take placewhere the edges of a finite beam lie within a dielectric.

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19

X. RECENT WORK

As mentioned in the introduction, current researchon the electromagnetic energy–momentum tensor relateslargely to practical applications, or to the extension of ex-isting knowledge into new domains, though there are stillmany who appear unaware that the Abraham–Minkowskicontroversy has been resolved. We present and discusssome of the more recent papers, to illustrate the presentstate of research in the field.

A. Extension of Existing Theory and Practical Applications

Obviously, any theoretical result requires verification.The ability of combined electromagnetic and materialenergy–momentum tensors constrained by conservationof linear and angular momentum to explain historicalexperiments such as those discussed in this paper is wellestablished. However, continued testing of such theoriesover an increasingly broad range of domains is an essen-tial part of the process of physics research. In this regard,we refer to the proposal of Antoci and Mihich (1998) todemonstrate the Abraham force at optical frequencies,and the work of Brito (2004) seeking violation of conser-vation of momentum, though his reported positive resultsfare poorly under independent scrutiny (Millis, 2004).Extending existing classical theory, Mansuripur (2004)

directly derived the force of electromagnetic radiation ona material medium from the Lorentz force law, an ap-proach which makes no pre-emptive assumptions regard-ing the value of the electromagnetic momentum. He re-produced the beam edge effects first discussed by Gordon(1973), but with the discovery that this force may be ex-pansive, rather than constrictive, for p-polarised light,an effect which awaits experimental verification. He alsodeduced that anti-reflective coatings experience a forcewhich attempts to peel them off their substrate. Likemany others, in the course of this paper he derived aunique “correct” expression for the electromagnetic mo-mentum density. There are almost as many such expres-sions as there are authors, and if we restrict physics todealing only with hypotheses which may be tested by ex-periment, then the question of the correct expression isinstead one of ontology. However, as physical predictionsare independent of the expression chosen (see Sec. VI),this in no way detracts from the significance of his otherfindings.Other recent work includes the experiment of

Campbell et al. (2005) to determine whether, when aphoton is absorbed by an atom within an atomiccloud, it should be considered as departing the medium,and therefore only transfer the momentum of hk to

the atom, in accordance with Hensley et al. (2001), orwhether it transfers the Minkowski value nhk, leavingbehind a disturbance in the medium, as proposed byHaugan and Kowalski (1982). The latter case was deter-mined to be correct, with significance for the constructionand operation of atom interferometers.

Using quantum theory, Garrison and Chiao (2004) de-veloped operators for the momentum of a photon in adielectric medium. Employing a quantisation schemeproposed by Milonni (1995), they arrived at a situ-ation they recognise to be directly analogous to theAbraham–Minkowski controversy. Their survey of theliterature included the recognition, which they attributedto Brevik (1979), that decomposition of the total energy–momentum tensor into electromagnetic and materialcomponents is arbitrary. However, due to the inconsis-tent nature of the available literature, this was statedas simply one possible position of many, rather thanan obligate consequence of momentum conservation.Garrison and Chiao proceeded to develop three momen-tum operators — Abraham and Minkowski momentumoperators, corresponding to classical momenta calculatedutilising the isolated electromagnetic energy–momentumtensors of Abraham and Minkowski, and a canonical mo-mentum operator, which is the generator of translations,and relates to the total energy–momentum tensor. Theyreanalysed the results of Jones and Leslie (1978), anddemonstrated that the experiment favours the canon-ical momentum operator over both the Abraham andMinkowski operators. This is to be expected, as no op-erators were employed which related to either the corre-sponding material momentum flux or the Maxwell stresstensor, and hence only the canonical momentum opera-tor accounts for the total flux of momentum within thesystem. Their analysis is novel, in that it demonstratesthat the results of Jones and Leslie are of sufficient accu-racy to distinguish between the total energy–momentumtensor, and the electromagnetic energy–momentum ten-sor of Minkowski in isolation. By doing so, they there-fore provide additional experimental verification that theMinkowski tensor, like the Abraham tensor, is incompletewithout a material counterpart. For any real medium theMinkowski material counterpart tensor will yield not onlythe Abraham term, but also terms giving rise to acoustictransients (Peierls, 1976) and a dispersion-related cou-pling between the electromagnetic wave and the mate-rial medium (Garrison and Chiao, 2004; Nelson, 1991;Stallinga, 2006a). This latter effect is responsible for theobserved discrepancy between the canonical momentumand the Minkowski electromagnetic momentum.

Garrison and Chiao concluded that the canonical (to-tal) momentum is most applicable for interpreting ex-periments, and question whether the use of Abraham

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and Minkowski operators can ever be appropriate. How-ever, they weaken their conclusion by citing the exper-iment of Walker et al. (1975, and Sec. IV.C above) asuniquely selecting the Abraham tensor, despite havingearlier cited Obukhov and Hehl (2003), who indirectlydisprove this assertion. They also present the argumentsof Lai (1980) as similarly favouring the Abraham ten-sor, though these have been refuted by Brevik (1982), asdiscussed in Sec. V. Garrison and Chiao do not exam-ine the experiment of Walker et al. or the suggestionof Lai (1981) in their quantum-mechanical formalism,which is a pity, since they might then have concludedthat once again, it is conservation of the canonical mo-mentum which governs the experiments’ behaviour.

Loudon et al. (2005) also investigate momentum trans-fer from a quantum perspective, looking at the absorp-tion of photons by charge carriers in a semi-conductor.They found that momentum transfer can be divided intothree separate components: momentum transfer to thecharge carriers, given by the Minkowski value nφhk andin agreement with previous experiment (Gibson et al.,1980), to the bulk of the semi-conductor, given by theAbraham value hk/ngc, and to the surface of the di-electric, yielding overall conservation of momentum. Aclassical treatment by Mansuripur (2005) again resultsin transfer of the Minkowski value of momentum to thecharge carriers, with the net action on the host semicon-ductor maintaining global conservation of momentum.

Feigel (2004) has looked at the transfer of momen-tum between matter and the electromagnetic field on thequantum scale, and predicted the intriguing phenomenonof fluid flow powered by quantum vacuum fluctuationswhich is dependent upon the high-frequency cutoff ofquantum field theory.

Several have found the electromagnetic energy–momentum tensor important in extension of the Casimireffect from vacuum-separated plates to those separatedby a dielectric matierial. Raabe and Welsch (2005) re-lated the importance of recognising that the Minkowskitensor on its own is incomplete, and Stallinga (2006a)suggested that consideration of the total momentum fluxdensity is most appropriate. Stallinga also demonstratedhow a classical technique utilising an artificial “auxiliaryfield” F may be useful in modelling dissipative systems.

Of interest for different reasons is the somewhat olderwork of Cole (1971, 1973) in representing the behaviourof massless photons in a material medium as that ofequivalent massive particles in vacuum. Due to a trans-formation effectively eliminating the material medium,all momentum is contained within the electromagneticenergy–momentum tensor, and in this special case, thecriteria of von Laue and Møller (see Sec. V) are fulfilled.In terms of the Abraham–Minkowski controversy, Cole is

in effect choosing an energy–momentum tensor in whichall active terms are placed within the electromagnetictensor and the material energy–momentum tensor is al-ways null. More recently, Antoci and Mihich (2000) havedrawn attention to a related treatment of the constitutiveequation of electromagnetism by Gordon (1923), validfor a medium homogeneous and isotropic in its local restframe.A similar approach is also of practical relevance in

modelling optical trapping. Since all forces acting due totransfer of momentum from fields and media must be con-sidered, and the trapped particle is typically suspendedin a dielectric medium, a separation of the field and ma-terial components of the momentum flux that allows thesurrounding medium to be ignored can simplify calcula-tion of the optical force. This separation requires thatthe surrounding medium carries none of the momentum,a condition which is adequately met by the Minkowskitensor provided both dispersion and the Abraham forceare negligible, the fluid medium has had time to reachequilibrium, and behaviours such as electrostriction andmagnetostriction, which are not supported by the elec-tromagnetic component of this tensor, may be neglected.The trapped particle can then be modelled as a dielectricparticle of refractive index nrelative = nparticle/nmedium infree space, with the beam delivering a momentum fluxdensity of nmediumS/c. This method is best suited tomonochromatic light or nondispersive media, where nmay be treated as a constant.For a rigid particle, the optical forces can be cal-

culated from the incoming and outgoing momentumfluxes (Nieminen et al., 2004). For a deformable particle,for example in an optical stretcher (Guck et al., 2001),one needs to include the mechanical elastic forces thatresist the field or medium forces acting to deform theparticle, as well as the field or medium forces themselves.While it may be simplest to assume the Minkowski mo-mentum, the alternative tensors are nevertheless physi-cally equivalent. Statements which are based on the in-terpretation of a single tensor, for example suggestingthat the deforming force arises because the momentumof a photon in a medium is nhk0, compared to hk0 invacuum, are therefore unwise.

B. Ongoing Controversy

The work of Penfield and Haus (1967),de Groot and Suttorp (1972), Gordon (1973), Mikura(1976), Kranys (1979a,b, 1982), Maugin (1980), andSchwarz (1992) effectively demonstrated that division ofthe total energy–momentum tensor into electromagneticand material parts is effectively arbitrary. Neverthe-

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21

less, many continue to attempt to identify a superiorexpression for the electromagnetic energy–momentumtensor. Often they are aware of the work of Penfieldand Haus, and de Groot and Suttorp, and one musttherefore suppose that these works, perhaps on accountof complexity or lack of sufficient emphasis, are provingineffective in conveying that such a distinction is at thevery least untestable. For Gordon, one may point to thelimited scope to which his treatment may be applied, andfor Mikura, Kranys and Maugin the problem appears tobe insufficient publicity.Whatever the cause, papers continue to appear which

attempt to demonstrate a uniquely privileged formfor the electromagnetic energy–momentum tensor (see,for example, Antoci and Mihich, 1997; Labardi et al.,1996; Mansuripur, 2004; Obukhov and Hehl, 2003;Padgett et al., 2003; Scalora et al., 2006). Padgett pro-posed an interesting experiment in the optical domain, inwhich a laser beam carrying orbital angular momentumtraverses, and transfers angular momentum to, a glassdisc. This experiment is in many ways analogous to thatof Ashkin and Dziedzic (1973, and Sec. IV.B above). InAshkin and Dziedzic’s experiment, pressure effects withinthe liquid were responsible for restoring equivalence be-tween the Abraham and Minkowski interpretations. InPadgett’s experiment, a similar role will be played byshear forces within the disc.The works cited above are far from alone in their con-

tinued fascination with the Abraham–Minkowski contro-versy. The topic frequently continues to be a point ofdiscussion at conferences, and the occasional physics col-loquium (Lett and Milonni, 2005), though awareness isgradually increasing that the distinction is of no func-tional significance, and hence the choice of energy–momentum tensor pair is essentially a question of per-sonal æsthetics.

XI. CONCLUSION

The original Abraham–Minkowski controversy, overthe preferred form of the electromagnetic energy–momentum tensor in a dielectric medium, has beenresolved by the recognition that division of the totalenergy–momentum tensor into electromagnetic and ma-terial components is arbitrary (de Groot and Suttorp,1972; Penfield and Haus, 1967). Hence the Minkowskielectromagnetic energy–momentum tensor, like theAbraham tensor, has a (frequently unacknowledged)material counterpart (Israel, 1977; Obukhov and Hehl,2003), and the sum of these components yields the sametotal energy–momentum tensor as in the Abraham ap-proach.

On these grounds, all choices for the electromagneticenergy–momentum tensor are equally valid and will pro-duce the same predicted physical results, as has beendemonstrated for a wide range of specific examples byMikura (1976), Kranys (1979a,b, 1982), and Maugin(1980). Nevertheless, awareness of the resolution of theoriginal controversy remains patchy, largely due to thefragmentary nature of the literature on the subject.In this paper, we have reviewed the controversy from

its initial formulation to the present day, incorporatingdiscussion of key experiments believed at one time oranother to have bearing on the relative truths of theAbraham and Minkowski models. We have discussed therealisation that any electromagnetic energy–momentumtensor must always be accompanied by a counterpart ma-terial energy–momentum tensor, and that the division ofthe total energy–momentum tensor into these two com-ponents is entirely arbitrary.A consensus has yet to be reached on the ultimate form

of the total energy–momentum tensor, but the limitationhere arises not from classical electromagnetism but frommaterials science, and we believe there is scope for fur-ther work in explicitly developing the theory of dispersivemedia, a subject touched upon in Sec. VIII.B.1.We believe a wider appreciation of the relationship be-

tween electromagnetic and material energy–momentumtensors could greatly benefit those working in this field,and with this colloquium we hope to assist in bringingthat wider appreciation into existence.

Acknowledgments

The authors would like to thank The Royal Society,Professor A. Ashkin, the American Physical Society, andthe National Research Council of Canada for permissionto reproduce the figures accompanying this paper. Thework of R.N.C.P was partly supported by an EndeavourInternational Postgraduate Research Scholarship fundedby the Australian Government Department of Education,Science and Technology.

References

Abraham, M., 1909, Rendiconti Circolo Matematico diPalermo 28, 1.

Abraham, M., 1910, Rendiconti Circolo Matematico diPalermo 30, 33.

Antoci, S., and L. Mihich, 1997, Nuovo Cimento B 112, 991.Antoci, S., and L. Mihich, 1998, in From Newton to Einstein

(A Festschrift in Honour of the 70th Birthday of Hans-Juergen Treder), edited by W. Schroeder (Science Edition,Bremen), eprint physics/9808002.

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22

Antoci, S., and L. Mihich, 2000, Nuovo Cimento B 115, 77.Arnaud, J. A., 1973, Opt. Commun. 7, 313.Arnaud, J. A., 1974, Am. J. Phys. 42, 71.Ashkin, A., and J. M. Dziedzic, 1973, Phys. Rev. Lett. 30,

139.Ashkin, A., J. M. Dziedzic, J. E. Bjorkholm, and S. Chu,

1986, Opt. Lett. 11, 288.Balazs, N. L., 1953, Phys. Rev. 91, 408.Blount, E. I., 1971, Bell Telephone Laboratories technical

memorandum 38139-9.Boffi, L. V., 1957, Electrodynamics of Moving Media, Sc.D.,

Dept. of Electrical Engineering, Massachusetts Institute ofTechnology, Cambridge, Massachusetts.

Brevik, I., 1970, Mat.-Fys. Medd. Dan. Vidensk. Selsk. (Den-mark) 37, 1.

Brevik, I., 1979, Phys. Rep. 52, 133.Brevik, I., 1982, Phys. Lett. A 88, 335.Brito, H. H., 2004, Acta Astronaut. 54, 547.Burt, M. G., and R. Peierls, 1973, Proc. R. Soc. London, Ser.

A 333, 149.Campbell, G. K., A. E. Leanhardt, J. Mun, M. Boyd, E. W.

Streed, W. Ketterle, and D. E. Pritchard, 2005, Phys. Rev.Lett. 94, 170403.

Chu, L. J., H. A. Haus, and P. Penfield, Jr., 1965, TheForce Density in Polarizable and Magnetizable Fluids,Technical Report, Massachusetts Institute of TechnologyResearch Laboratory of Electronics, Cambridge, Mas-sachusetts, URL http://hdl.handle.net/1721.1/4392 .

Cole, K. D., 1971, Aust. J. Phys. 24, 871.Cole, K. D., 1973, Aust. J. Phys. 26, 359.Dallenbach, W., 1919, Ann. Phys. (Leipzig) 58, 523.Einstein, A., and J. Laub, 1908, Ann. Phys. (Leipzig) 26, 541.Einstein, A., and J. Laub, 2005, Ann. Phys. (Leipzig) 14, 327,

[Reprint of Einstein and Laub (1908)].Fano, R. M., L. J. Chu, and R. B. Adler, 1960, Electromag-

netic Fields, Energy and Forces (John Wiley & Sons, NewYork).

Feigel, A., 2004, Phys. Rev. Lett. 92, 020404.Garrison, J. C., and R. Y. Chiao, 2004, Phys. Rev. A 70,

053826.Gibson, A. F., M. F. Kimmitt, A. O. Koohian, D. E. Evans,

and G. F. D. Levy, 1980, Proc. R. Soc. London, Ser. A 370,303.

Ginzburg, V. L., 1973, Usp. Fiz. Nauk 110, 309, [Sov. Phys.Usp. 16, 434 (1973)].

Gordon, J. P., 1973, Phys. Rev. A 8, 14.Gordon, W., 1923, Ann. Phys. (Leipzig) 72, 421.de Groot, S. R., and L. G. Suttorp, 1967, Physica 37, 284,

297.de Groot, S. R., and L. G. Suttorp, 1968, Physica 39, 28, 41,

61, 77, 84.de Groot, S. R., and L. G. Suttorp, 1972, Foundations of

Electrodynamics (North-Holland, Amsterdam, The Nether-lands).

Grot, R. A., and A. C. Eringen, 1966, Int. J. Eng. Sci. 4, 639.Guck, J., R. Ananthakrishnan, H. Mahmood, T. J. Moon,

C. C. Cunningham, and J. Kas, 2001, Biophys. J. 81, 767.Gyorgyi, G., 1960, Am. J. Phys. 28, 85.

Haugan, M. P., and F. V. Kowalski, 1982, Phys. Rev. A 25,2102.

Hensley, J. M., A. Wicht, B. C. Young, and S. Chu, 2001,in Atomic Physics 17, Proceedings of the 17th Interna-tional Conference on Atomic Physics, edited by E. Ari-mondo, P. D. Natale, and M. Inguscio (American Instituteof Physics, Melville, New York), pp. 43–57.

Israel, W., 1977, Phys. Lett. B 67, 125.Israel, W., 1978, Gen. Relativ. Gravit. 9, 451.Jackson, J. D., 1999, Classical Electrodynamics (John Wiley

& Sons, Inc., New York), chapter 12.10, 3rd edition, pp.607–608.

James, R. P., 1968a, Force on Permeable Matter in Time-Varying Fields, Ph.D., Dept. of Electrical Engineering,Stanford University.

James, R. P., 1968b, Proc. Natl. Acad. Sci. USA 61, 1149.Jones, R. V., 1951, Nature (London) 167, 439.Jones, R. V., 1978, Proc. R. Soc. London, Ser. A 360, 365.Jones, R. V., and B. Leslie, 1978, Proc. R. Soc. London, Ser.

A 360, 347.Jones, R. V., and J. C. S. Richards, 1954, Proc. R. Soc. Lon-

don, Ser. A 221, 480.Kranys, M., 1979a, Can. J. Phys. 57, 1022.Kranys, M., 1979b, Phys. Scr. 20, 685.Kranys, M., 1982, Int. J. Eng. Sci. 20, 1193.Labardi, M., G. C. La Rocca, F. Mango, F. Bassani, and

M. Allegrini, 1996, J. Vac. Sci. Technol. B 14, 868.Lai, H. M., 1980, Am. J. Phys. 48, 658.Lai, H. M., 1981, Am. J. Phys. 49, 366.Lai, H. M., 1984, Phys. Lett. A 100, 177.Lang, M. J., and S. M. Block, 2003, Am. J. Phys. 71, 201.von Laue, M., 1950, Z. Phys. 128, 387.Lett, P., and P. Milonni, 2005, How Does The

Photon Get Its Momentum?, physics colloquium,University of Toronto, Nov. 17th 2005, URLhttp://www.physics.utoronto.ca/~colloq/2005_2006.html.

Livens, G. H., 1918, The Theory of Electricity (UniversityPress, Cambridge).

Loudon, R., 2002, J. Mod. Opt. 49, 821.Loudon, R., S. M. Barnett, and C. Baxter, 2005, Phys. Rev.

A 71, 063802.Mansuripur, M., 2004, Optics Express 12, 5375.Mansuripur, M., 2005, Optics Express 13, 2245.Marx, G., and G. Gyorgyi, 1955, Ann. Phys. (Leipzig) 16,

241.Maugin, G. A., 1980, Can. J. Phys. 58, 1163.Mikura, Z., 1976, Phys. Rev. A 13, 2265.Millis, M. G., 2004, Prospects for breakthrough propul-

sion from physics, NASA ref. TM-2004-213082, URLhttp://gltrs.grc.nasa.gov/reports/2004/TM-2004-213082.pdf.

Milonni, P. W., 1995, J. Mod. Opt. 42, 1991.Minkowski, H., 1908, Nachrichten von der Gesellschaft der

Wissenschaften zu Gottingen, 53.Minkowski, H., 1910, Math. Ann. 68, 472, [Reprint of

Minkowski (1908)].Møller, C., 1952, The Theory of Relativity (Oxford University,

Oxford), 1st edition.Møller, C., 1972, The Theory of Relativity (Oxford University,

22

http://hdl.handle.net/1721.1/4392

http://www.physics.utoronto.ca/~colloq/2005_2006.html

http://gltrs.grc.nasa.gov/reports/2004/TM-2004-213082.pdf



23

Oxford), 2nd edition.Nelson, D. F., 1991, Phys. Rev. A 44, 3985.Nieminen, T. A., N. R. Heckenberg, and H. Rubinsztein-

Dunlop, 2004, in Optical Trapping and Optical Microma-nipulation, edited by K. Dholakia and G. C. Spalding (So-ciety of Photo-Optical Instrumentation Engineers, Belling-ham, WA), volume 5514 of Proceedings of the SPIE, pp.514–523.

Obukhov, Y. N., and F. W. Hehl, 2003, Phys. Lett. A 311,277.

Padgett, M., S. M. Barnett, and R. Loudon, 2003, J. Mod.Opt. 50, 1555.

Pauli, W., 1921, in Encyklopadie der mathematischen Wis-senschaften (B. G. Teubner, Leipzig), volume V19.

Pauli, W., 1958, Theory of Relativity (Pergamon, New York).Peierls, R., 1976, Proc. R. Soc. London, Ser. A 347, 475.Peierls, R., 1977, Proc. R. Soc. London, Ser. A 355, 141.Peierls, R., 1991, More Surprises in Theoretical Physics

(Princeton University Press, New Jersey).Penfield, P., Jr., and H. A. Haus, 1967, Electrodynamics of

Moving Media (M.I.T., Cambridge, Massachusetts).Raabe, C., and D.-G. Welsch, 2005, Phys. Rev. A 71, 013814.Robinson, F. N. H., 1975, Phys. Rep. 16, 313.Scalora, M., G. D’Aguanno, N. Mattiucci, M. J. Bloemer,

M. Centini, C. Sibilia, and J. W. Haus, 2006, Phys. Rev. E73, 056604.

Schwarz, S., 1992, Int. J. Eng. Sci. 30, 963.Skobel’tsyn, D. V., 1973, Usp. Fiz. Nauk 110, 253, [Sov. Phys.

Usp. 16, 381 (1973)].Stallinga, S., 2006a, Phys. Rev. E 73, 026606.Stallinga, S., 2006b, Optics Express 14, 1286.Tamm, I., 1939, Journ. Phys. (Moscow) 1, 439.Tang, C. L., and J. Meixner, 1961, Phys. Fluids 4, 148.Tolman, R. C., 1934, Relativity, Thermodynamics and Cos-

mology (Oxford University, Oxford), §104.Walker, G. B., D. G. Lahoz, and G. Walker, 1975, Can. J.

Phys. 53, 2577.Walker, G. B., and G. Walker, 1977, Can. J. Phys. 55, 2121.

23

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