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I . BREVI K ELECTROMAGNETI C ENERGY-MOMENTUM TENSO R WITHIN MATERIAL MEDI A 2 . DISCUSSION OF VARIOUS TENSOR FORM S Det Kongelige Danske Videnskabernes Selska b Matematisk-fysiske Meddelelser 37, 1 3 Kommissionær : Munksgaar d København 1970 Synopsi s This paper represents the second part of a study of the electromagnetic energy-momentu m tensor within a material medium . Similarly as in the first part, essentially a macroscopical poin t of view is adopted, and emphasis is laid upon the comparison with experiments, both in the case of static fields and in the case of time-varying fields within bodies at rest and in relativisti c motion . For the main part the relative behaviour of Minkowski's and Abraham's tensors is studied, but some attention is also given to the tensors introduced by Einstein and Laub, d e Groot and Suttorp, Beck and Marx et al . Deductive procedures are employed, characteristi c effects are studied, both within media at rest and in motion, and some attention is given to a critical analysis of earlier treatments . Our main conclusion is that Minkowski's and Abraham' s tensors are equivalent in the usual physical cases, while the remaining tensor expressions see m to run into conflict with experirnental evidence . PRINTED IN DENMAR K BIANCO LUNOS BOGTRYKKERI A/S 1 . Introduction and Summar y In a previous paper ( ' ) -hereafter referred to as I-we discussed th e application of MINKOWSKI ' S energy-momentum tensor in phenomenologica l electrodynamics . The medium was assumed to be homogeneous, transparen t and usually also nondispersive . Since the essential differences between th e various competing tensor forms are present also in the most simple media , the above restrictive assumptions were legitimate in relation to the mai n purpose of the investigation, namely to examine whether MINKOWSKI's tensor is appropriate to use in the most common and simple situations . And the affirmative answer to this question made it just convenient to restrict the treatment so as to incorporate MINKOWSKI's tensor only. In the present paper we shall consider also other tensor forms, so let u s first write down some expressions . The rest inertial frame of the mediu m shall be denoted by K°, while the inertial frame in which K° moves with the uniform velocity v, shall be denoted by K. MINKOwSKI ' s tensor reads Sk = - Ei Dk HH B k + bik( E . D + H . B) S4k=i(ExH) k , S koi(DxB)k , S44=- ;(E .D+H•B), (l .la ) (1 .lb) or, in covariant form, Siv = F/A Hva - 4 åµv Faß Haß (1 .2 ) (for notation, see I) . Perhaps the main reason why MINKOwsKI's tensor often has been rejecte d and instead replaced by some other tensor form is the asymmetry of th e former, which is present even within isotropic media . The symmetry requirement is met by the following tensor, which we shall call ABRAHAM ' S tensor, Sk° - - ( E° Dk + ET D2) - + ( H° Bk + 4E2) -I- åik(E° . D ° + H° • B°) Sko = SIP = i(E° x H°)k, St? = - 2(E° • D° + H° B° (1 .3a ) (1 .3b) 1* (here given in K°), although this symmetrized form of the stress tensor SI for anisotropic media seems to have been given first by H . HERTZ(2 ) . When the body is isotropic, the force density in K read s n 2 - 1 Ô~MO f A0 = f M0 + C2 f4O , f4 0, (1 .4 ) where n is the refractive index . We shall often be concerned with this tenso r in the following chapters . Its covariant form can be written as S~v = Sir, + FfL 12 F« F« Vu }Vv , (1 .5 ) c where x = (e,u - 1)/c 2 = (n 2 - 1)/c 2 , Fa = Fav Vti, and VI, = y (v, ic) . Another proposal was put forward by G . MARX and collaborators( 3 ) . They examined a simple radiation field travelling through an isotropi c medium, and came to the conclusion that ABRAHAM ' S tensor, describing th e electromagnetic field, must be supplemented with a mechanical tensor t o give the symmetrical " radiation" tensor S its , describing the total system : radiation plus connected mechanical field . In K° the radiation tensor i s given by S2k0 = n 2 SakO = n 2 Szk °, S4° = Sv4 = S v, (1 .6 ) 4 for all v between 1 and 4 ."' The covariant expression can be writte n S s,,,, = n2 1S + x2 [VFva Fa + zV~Vv(xFaFa+ ,un a~Faa)] (1 .7 ) A . EINSTEIN and J . LAUB(5 ) have also examined the problem ; by means o f simple examples they constructed an expression for the force density in K ° which corresponds to the following components of the energy-momentu m tensor (1 .8a ) Sak° = - El' D70, H°Bk + z Sak (E02 + H° 2) Sok° = Sko = i(E° x H°)k . (1 .8b ) The energy density component was not given . The last tensors we shall mention here are due to S . R . DE GROOT an d L . G . SUTTORP (6) . These authors have examined the problem from a purely * F . BECK(') has also introduced a tensor which, however, in the case of a radiation field coincides with MARx ' s radiation tensor . Therefore we shall not pay any special attention t o this form in the following sections . 5 Nr . 13 microscopical point of view, and published recently a series of papers o n the subject . (See also I, section 7 .) They give two tensor expressions, de pendent on whether the total interaction between field and matter is take n into account or not . In the case of an isotropic medium their first proposa l reads in K° Sk° = E° Dk - H° Bk + $~ k ( ~ S2 = Sk4 = i(E° X H° )k , S44 E02 - -I- 2 B°2 - .] (E°2 M° . B°) + B02 ), (1 .9 a) (1 .9 b ) where M° = B° - H° . It is apparent that for M° = 0, the components (1 .8 ) of the EINSTEIN-LAUB tensor agree with the corresponding components of th e DE GROOT-SUTTORP tensor (1 .9) . The second tensor expression proposed by DE GROOT and SUTTORP . wa s defined as the difference between the total energy-momentum tensors wit h and without external electromagnetic fields . This tensor thus corresponds t o taking the whole interaction between field and matter into account . B y omitting the variations of the material constants with density and temperature, as we mainly do throughout our work, we find that their second fiel d tensor agrees with ABRAHAM ' S tensor within an isotropic body . There exist also other proposals that have been put forward, and we shall have the opportunity to comment upon some of them in the detailed considerations later on . Mostly we shall be concerned with the relative merits of ABRAHAM ' S and MINKOwSKI ' s tensors, since these tensors, combined wit h their appropriate interpretations, are found to be both adequate and equivalent in most of the simple physical situations considered . Further introductions to the subject are given in the books by C . MØLLER (7) and W . PAuLI( 8 ), and in the review article by G . MARx (9) . The main task of the subsequent exposition can be conveniently divide d into three parts . Firstly, we want to apply some deductive methods in orde r to see how the various tensors adapt themselves to the formalism . As indicated already in I it must be borne in mind that the power of this kind o f method is restricted in the sense that the expressions one obtains are no t unique . Secondly, we wish to examine the applicability of the variou s tensor forms to the description of definite phenomena . The description o f the experiments is here a crucial point . Thirdly, we shall spend some effor t to comment upon parts of the earlier literature . There has been published a large number of papers on the subject, which are often mutually contradictory and moreover scattered over a number of different journals . We fin d it therefore of importance to point out some crucial points in the various 6 Nr . 1 3 derivations as an attempt to find the deeper reason why the results ar e seemingly incompatible . Throughout this work we take a phenomenological point of view an d refer only occasionally to the simple microscopical treatment in I . This i s done for practical reasons, a thorough scrutiny of the microscopical aspect s would require a separate treatment . However, we think there is also a reason of principle why it is sensible first to choose the macroscopical lin e of approach in order to obtain a satisfactory description of the physica l phenomena : In the simple cases considered, the results obtained by mean s of these macroscopic or semi-macroscopic methods are both consistent an d moreover fit the observed data in an excellent way . From a pragmatic point of view the macroscopical kind of method is therefore not only a possibl e kind of approach but in fact the appropriate one as a first step, and microscopical methods with their complicated formalism should properly b e considered to represent a later stage of the development . Let us now review the subsequent sections . Section 2 is devoted to a n analysis of electrostatic fields . We considor again the variational metho d which was employed in section 3 of I, and show how MINI{owsKI ' s an d ABRAHAM ' S tensors emerge from the formalism in an equivalent way . It i s found that, as far as a dielectric body is surrounded by a vacuum or a n isotropic liquid, no experiment testing electromagnetic forces or torques o n the body can decide between these tensors . The two tensors correspon d merely to different distributions of forces and torques throughout the body : According to MINxowsIu the torque is essentially a volume effect, describe d by the tensor asymmetry, while according to ABRAHAM the torque is describe d completely in terms of the force density. We consider a typical example, i n which ABRAHAM'S torque naturally comes out as a surface effect . In the remainder of section 2 we discuss to some extent the EINSTEIN LAUB (or the DE GROOT-SUTTORP) tensor . It is found that also in this cas e no force or torque experiments on a body surrounded by a vacuum or a n isotropic fluid represent a critical test for the tensors in question . However , there is actually one effect which represents a critical test, namely the pres sure increase in a dielectric liquid because of the field . In order to apply th e theory to this case it is necessary to extend the variational method mentione d earlier (the HELMHOLTZ method) so as to include also the electrostrictio n effect, although we are otherwise ignoring this effect in our work . S . S . HAKIM and J . B . HIGHAM have tested the pressure increase experimentally , and they found that the HELMHOLTZ force describes the observed data ver y well . On the contrary, the pressure increase predicted by the EINSTEIN-LAUB Nr . 13 7 force (which is also called the KELVIN force) was found to be in disagree ment with the experiment. In section 3 we continue the consideration from I, section 6 concernin g the propagation of an electromagnetic wave within an isotropic body at rest . By means of the semi-macroscopic method that we are adopting, and b y taking the radiation pressure experiment due to R . V . JONES and J . C . S . RICHARDS into account, we find that ABRAHAM ' S and MINKOwsKI ' s tensors ar e equivalent in the following sense : ABRAHAM'S force density excites the constituent dipoles of the material and produces a mechanical momentum whic h travels together with the field . If we count this mechanical momentu m together with ABRAHAM'S momentum as a field momentum, we obtai n MINxowSKI's tensor . By considering the situation in the frame where th e mean motion of the constituent particles vanishes we find that, in the case o f an infinite medium, the energy-momentum tensor of the total system can b e written as the sum of ABRAHAM'S tensor and the mechanical tensor in th e absence of fields . We continue section 3 by discussing an example in which the boundar y between two media is involved . Finally we consider alternative tensor forms , and find that the radiation pressure predicted by the radiation tensor is i n disagreement with the JONES-RICHARDS experiment . In section 4 we discuss possibilities for torque experiments, especiall y when MINKowsxl ' s or ABRAHAM ' S tensors are taken as field tensors . For a stationary optical wave in interaction with a dielectric body we find that th e two tensors will always yield the same value for the torque . Thereafter w e propose an experiment involving a low-frequency combination of electri c and magnetic fields . This experiment should be appropriate for the detectio n of ABRAHAM'S force, which is hidden in the case of optical fields . Finally it i s concluded that the case of an optical field travelling through a dielectric bod y immersed in a dielectric liquid should represent a possible means for a further experimental check of the radiation tensor and the EINSTEIN-LAU B tensor . Section 5 is devoted to a critical review of some parts of the earlie r literature, especially those parts which seem to run into conflict with ou r own interpretations . We are otherwise commenting upon passages fro m earlier treatments also in our ordinary exposition of various topics, but ther e remain interesting arguments which cannot so naturally be dealt with i n the ordinary treatment . We think such a critical analysis is desirable in a study of the present problem, since an important part of the task is just t o clear up a situation which is confused by mutually contradictory opinions . 8 Nr . 1 3 For the main part we discuss gedanken experiments which have been pu t forward to support either MINxowSKI ' s or ABRAHAM ' S tensor, and show ho w these situations are to be explained with the use of the formerly rejecte d alternative . In the remaining part of the section we mainly discuss som e aspects of the EINSTEIN-LAUB paper . In the subsequent sections we discuss topics connected with relativity , and, except for the last section, limit the consideration to the case of isotropi c media . Section 6 is devoted to a study of the torque acting on a moving bod y when an electromagnetic wave is travelling within it . We first calculat e ABRAHAM ' S and MINKOWSKI ' S torque expressions when the body is assumed infinitely extended, and show thereafter that both these expressions ar e relativistically consistent . In this context we draw into consideration an analogous situation encountered in relativistic mechanics : An elastic bod y subjected to stresses in its rest system may in other inertial systems require a torque in order to maintain steady motion . A similar situation is found to b e present also here in electrodynamics : We require stady motion of matter plus field and find that there must then exist a rate of change of electromagnetic momentum which is just equal to the previously calculated torque , with the opposite sign . If the body is finite, we find that the most natural division of the tota l angular momentum into a field part and a mechanical part is obtained wit h the use of ABRAHAM ' S tensor for the field . Section 7 contains a discussion of various relativistic phenomena . W e begin by considering the velocity u = S/W of the energy in an optical wave . In section 9 of I we found that u transforms like a particle velocity if MIN g owsar's tensor is used . We now find that ABRAHAM ' S tensor cannot fulfil th e transformation criterion due to the fact that this tensor does not describe th e total travelling wave . We analyse the background for the transformatio n criterion, and give a rather general form of a tensor that fulfils it . The radiation tensor falls within this category . Next we consider the relativistic centre of mass of a finite, but practicall y monochromatic, field . In section 12 of I we found that the various centre s obtained with the use of MINKOWSal ' s tensor in general do not coincide whe n considered simultaneously in one frame . Actually, by considering in the rest frame K° the centres of mass obtained by varying the direction and magnitud e of the medium velocity, we found that they are located. on a circular dis k lying perpendicular to the inner angular momentum vector in K° with centr e at the centre of mass in K° . Now the various centres of mass are found t o behave in exactly the same way if the ABRAHAM tensor or the radiation tenso r is adopted . Nr . 13 9 The CERENK0V effect is thereafter briefly analysed in the inertial fram e in which the emitting particle is at rest . From a study of the momentu m balance in this situation, I . TAMM has given preference to MINKOwSKI' s tensor . We show how the momentum balance appears with the use of ABRAHAM ' S tensor . Section 7 is closed by some further remarks upon the literature . In the last section we employ a variational method which implies th e application of curvilinear coordinates as a formal remedy . For a close d system this method in general leads to a determination of the energy-momentum tensor, but the method is shown to leave a certain ambiguity here due t o the fact that the LAGRANGIAN leading to the electromagnetic field equations corresponds to a non-closed physical system . Section 8 is rather detailed , since this subject has caused some confusion . Finally we consider again the Sagnac-type experiment due to C . V . HE ER , J . A . LITTLE and J . R . Bunt, which was discussed in section 9 of I . We find that this experiment, although it gives an excellent verification of the pre dictions of macroscopic electrodynamics, does not represent a critical tes t for MINKOwsxr's tensor, such as it was originally claimed . In fact, the experiment is found to be explained equivalently also by ABRAHAM ' S tensor an d the radiation tensor . The Appendix gives in tabular form a summary of the behaviour of th e various examined . energy-momentum tensors in some physical situations . 2 . Static Field s We begin with an examination of the various tensors applied to th e simplest physical case, namely the static fields . Actually, only electrostati c fields shall be considered since, for the simple case with linear inductiv e magnetization here considered, the corresponding results in the magneto static case can be taken over by analogy . In this section we first consider the important point concerning the relative behaviour of MINKOwsKI ' s an d ABRAHAM ' S tensors, and show how they in general lead to equivalent experimental results . Thereafter we consider various other tensor possibilities . Since all quantities are taken in the rest frame, the superscript zero on the m shall simply be omitted . IY7inkowski's versus Abraham's tensor From (1 .la) and (1 .3a) it is apparent that MINaowSKI ' s and ABRAHAM ' S tensors are equal in the electrostatic case for isotropic media . We therefor e generalize the situation and consider the same physical system as in I, 10 Nr .1 3 section 3, namely a dielectric, anisotropic medium containing an electri c field which is produced by some external devices . The linear relatio n E i = iikDk is assumed to be valid . By varying the free energ y g- _ JE . DdV (2 .1 ) and equating - d. /dt to the rate of mechanical work f f • udV exerted by th e volume forces, we found in = f', wher e If f A = 0E + - D .iDkv?7ik - Ô k (ED k - E k D) . (2 .2 ) This corresponds to the stress tenso r Sik = - z(Ei Dk +Ek Di )+ S ik E D. (2 .3) By comparison with (1 .3a) it is thus evident that we have obtained ABRAHAM ' S tensor . However, by invoking the "dipole model" and assuming th e existence of a torque density T. = D x E with a corresponding extra con tribution f r • (dq)/dt)dV to the rate of mechanical work (q) being the rotational angle), we found instead MINKOwsKI ' s result f m = OE + Dk V ry7ik (2 .4 ) Sik = - Ei Dk +ZSik E•D . (2 .5 ) According to this description, the result is dependent explicity on the assumption of an extra torque density . In order to make a more distinct comparison between the two tenso r forms, it is convenient to reformulate the balance equation in terms of th e rotational angle q) rather than the velocity u = ds/dt . Since f • s = (r x f) • 9) we have from (I,3 .10,9 ) f [r x (~,E -I D i Dk V nik)] dV + 1(D x E) • q)dV (2 .6 ) = J(rxfYdV+JrdV. where f and r are as yet unspecified . As ip is arbitrary, we obtai n rxf+r=r x (eE+-'_,-DiDkVnik)+DxE . (2 .7 ) This relation is fulfilled directly with MINKOWSKI ' s tensor, and only then . However, let us add the vanishing quantity -Z f (Ei D•n - E•nDi )si dS, (2 .8 ) con(' taken over the external conductors that produce the field, and let us combine f (D x E) • TcIV - f (Ei D • n E• nDi)sidS - ; f ak (EDk f [r x ô k (ED k - Ek D)] • cpdV. - Ek D) • sdV = - l l tond (2 .9 ) J Then (2 .6) is equivalent t o f[r x (e E + z DiDk0~7ik - a k (EDk = J(rxf)dV+Jr . Ek D))]'T d V dV, and we obtain now f = f-4 , z = 0, i .e . ABRAHAM ' S tensor . In this case th e torque is described in terms of the force density, while in the former case i t was described by the asymmetry of the stress tensor . We must conclud e that, as far as the dielectric body is surrounded by an isotropic mediu m (here vacuum), no unambigeous answer can be given for electrostatic systems . And this result is connected with the fact that the total body torque is th e same for both tensors in this case : We may put the torque formula into th e form Ni = f (x ifk - x kfi + S ik - Ski) dV = - f (r x Sn ac )t dS, (2 .11 ) surfac e where Sni = Sik n k . Thus the total torque can be evaluated from the vacuu m values of the field, and MlNxowsiu's and ABRAHAM ' S tensors must yield th e same result . Similarly, the total body force can also be put into a for m which involves the vacuum field values only ; by starting from the balanc e equation for total momentum we obtain readily for the total body forc e Fi f Sni t dS, (2 .12 ) surfac e in accordance with (2 .11) . It should be emphasized that in order to obtain MINIiOWSKI ' s tensor i n the first procedure above, we had to take into account the existence of extr a body torques with the density D x E . In the second procedure, however , the equivalence between MINKOWSKI ' s and ABRAHAM ' S tensors was demon- 12 Nr . 1 3 strated simply by adding the vanishing term (2 .8) in the energy balance . The additional assumption concerning the torque D x E will thus lead to a n equivalent description with respect to observable effects for the whole dielectric body, only the distribution of torques and forces within the bod y will in general be different . It is clear that the above reasoning will not be changed if we assume tha t an isotropic, dielectric liquid fills the space between the body and the conductors, since MINxowsai's and ABRAHAM ' S tensors are equal in such a liquid . The arguments hitherto have dealt with the dielectric system considere d as a whole . If several insulators are present between the conductors, the n the torque acting on an individual insulator is still independent of which tensor we use . That follows immediately from the fact that we obtain ex pressions like the last term in eq . (2 .11) for each insulator in question . An exampl e For the sake of illustration, let us consider again the same physica l situation as in I, section 3 : A dielectric sphere is located in a homogeneou s electrostatic field such that the principal axes of the sphere coincide with th e coordinate axes. The external field is given as E° = (El, E°2 , E3) . With the use of MINxowsxr's tensor, we obtained in I for the single nonvanishing component of the torqu e N 3 = f (Sil S2 )dV body = f (D (2 .13 ) x E)3 dV = (p x E°)3, bod y where p = 3V[(ei - 1)4/(r1 + 2), (E2 - 1)E2/(e2 + 2),0],V being the volume of the sphere . According to (2 .13), it is natural to interpret the effect as a volume effect . Let us now insert ABRAHAM ' S tensor into the torque formula (2 .11) so as to obtai n Snac)]3 dS N3 = (r x fA ) 3 dV + J [r x (Sn I J body surrace - f surrace (r x Snac)3 dS =(p x E°)3 ' l (2 .14) I f J The expressions (2 .14) and (2 .13) are equal, as they should be . But fo r tensor the volume effect vanishes, as is apparent also from th e fact that fA = 0 in the homogeneous field in the body . In this case it is natura l to interpret the effect as arising from the volume forces in the boundary layer . ABRAHAM ' S 13 Nr . 13 Olher tensor forms Let us now examine the various other tensor proposals mentioned i n section 1 . The radiation tensor due to MARx el al is defined for radiatio n fields within isotropic media only, and shall not be considered here . But there remains the EINSTEIN-LAUB tensor (1 .8a) and the DE GROOT-SUTTOR P tensor (1 .9a), which actually are seen to be equal in the electrostatic case . The force density is f E = PE+ (P•V)E, (2 .15 ) which is different from both (2 .4) and (2 .2) . This force is also called th e KELVIN force . The difference is expected to be connected with the fact that the force densities (2 .2) and (2 .4) were obtained from a variational principl e based on the free energy in the form (2 .1), which includes the interactio n energy between field and matter . And this energy is not directly compatibl e f with the energy 2 . E 2 dV following from (1 .9b) . As regards the possibility for an experimental check of the force (2 .15 ) we have first to point out that, as far as the dielectric body is surrounded b y a vacuum, the total body force and torque obtained from S k must both b e equal to those obtained from the two tensors considered earlier . That this is so follows immediately from (2 .11) and (2 .12) ; the effects can be calculate d directly from the vacuum tensor . We therefore next have to consider th e situation where the body is surrounded by an isotropic liquid. There exist certainly electrostatic effects for which the influence of a dielectric liquid i s essential ; we may think of the rising of a liquid between two charged con denser plates partly dipped into the liquid ( ' 0 ), or the force acting on a grounded metal sphere immersed in a liquid and surrounded by an in homogeneous field . However, none of these experiments represent critical tests for the validit y of either MINKOWSKI ' s or EINSTEIN ' S force . This can be seen in a simple wa y by first noting that the force difference is a gradient term : fE = V(E P)+PE+ Compared to MINKOWSKI ' s tensor, extra isotropic pressure Di DkV' = zV(E• P)+fM . (2 .16) EINSTEIN ' S tensor thus gives rise to a n i E . P. (2 .17 ) p E _ pM = In accordance with (2 .11) and (2 .12) the total force and the total torque o n the solid body are determined by the values of Sn in the liquid just outsid e the body. We have 14 Nr . 1 3 Sn = Sn - z n(E• P), (2 .18) but the effect from the last term in (2 .18) (acting outwards) is just balance d by the extra pressure (2 .17) which the liquid exerts on the solid . Henc e MINKOwsKI ' s and EINSTEIN ' S tensors give the same values for the body forc e and torque . This compensation effect is the direct reason why a measuremen t of the total force on a metal sphere in the liquid represents no critical test : With EINSTEIN ' S tensor there are additional forces in the boundary layer o f the sphere which just counterbalance the additional forces in the liqui d tending to press the liquid into regions of higher field .{) If we suppose tha t the system producing the inhomogeneous electric field (for instance a small , charged metal sphere) is maintained at constant charge when it is surrounde d by the dielectric liquid, we find that the total force FM = FE on the test spher e will drop in the ratio 1 /e in comparison with the total force in the absenc e of the liquid, FM = (1/e)F vac In the remaining example mentioned above, where two parallel con denser plates are partly immersed in a dielectric liquid, the main reason fo r the equivalence is simply the compensating forces in the liquid itself : Th e total electromagnetic force in the liquid between the condenser plates whic h balances the gravity force at equilibrium is found by integrating the forc e density over a volume which starts in a domain of the liquid where th e field vanishes and ends just above the surface where e = 1 . Thus the effect from the gradient term in (2 .16) vanishes, and a measurement of the heigh t of the liquid between the condenser plates cannot serve as a means to deter mine the validity of either fE or fm . This point has been emphasized also b y S . S . HAIUMcll > . [As stated above, MINKOWSKI's and EINSTEIN'S tensors must be equivalent also with respect to the torque on the body . Actually, this latter kind o f equivalence can be seen already by inspection of the expressions (2 .5) an d (1 .8a) . For the difference between the tensors is contained entirely in th e terms multiplying Gik , and the torque effect from such a term is foun d simply by integrating - + E•D(r x n) and - 2E2 (r x n), respectively, wher e the field variables refer to the fluid, over the body surface . If the body is a sphere, it follows immediately that this torque effect vanishes . Further, th e same result also applies if the body does not have a spherical form : In thi s case we may lay a fictitious spherical surface in the fluid outside the bod y so that r x n = 0 on the surface, and from the stability of the fluid it follow s that the torque exerted on the fictitious surface from the outside must b e * We are as usual assuming a rapid but continuous variation of e across the boundary layers . Nr . 13 15 equal to the torque acting on the real body surface . In all cases the bod y torque is determined entirely by the first terms in (1 .8a) or (2 .5) . ] While MINKOWSKI ' S and EINSTEIN ' S tensors thus lead to the same ex pressions for forces and torques, we shall now see that there actually exist s another effect which is measurable and which represents a critical test of th e two tensors, namely the pressure increase in a dielectric non-polar liquid because of the field . Let us then first point out which electromagnetic force s may produce this excess fluid pressure . MINKOWSEI ' S force density is, i n accordance with (2 .4), fm = eE- 1E 2 VE, (2 .19 ) and so the only pressure-producing term within the fluid, where e = 0, i s the term - z E 2 VE . This term is of importance in the boundary region betwee n two media . We shall, however, in the following confine ourselves to situations where this term is of no importance, as for instance the situatio n where a charged condenser is completely immersed in the liquid .*) Th e condenser is moreover imagined placed horizontally, so that the gravity effect can be ignored . The next kind of force which may yield an increased pressure effect i s the electrostriction force. We have hitherto ignored the electrostriction in our work, it has usually no influence upon measurable quantities, but at thi s point it is indispensable . We then start again from the free energy (2 .1) and carry through the variational procedure similarly as in sect . 3 of I, but no w with the inclusion of terms showing the dependence of s on the mass density e n, . For definiteness we shall continue to call the expression (2 .19) MINKOWSKI'S force, while the complete force expression shall be denoted as HELMHOLTZ ' force f H = oE i-E Z pE+ z0 E2 om de dom (2 .20) For the simple non-polar liquids here studied we may eliminate the mas s density be means of the CLAusIus-Mosso I relation (e - 1)/(E + 2) = const . Pm , and so (2 .20) yields the following expression for the excess pressure , produced by the field : dprr = - 10° _ ( s-1)(e+2)E2, (2 .21 ) * However, even in such a case Ve will not be exactly equal to zero ; e will increase some what in the domain between the condenser plates if the fluid pressure here increases due t o some other kind of force. With the simple non-polar liquids and moderate pressure change s that we shall be considering (Op of the order of one atmosphere), the influence from Vs on th e force is, however, negligible . See refs . it, 12 or International Critical Tables . 16 Nr . 1 3 where po is the fluid pressure when the field is turned off, thus correspondin g to a slightly diminished mass density . Finally we turn our attention to the EINSTEIN force (2 .16) . Since MIN KOWSKI ' s force yields no pressure effect in the physical situations we consider , it follows immediately from (2 .17) that 4pE - pE _ p o = (r - 1)E 2 . (2 .22 ) It is clear from eqs . (2 .21) and (2 .22) that an experimental detection o f the excess pressure represents a critical test of HELMHOLTZ ' and EINSTEIN ' S force expressions . Now this kind of experiment has actually been per formed by S . S . HAKIM and J . B . HmHAM( 12 ) . They used an ingenious metho d based on the fact that the excess pressure which the field produces gives ris e to a slight compression of the liquid and so increases its refractive index . This increase was determined experimentally by means of a TOEPLERSCHLIEREN optical technique, i .e . by a measurement of the angular deflection of light rays passing through the liquid . The experimental result s were found to be in agreement with the formula (2 .21) within limits o f accuracy of ± 5 o /o, while they disagreed completely with the formula (2 .22) . The HAKIM-HIGHAM experiment thus yields the important result that th e fluid pressure p in the presence of the field can be identified with the HELMHOLTZ pressure p H . Hence we can draw the conclusion that the validity o f the HELMHOLTZ variational method used above, based on the free energ y (2 .1), is confirmed experimentally . It has sometimes been argued that on e has the freedom to define the force density f and the pressure p arbitrarily, also in the electrostatic case, apart from the single restrictive condition tha t the relation f = Vp must be satisfied . We think however that the experimen t clearly demonstrates that there is no room for this kind of arbitrariness i n the electrostatic case within a dielectric liquid : By an integration of the forc e density over a volume element one must obtain the total electromagnetic forc e on that element which is compensated by the external pressure force actin g on the surface . Since the excess pressure predicted by HELMHOLTZ' forc e expression has been verified experimentally, one should not introduc e different definitions for pressure and force that would destroy this cor respondence . We also refer to another, theoretical, work(") by HAMM in which th e HELMHOLTZ force is derived under essentially the same assumptions a s those inherent in the usual derivation of the CLAu5IUS-MossoTTI equation . Further, HAKIM was able to show that the EINSTEIN force runs into conflic t with the CLAuslus-Mossorri equation . 17 Nr . 13 Since the electrostatic contribution to the force consists in a gradient ter m it follows immediately, as indicated above, that the electrostriction wil l yield no observable effect upon the electromagnetic force or torque acting o n a test body . The gradient form implies that there is always a balance betwee n two equally large and oppositely directed forces at the body surface . Fo r this reason HELMHOLTZ ' force can usually be replaced by MINKOwsKr ' s force, as we have done in our work . It is instructive to give the expression for the total stress tensor Ti t corresponding to both the liquid and the field : Tik = pHå = p° ak a åk - Ei Dk + E 2 bik (E E i Dk + a k Tik b ak E ' D , =O . - Pm dE l dPm ) (2 .23a) (2 .23b ) (2 .23c ) These equations obviously do not apply to the domains in space wherein external bodies have been placed . Note that the validity of eq . (2 .23b) i s dependent on the fact that we have confined ourselves to systems for which the excess pressure is due entirely to the electrostrictive force . If on the othe r hand we had considered a situation in which also the term - 2E 2 ps in th e force had a pressure-producing effect (as for instance the situation where th e vertical condenser plates are partly immersed in the liquid), the fluid pres sure p H appearing in (2 .23a) would no longer have been determined by th e simple equation (2 .21) . Now we have considered the pressure as a function of the zero-fiel d pressure p° and the squared electric field E2 . It is however possible to regar d the pressure as a function of the mass density Pm only, where the latte r quantity includes also the contribution from the compressional potential energy set up by the electromagnetic forces . We can write the total fre e energy density Ftot as the sum of a mechanical part Fmeeh and an electromagnetic part F = z E•D : F tot = Fmech (Pm) (2 .24 ) where Pm = em + 4 Pm ' P°m denoting the zero-field value and 4Pm denotin g the increase on account of the field . The pressure is then derived accordin g to the familiar formula l Fmeeh ) _ ( Pm (2 .25 ) p aPm1 Mat.Fys .Medd.Dan .Vid.Selsk . 37, no . 13 . 2 18 Nr . 1 3 Thus, although the amount of compressional potential energy transferre d to the material from the electrostrictive forces is very small, it is nevertheles s important to include also the electrostrictive contribution to gm when derivin g the pressure according to (2 .25) . Otherwise, if the expression (2 .25) i s calculated simply when the field is switched off, one will obtain the pressur e p° . Obviously it is not the electric field per se which is of main importance ; we may well assume that the field is absent in calculating (2 .25), but the n we have to imagine the presence of some other kind of external force whic h produces the same value of the density at each point . We now turn to a comparison of the above results with those obtaine d by DE GROOT et al . As mentioned already in section 1, DE GROOT and SUTTORP 06 have introduced also a second form of the electromagnetic energymomentum tensor, which is assumed to describe the whole interactio n between matter and field . This tensor form is in agreement with ABRAHAM ' S expression when the latter is supplemented with the appropriate electrostrictive and magnetostrictive terms, and when the terms involving the derivatives of the material constants with respect to the temperature are omitte d (these temperature-dependent terms being negligible in the case of non-pola r media) . We then first note the interesting result that the second tenso r introduced by DE GROOT and SUTTORP is in accordance with the HELMHOLTZ force in the electrostatic case, and thus is in agreement with ou r interpretation above . Now, since this tensor is assumed to describe the whol e interaction between field and matter, it is constructed as the differenc e between the total (field plus matter) tensor in the presence of the field, an d the total tensor in the absence of the field but at the same value of the densit y (and the temperature) . This last statement is presumably to be understoo d so that the total mass density em (including the contribution from the compressional potential energy) is required to be kept constant, independent o f the field, the authors thus implicitly presupposing the existence of some extr a kind of force to maintain the compressional energy when the field is switched off . By looking at the theory in this way we find that their mechanica l stress tensor can be written as p H åfk , the force balance thus readingfH - pp H , in accordance with our result earlier obtained . However, in spite of this formal agreement between the results it turn s out that the two procedures are essentially different . (Apart from the alread y cited papers by DE GROOT and SUTTORP, see also similar treatments b y MAZUR and DE GROOT (13, 17 ) .) Let us here therefore sketch some importan t parts of the mathematical formalism . The authors employ the following , rather unusual, balance equation for free energy per unit mass 19 Nr . 13 d(emlFtot) - pd(Pm 1 ) I E d(Cm'P) . (2 .26 ) Here we have omitted a temperature-dependent term . We shall not penetrate into the background of this equation, but mention that it is connected wit h the adoption of ; E 2 as the electrostatic energy density . Eq . (2 .26) is inte grated at constant o m to give F tot = FmechOm) +2 E . P, (2 .27 ) where Fmech is the free energy density in the absence of the field, but at th e same mass density . The authors then invoke eqs . (2 .26) and (2 .27) t o calculate the pressure [00;iFtot) aeml de = pH + z E• Pz E2Qm e ('P) dC m (2 .28 ) This pressure p is now identified with the EINSTEIN pressure p E and the expression (2 .28) is inserted into the force balance fE = pp E . The force f E can be expressed in terms of the field quantities by means of eqs . (2 .16) an d (2 .19), and by comparing with the expression (2 .20) for the HELMHOLT Z force one sees that fE = fH + D(E•P de E2Pm domj (2 .29 ) Thus, by using eqs . (2 .29) and (2 .28) the authors obtain that the forc e balance fE = ppE can alternatively be written f H = vpH , as previously mentioned . Correspondingly, the identification of the pressure p in eq . (2 .28) with the EINSTEIN pressure pE is in accordance with eq . (2 .29) . At this stage it should be clear what in reality distinguishes the metho d employed by DE GRooT et al from the method we have employed earlier i n this section . First, the expression (2 .27) for the free energy density differ s essentially from the expression (2 .24) and hence does not correspond t o the free energy density - E D for the field . The latter density was used in th e variational principle based on eq . (2 .1), and it must be equal to the wor k exerted per unit volume in building up the field . Secondly, a relation of the form (2 .28) is incompatible with our earlier interpretation according t o which the pressure is a function of the total mass density alone, the fiel d playing only a secondary role in establishing the compressional force . Instead of calculating the pressure as a partial derivative of the type (2 .28 ) whose physical meaning does not appear quite clear to us, we have instea d 2* 20 Nr . 1 3 employed the usual method according to which the pressure gradient an d the electromagnetic force emerge from a variational principle wherei n respectively the mechanical free energy density F in' and the field fre e energy density F = ;E • D are varied . Thus, after variation of the mechanica l part, the fluid pressure can be written simply as a partial derivative in th e form (2 .25), but this quantity is not explicitly dependent on the field . If w e instead had inserted the total free energy density Ftot into the variationa l integral we would have obtained the resulting force density equal to zero, i n accordance with the fact that the system consisting of matter plus field is a closed system . The results obtained in this section can be summarized as follows : Th e variational method based on the energy (2 .1) can lead both to MINKOWSKI' s and ABRAHAM ' S tensors, and as far as the dielectric body is surrounded b y an isotropic medium (vacuum or liquid), no experiments testing forces o r torques can decide between them . These tensors correspond only to different distributions of forces and torques throughout the body . Within an isotropi c medium the tensors become equal, and the increased pressure effect predicte d in a dielectric liquid (including the electrostriction effect) has been verifie d experimentally . The other proposal considered, but forward among others by EINSTEI N and LAUB (as well as DE GROOT and SUTTORP in their first proposal), i s different from the above two expressions even in the isotropic case . Th e extra pressure effect predicted by this tensor does not agree with experiment . As usual, we have in this section confined ourselves to the macroscopi c approach . It seems to be a rather common feature, however, that the microscopic treatments that have been given in this field favour the force expressio n which we have called EINSTEIN'S force . Apart form the already cited paper s by MAZUR and DE GRooT( 13 ), DE GROOT and SUTTORP(6 ), we may refer als o to a paper by KAUFMAN (14) , in which similar conclusions have been drawn . We shall not, however, go into further considerations at this point . 3 . Consideration of an Electromagnetic Wave in an Isotropi c Body at Res t We now turn our attention to simple time-varying fields within a dielectri c medium at rest . In the first part of the section we rely upon the semi-microscopical arguments from I, section 6 to point out the connection betwee n MINKOwsKI's and ABRAHAM ' S tensors for a plane wave travelling within an Nr . 13 21 isotropic and homogeneous body ; thereafter the considerations are illustrate d by an example where also boundaries are involved . Finally, we examin e alternative tensor proposals . We recall the essential parts of the procedure for constructing the energy momentum tensor in I : The energy density was taken to be the sum of th e electrostatic and magnetostatic energy densities ; correspondingly, the stres s tensor was constructed as the sum of the electrostatic and magnetostati c stress tensors derived by the usual energy variational method . From th e energy density in the form W = (E • D + HR) and from the fact that th e four-component of force, /4, vanishes within the dielectric, we deduced th e expression S c(E x H) for the energy flux . Assuming the relation S = c'g , expressing PLANCK ' S principle of inertia of energy, to be valid also for th e electromagnetic field, we further found the momentum density g = (1 /c ) (E x H) . In accordance with (1 .3) it is apparent that these components for m ' ABRAHAM S tensor . If the remaining part of the total system (the mechanica l part) is described by an energy-momentum tensor Um) , the present divisio n of the total system into electromagnetic and mechanical parts may be ex pressed by the equation - ay S v = fû = av Up , (3 .1 ) The covariant form of Sit, is given in (1 .5) . ABRAHAM ' S tensor has been advocated by many authors, and we also agree that it represents a full y adequate description of phenomenological electrodynamics . It must b e borne in mind that we are neglecting electrostriction and magnetostrictio n effects ; these effects would lead to additional terms in the tensor components . Actually we find, in the time-dependent case as well as in the static case , that if ABRAHAM ' S tensor is augmented by the electrostrictive and magnetostrictive terms the resulting expression is just equal to the second tenso r expression given by DE GROOT and SUTTORP (apart from terms involvin g the derivatives of the material constants with respect to temperature) . It must be borne in mind however, that the present problem is to som e extent a matter of convenience, and the question arises whether there are alternative tensors which can equally well be justified on the basis of (3 .1) . Our next task is thus to examine the effect induced in the mechanical tenso r Utz, on account of the force T,,A . According to (1 .4) this force has only on e nonvanishing component, namely a fluctuating component in the direction o f propagation of the plane wave . We take this direction as the x-direction ; i f the velocity of the constituent dipoles in the x-direction is denoted by u l , 22 Nr . 1 3 we found in I that the contributions to the components Uzk and U44 becaus e of this velocity component are at most of the order (ul f c) 2 , which are negligibl e me quantities . On the other hand, the components U14 - - U41 = isglth = iCOm ~ u1 are of the first order in ul f c and may thus be appreciable . By invoking the JONES-RICHARDS experiment( 15 ) we actually determined the induced mechanical momentum density as gmech = n2 1 (E x H) (3 .2 ) C in the case of an optical wave . This mechanical momentum runs alway s together with the field . Simply by including (3 .2) in the field momentum density we obtained MPIKOwsK1 ' s value g m = (1/0(D x B) . This is the tota l electromagnetic and mechanical momentum density associated with a propagating optical wave . Further, this interpretation means that the matter i s set into a small motion with the velocity u 1 when the field passes through it ; the flux of mechanical energy Sr' = - icU41 icU14 being present be cause of this motion must naturally be included in the mechanical tensor . Note that f4 = 0 (fl u l being negligible), so that a v U4, = O . If we suppose that the optical wave travels within an infinite medium , so that there are no forces in the boundary layers to cause stresses in th e material, the components L Tit the stress tensor are equal to their values a t zero field . In more general cases, the components Utik have to describe th e elastic stresses which are set up because of the electromagnetic forces at th e boundaries . Further considerations on these topics are contained in I, section 6, bu t we shall here write down the tensor scheme which pertains to MINKowsxi' s tensor : The field is described b y sm ~ ( 'a, A s, k , ), a v S v = 0, (3 .3 ) ~44 and if T,uv is the energy-momentum tensor of the total system the mechanica l part is described by 0 lU2k , (3 .4 ) T,czv SM Fi U4k U44 ' where U44 = - f m c2 . The symmetrical and divergence-free tensor T ., i s thus divided into two asymmetrical but divergence-free tensors describin g the electromagnetic and mechanical parts of the system . We emphasize Nr . 13 23 again that the reason why this kind of division is convenient lies entirely i n experience . Further, although the division of course does not affect th e angular momentum conservation law for the total system, the asymmetry o f the partial tensors gives rise to unfamiliar aspects for the angular moment a of the two subsystems . It is instructive to consider the system not only in the frame K°-th e original rest frame-but also in the frame K ' in which the mean velocity o f the matter is zero . In this frame all tensor components retain their old value s from K°, except for the components U4'k - Uk4 whose average values ar e zero . Apart from fluctuating terms the above two kinds of splitting the n become equivalent : The field is described by the same ABRAHAM tensor a s in the frame K°, and the remaining matter system is described by the tenso r Up which, in the case of an infinite medium, can be taken to be equal to th e energy-momentum tensor at zero field . If the medium is finite, the components Uik must describe also any mechanical stresses that may arise . With the omission of electrostrictive and magnetostrictive terms we thus obtain in the frame K ' a division of the total energy-momentum tensor into an electromagnetic and a mechanical part in a way which is in agreement with th e division that has been proposed by several other authors(s, 16, 17) in the rest frame . The new element of our analysis is essentially that this kind of divisio n is interpreted not to run into conflict with MJNKOWSKI's tensor, due to the fact that the experiments lead us to distinguish between the original res t frame K° and the frame K ' in which the mean velocity vanishes . Further, there is still another aspect which should be emphasized i n connection with the comparison between MINKOWSKI ' s and ABRAHAM'S tensors : ABRAHAM'S force density is the real force acting on a unit volume, i .e . th e force on the matter itself as well as on any charges and currents presen t within the volume . This force is compensated by the mechanical stresse s plus the inertial force, in accordance with the relatio n ft = å k Uik + (a/at) gmech (3 .5 ) MINK0w5KI ' s force, on the other hand, amounts to counting the inertial forc e together with the proper force : fM = fA - (a/at)g eeh = akUik, (3 .6 ) and it has thus a less direct physical meaning than ABRAHAM'S force . MIN xowSRI's force does not contain any term which corresponds to the magneti c force on the polarization currents, this term is hidden in the field momentum . 24 Nr . 13 The non-appearance of such a magnetic force term has represented a n obstacle for the acceptance of MINK0WSKI ' s tensor, as reflected for instanc e in EINSTEIN and LAUB ' S article . (5) Example involving the boundary between two medi a By the above analysis we have come to the important conclusion tha t the propagation of an electromagnetic wave through matter is convenientl y described by MINxowsKI's tensor in such a way that the rest of the syste m (the mechanical part) may usually be ignored . In this subsection, however, we shall examine the total momentum and motion of centre of mass for a total system when boundaries are involved ; in this case all kinds of momentum and energy flows have to be taken into account . Imagine a plane wave with E = Eoe,sin(kox-wt) that falls in fro m vacuum towards an isotropic and homogeneous insulator at zero angle of incidence. We take the boundary as the plane x = 0, and put for simplicit y e = u = n so that the reflected wave vanishes . We may consider a certai n part of the plane wave, say of length 1o and cross section unity, and examin e the consequences of the application of different forms of the momentu m expressions. (The length to is then required to be much smaller than th e width L of the body over which the field travels .) But it is more convenien t simply to consider the field as a wave parcel with . length to and cross sectio n unity, where to « L, so let us look at the system in this way . The total field energies in vacuum and in the body are equal, o = lo Eår 2 = nlEg/2 = , where 1 and refer to the body . By taking the divergenc e of ABRAHAM ' S tensor we obtain 2 fA = - zE 2 pe-~H 2 p,u + n c 1 ~t (Ex H) (3 .7 ) (cf.(1 .4)), valid also over the boundary if one assumes a continuous variatio n of e and u . We shall first use this force in a computation of the variou s momenta . As E = H everywhere, the surface force during the penetratio n period is (1 - n)E 2 , and so the total momentum component in the x-directio n transferred to the body on account of this force is G'f _ 1,/c c ( n2 - 1)E, Jsin 2 wtdt = 1 0 c where we have integrated over the penetration period . n (3.8) 25 Nr .13 According to our earlier results, the effect of the last term in (3 .7) is to excite a mechanical momentum in the body : G mech = n 2 c n f EHdx = 2 (3 .9 ) nc o Finally, the electromagnetic part i s = 1 f EHdx = G el .m. 1 ~i. ne cJ (3 .10 ) 0 Collecting these terms the balance of total momentum can be checked : Gsurf Gmech + Gel .m . :Ye = G vao c (3 .11 ) ' where G° ac is the magnitude of the momentum of the incoming field . This simple analysis exhibits the behaviour of the various momentum parts . If we instead had started from MINKOwsxl ' s tensor, the last term in (3 .7 ) would have been absent . In this case the momentum component Geer' supplie d by the forces in the boundary layer, plus the field momentum G m = Gel . ' + Gmech = would have added up to give the total momentum /c . Let us also examine the centre of mass velocity for the total system . Denoting the coordinates of the centre of mass by X = (X, 0, 0), we hav e '''JP, dtXtot = dt ~ 1 ÇX tot~ W tOt dVl is -tot L a t (.xS to 4 ,,)dV, (3 .12 ) åxv since the contribution to (3 .12) from v = 1,2,3 vanishes when the boundar y surface of the integration volume is chosen sufficiently far away . Hence d dtX tot 2 va c G c 1 ` tot ytot S dV = ,tot" = Jtot c, (3 .13) corresponding to the fact that the parcel travels with the velocity c before i t strikes the body . It should be noted that in (3 .13) Stet includes also th e mechanical energy flux Smech due to the small motion of matter induced b y the field . Since the body has a finite extension L in the x-direction then, durin g the period when the wave parcel leaves the body, the effect on the body i s 26 Nr . 1 3 equal and opposite to that during the entrance period . Further, the motio n of matter described by Smech is considered to be absent when the wave ha s left, so the body will stay at rest . Since the length of the parcel is small, i t can be considered to have remained a time r = Ln/c in the body . Let M an d $ denote the total mass and displacement of the body in the x-direction ; we then find from (3 .8), (3 .9) and the relation M /z = G surf + Gmech that _ (/Mc 2 )(n- 1)L . The gedanken experiment above is one of those considered by N . L . BALAZS(18 ) . We cannot agree to his conclusion, however, when he claim s the correctness of S,, in contrast with Sµv by an analysis of the total momen tum and centre of mass . Let us apply his procedure to the above case : Th e equation of momentum balance is given in the for m Gvac = G' + M4/7. , (3 .14 ) where G ' is the magnitude of the field momentum in the body which is t o be determined . Further, the law of conservation of the centre of mass velocity is written as ~Pca = A°ca /n + Mc 2 . (3 .15 ) From these equations he obtains G ' = /(nc), which agrees with ABRAHAM ' S expression only. But by comparison with our previous treatment it is apparent that th e balance equation (3 .14) is incomplete . Eq . (3 .15) is valid for both tensors , and leads to the expression for found above . But (3 .14) implies that th e magnitude of the mechanical momentum be given by Me/r, which ,in accordance with (3 .15), (3 .8) and (3 .9), is equal to G 6urf + Gmech This i s an assumption which is compatible with ABRAHAM ' S expression only ; we se e from (3 .14) that G ' = Gvac _ Gsnrf _ Gmech = Gel .m . when the balance equatio n (3 .11) is taken into account . Finally we should mention that in an examination of an example simila r to the one above, L . G . GuLLwica (19 ) has claimed that ABRAHAM'S momentu m density is satisfactory while MINnowsKI's momentum density leads t o inconsistencies with respect to the momentum balance . His argument i s essentially tantamount to saying that, in the situation above, the relation s gvac gvac gM, determine the validity of the ABRAHAM expression . gA It is however evident that in order to check the momentum balance over th e boundary one should integrate the equation akSik + agi/Ol = - fi in questio n over a volume which includes a part of the boundary, and thus one must Nr . 13 27 consider instead the momentum flow described by the components Sz k Moreover, the surface force must also be taken into account . The paper ha s been criticised also by P . PENFIELD(20, 21) . Other tensor form s It is convenient to collect the remarks on the alternative tensor forms i n this final subsection . First we recall that the DE GROOT-SUTTORP tensor (1 .9) must describe essentially another part of the total system than the part whic h we have made to correspond to the electromagnetic energy-momentu m tensor. This follows from a comparison between the energy density (1 .9b) and the energy density W = ( ED + HR) on which we have based our derivations (cf . also the HAKIM-HIGHAM experiment mentioned in section 2) . Next, the EINSTEIN-LAUB tensor (1 .8) is in conformity with the expression s (1 .9) when ,u = 1 . The most interesting alternative in relation to the topic s considered in the present section is the radiation tensor (1 .6) introduced b y MARX and his collaborators ; we recall that this tensor was defined fo r radiation fields only . The essential point in the construction of the radiatio n tensor can be visualized by an inspection of the equation (3 .1) : One assumes that the effect of the force f A is not to create a mechanical momentum, described by the components UI4 , but rather to form stresses, described by the components U 7, . Eq . (3 .1) can then be written as å 7,(S,k + U2k) + ag^/at - 0 , leading to U2k = ( n-2 - 1)Sk, in accordance with (1 .6). However, the main reason why we have not constructed the theory in this way is simpl y the result of the JONES-RICHARDS experiment, to which we have alread y referred repeatedly . As we pointed out in the rather detailed consideratio n in I, section 6, it was essential for the validity of the derived formulas that the electromagnetic energy-momentum tensor in question be a divergencefree quantity in the interior of the body . Since the radiation tensor just ha s this property, and since the relation between the momentum flow component s is Sdk = = (1 /n2)Sk , it follows that the radiation pressure predicted by th e radiation tensor is equal to 1/n2 times the MINKOWSRI radiation pressure . By a comparison with the observed data we are thus in a position to dra w the decisive conclusion that the characteristic assumption inherent in th e derivation of the radiation tensor should be rejected . Note that the electrostriction effect will have no influence on this result . Although it should therefore not be of importance to go into a detaile d examination of the use of the radiation tensor in the example considered i n the above subsection, let us yet note the following points . The force density Nr . 1 3 28 can no longer be written as (3 .7), since this expression will violate the law o f conservation of momentum . This is so because the last term in (3 .7) is n o longer associated with a mechanical momentum, and hence the total momentum after the wave has entered the body is Gsnrf + G el ' . G"° . I n order to fulfil the momentum conservation law the force density must b e defined as ff a„S v, where the stress components are not the sum of th e electrostratic and magnetostatic stress components . If we define Si _ (1/n 2 (x))SM also in the spatially dispersive region in the boundary layer , we find that the momentum induced by the surface forces is ( Gs) surf = (1 - 1/n) '/c . The interesting aspect here is that the quantity (G s ) 8u " ha s the opposite sign of the quantity Gsurf calculated earlier in eq . (3 .8) ; whil e the surface force following from the radiation tensor acts inwards to th e body the surface force following from ABRAHAM ' S and MINKOwsII ' s tensor s acts outwards from the body surface . We are not, however, aware of a direc t experimental test of this effect (cf . the last part of the next section) . 4 . Discussion of some Possibilities for Experiment s In this section we examine experimental situations in which time dependent fields exert torques on dielectric bodies at rest . As usual we firs t focus our attention on the relative behaviour of MlNxowsxu's and ABRA HAM'S tensors . In the first class of experiments considered-the interaction between a stationary radiation field and a dielectric body-the result is that the two tensors lead to the same answers . Thereafter, an example is given of a second type of experiments in which the difference can be measured . Finally, we propose a critical experiment testing the radiation tensor and th e EINSTEIN tensor . Proof of equivalenc e As an example of an experiment which traces the angular momentu m interaction between an electromagnetic wave and a dielectric body, the ol d G . BARLOW experiment( 22 ) should first be mentioned . He made a carefu l measurement of the torque produced by a beam of light in oblique refractio n through a glass plate, and obtained good agreement with the theory . W e refer also to the famous R . A . BETH experiment (23 ), in which the existence o f angular momentum in a light wave was detected by letting the wave pas s through an anisotropic crystal . The latter experiment has more recently bee n repeated by N . GARRARA (24) with the use of centimetre waves . These ex- Nr . 13 29 periments consisted in letting a stationary wave interact with the body an d then measuring the deflection when equilibrium was established betwee n the electromagnetic torque and the mechanical torque exerted by a torsiona l suspension . However, we need not go into detailed considerations of thes e situations in order to test the relative behaviour of MINaowSKI ' s and ABRAHAM's tensors, since we will find the torque N A = N M, just as we did in th e static case . Instead we present a simple argument which shows in genera l that in a wave-dielectric body situation the two energy-momentum tensor s yield the same value for the torque . Consider then a stationary high-frequency wave interacting with a dielectric body (in general anisotropic) . The body is assumed so heavy tha t no macroscopic motion needs to be taken into account . If the angular momentum of the internal field in the body is denoted by M i , the torque N can be written as N = dM' /dt - dM i /dt . (4 .1 ) It can readily be seen that each of the two terms on the right hand side o f this equation is the same for ABRAHAM ' S or MINKOWSPI ' S tensor . In both cases the energy flux is given as c(E x H), therefore the direction an d velocity of the travelling field energy is the same, and it follows that th e first term on the right of (4 .1) is also the same . Further, since we assum e that the field is stationary, we can simply put dM i /dt = O . Hence NA =WI = - dM°"/dt : The two energy-momentum tensors are equivalent wit h respect to torque effects since these effects are determined in terms of th e vacuum field . (Alternatively, we may consider a wave packet in interaction with th e body during the time period t = 0 to t = T, during which the field is assume d to be stationary . Then we can require on physical grounds that N be independent of T at any time t, also in the small transient period when the fiel d leaves the body . We now assume only that dM i /dt must be equal to som e constant during the stationary interaction period, since each component i s proportional to the averaged energy density of the incoming wave . Whe n t > T, one has M i = 0, but then dMi /dt can be made arbitrarily large in th e transient period when the wave leaves the body, by choosing T large . Thes e features are incompatible with the condition (4 .1), hence dMi/dt = 0 in th e stationary interaction period .) 30 Nr . 1 3 Proposal of an experimen t In the preceding we considered an electromagnetic wave in interactio n with a dielectric system . Now there exists the possibility of combinin g electric and magnetic fields in a way which, in principle, makes it possibl e to bring out explicitly the effect arising from ABRAHAM ' S force . We shall give a proposal similar to one put forward by MARX and GvöRGyI( 3 ) . A cylindric dielectric shell of isotropic matter with large s is suspended between the surfaces of a cylindric capacitor so that, in the absence of fields, the shel l can oscillate about its axis (z) with a frequency wo . The internal surface o f the capasitor is then charged to the amount q per unit length, and a homogeneous magnetic field Hoe-iwt is impressed parallel to the z-axis . We suppos e that the wavelength which corresponds to the frequency w is large compare d with the dimensions of the system, so that within the internal, massiv e cylindric conductor, we may write V x H = aE/c, where a is the conductivity . Taking into account that the penetration depth into the conductor is appro ximately equal to Vc2 /wa, which is a large quantity when w is small, an d putting ,u = 1, we obtain within the internal region of the conductor H = Ho é iwt tw E= 2c ~ Within the dielectric shell Er this domain . Thus A f~ = rH,o é iwt (4 .2 ) = q/(2ver), while eqs . (4 .2) remain valid also i n s -1gwHo s-1 a s ôt(rrHz) e Sin wt, (4 .3 ) 27CCr when we take the real part . Hence the torque component is N3 = rfq,dV s -1gHo V = s wsinwt = Kwsinwt, (4 .4 ) 27rc where V is the volume of the body . We have ignored the surface forces sinc e these act in the same directions as - Vs and hence have no influence on th e oscillations . The equation of motion can be written a s K cp+yg +wôq~ = j w sin wl , (4 .5) where y is the damping constant and I the moment of inertia about th e z-axis . The largest oscillations occur when w = wo and are given by 31 Nr . 13 K ~ = - j cos wo t . Y (4 .6 ) This effect can in principle be measured . With a direct use of MINKOwSKI ' S tensor one obtains no force that can account for these oscillations, an d MINKOWSKI ' S tensor is thus inappropriate in the present case . (It must b e emphasized that the previous derivation of MINKOwsIu ' s tensor for time dependent fields in isotropic media applies properly only to the case o f radiation fields . ) As far as we know, the experiment has not been performed . We emphasize the essential difference between this situation and thos e considered in the above subsection : At a given instant, the force component f causing the torque does not vanish when integrated over the volume . Further, it is now the total time oscillations themselves which are detecte d and not, as in the previous situation, their effect after integration over a tim e which is large in comparison with the oscillation period . Other tensors Let us consider again the system of a stationary wave field and a dielectri c body studied in the first of the subsections above, and first employ th e radiation tensor This tensor has been derived for the case of isotropi c bodies only, so we shall accordingly assume the body to be isotropic . It i s immediately apparent that if the wave comes in from vacuum, interacts with the body and then enters into vacuum again, we can apply just the sam e argument as before to conclude that the radiation tensor yields the sam e value for the torque as MINKOwsKI ' S and ABRAHAM ' S tensors . But a simpl e calculation shows that the direction and magnitude of the surface force wil l in general be different from what we obtained in the previous cases ; it i s only the total torque itself that remains unchanged . (For instance, if a n appropriately polarised optical wave falls obliquely inwards to the body a t BREWSTER ' S angle of incidence such that the reflected wave vanishes, it ca n be verified that the surface force acts in a direction parallel to the surface , instead of in a direction outwards along the normal vector, as obtained fro m MINKOWSKI ' S or ABRAHAM ' S tensor .) It has sometimes been claimed that th e BARLOW experiment( 22 ) mentioned above, involving a measurement of th e torque exerted by a light wave on a glass plate, should actually provide a n experimental test of the direction and magnitude of the surface force . But we think that this is not so, although BARLOW himself interprets the effect i n a way corresponding to MINKowsKl ' s or ABRAHAM ' S tensor . The only thin g S . 32 Nr . 1 3 measured is the total torque, which is explained equivalently by all tensor s in question . However, an obvious generalization lies at hand in order to change fo r instance the BARLOW experiment into a critical experiment with respect t o the radiation tensor, namely, to immerse the body into an isotropic dielectri c liquid . The radiation tensor has a value different from the two other tensor s mentioned in the liquid, and so a torque measurement can be crucial . I n order to derive the appropriate torque expression it is convenient to write the general formula (cf. (I, 1 .7) ) Nf =f Sik (x,fk xkfa Ski) dV (4 .7 ) body in the following compact form : N= f (r x Slig) dS surface (r x - dl f (r x g) dV (4 .8a ) body Slig) dS . (4 .8b ) surface For an optical wave the last integral in (4 .8a) vanishes because the field i s assumed to be stationary and the body remains practically at rest, and th e surface integrals are taken in the liquid just outside the body . By means o f (1 .6) and (4 .8b) we find the result N s = (1/n 2 )N M = (1/n2 )NA , where n i s the refractive index of the liquid . The surface integral in (4 .8b) can b e evaluated in the actual experimental situation with one of the tensors inserted, and one can thus check the tensors by a comparison with the observe d torque . As the next point we consider the EINSTEIN-LAUB tensor Su„ applied t o the same situation . (For optical fields we can put,u = 1, and it is then apparen t from (1 .8) and (1 .9) that the EINSTEIN-LAUB tensor and the DE GROOTSUTTORP tensor are in agreement .) This tensor is defined also for anisotropi e media . We evaluate this case most simply by noting the following relatio n in the liquid which surrounds the body : S ,n = - n (E . P), f (r x n) E • (4 .9 ) so that (4 .8b) yields NE = NA +4 surface PdS, (4 .10 ) Nr . 13 33 where the surface integral is taken in the liquid . EINSTEIN'S tensor thus leads to still another value for the torque, which might be tested experimentally . The dielectric shell-experiment considered in the second subsectio n above is not of direct importance for the radiation tensor since this tensor i s defined for radiation fields only . However, it can readily be seen that bot h the radiation tensor and EINSTEIN ' S tensor lead to ABRAHAM ' S value (4 .4 ) for the torque . To this end we need only examine eq . (4 .8a), where now the last term is non-vanishing and where S,L° is replaced by Snae in vacuu m outside the shell . Since gS = g E = gA it follows that NS = NE = NA . (Moreover, the value (4 .4) can be checked by inserting the field value s (4 .2) and the expression for E r into (4 .8a) .) This experiment is therefor e not a critical test of the relative behaviour of the three tensors mentioned . In this case it does not seem either to be an appropriate generalization t o immerse the system into a dielectric liquid . 5 . Some Remarks on the Literatur e Together with the exposition of the various topics we have met up til l now-both in I and in the present paper-we have found it desirable t o include also some remarks pertinent to essential passages in earlier works o n the subject . The literature is however large, and there remain important part s of it that could not naturally be considered or even touched upon in th e preceding exposition . We have therefore reserved the present section for a critical review of some earlier (phenomenological) treatments, especially those which seem to be incompatible with the interpretations given above . We think that this avenue is natural to follow, since the present problem i s not only a deductive task but also a matter of clarification of a confuse d situation . Evidently we cannot give a detailed scrutiny of all the relevan t papers of phenomenological nature, but shall rather be concerned with illustrative examples . For a large part we shall be concerned with the analysi s of criteria . The present section represents the end of our nonrelativisti c treatment ; from the next section on we concentrate upon topics connecte d with relativity . In the two first subsections we consider two gedanken experiment s which have been put forward . The idea behind these gedanken experiment s has been that by comparing with results obtained from physical conservatio n laws, one should be able to decide which energy-momentum tensor i s correct . In both the cases we shall consider, MINKOwSaI's tensor has bee n Mat .Fys.Medd.Dan .wid .Selsk . 37, Do . 13 . 3 34 Nr . 1 3 claimed to be preferable, as a result of a study of the conservation equation s for momentum . We shall show how these experiments can be equivalentl y described with the use of ABRAHAM ' S tensor . In the subsequent subsectio n some aspects of the CERENKOV effect are considered, and finally we mainl y dwell on arguments favouring other tensors than MINliowsKI ' s and ABRAHAM ' S expressions . Propagation of discontinuities In two papers A . RuBINowlcz (25) investigated the situation where electromagnetic discontinuities are propagated through an isotropic body at rest . The conservation equations for energy and momentum are integrated over a domain E in four-space bounded by the hyperplanes 6o, 61 and a3 ; o corresponds to the three-dimensional volume Vo which at the time to i s enclosed within the two-dimensional surface eh ; a l corresponds to the volum e V1 at t = t 1 > to, and a3 is the connecting time-like hypersurface . The surfac e 0 is considered moving inwards with the velocity u = c/n in the direction o f its normal . Then imagine a two-dimensional surface 0*(t) across which the field is discontinuous : E1 =E,H1 =H ; E2 =E+AE,H2 =H+4H. (5 .1 ) Here 1(2) denote the inner (outer) side of Ø* . For simplicity, we suppos e 0, also to move together with the field, with the velocity u = c/n in th e direction of its normal . RuBINowucz integrates the energy conservation equation over E and find s that Ø* is associated with no source of energy when either of the two energymomentum tensors is inserted . We therefore turn our attention to the momen tum conservation equation written in the following form (our notation) , where the time derivative is taken along the moving volume element : a k (Sik and integrate over E : r 1 d - gi ux) + ( gi dl'') = -fi dVdt t, r J)v I+ J dt v, (5 .2) te r( (Si -uk )n k dS = J Ø+Ø * dVdt . (5 .3 ) The contribution from Ø` to the left hand side of (5 .3) can be written i n vector form, according to RuBiNowicz, as Nr . 13 35 r f dt J [(Sn +gu) 1 + (S r, - gu) 2 ]dS = t o ~ dt to rn 2 J n 4H+ 4E x P -[Ex H]dS . (5 .4 ) Ø* Here, S r, is a vector with components Sn2 Sik nk , and n is taken to point outwards from the integration domains ; p is a number, such that p m = n 2, p A = 1 . Hence RuBINowrcz concludes that Ø, is associated with no sourc e of energy or momentum as far as MrNKOwsKl ' s tensor is employed, in contrast to what is the case with ABRAHAM ' S tensor, since (5 .4) then is non vanishing . This feature is claimed to favour the former expression . Let us, however, examine the case p = 1 . We see that the contributio n (5 .4) is not yet complete since the effect arising from gmech has not bee n incorporated . This effect is connected with the term (n2 - 1)fc 2 aSfat in f. Hence, the amount on the left of (5 .3) is to be augmented b y ~t (Si dV) - p V . (Sti u) dV d t J n 2 2 2 (5 .5) t, [_j)si d v 1 I dt I J Su•ndS . J to Ø + From (5 .5) we see that the contribution from Ø'' equals, in vector form , n2 2 1 (' C2 J to dl J (S1 - S2 ) udS, (5 .6 ) Ø* which, together with (5 .1) and (5 .4), yields MINxowsKI's result . We see agai n that the choice between MINKOwSKr ' s and ABRAHAM ' S tensors is mainly a matter of interpretation . Induced motion of a ferromagnetic test body Let us next examine the gedanken experiment recently considered b y COSTA DE BEAUREGARD (26) . The arrangement is rather similar to the one w e considered earlier in the second part of section 4 : A ferromagnetic shell with mean radius ro, thickness b and length a is subjected to forces arisin g from a short current pulse in a rectilinear wire placed along the symmetry axis (z) of the shell . Besides, the wire is charged to a constant charge q per unit length and hence gives rise Lo the radial electric field E r = q/(2mr) , 3* 36 Nr . 1 3 when c is put equal to 1 . When the current is flowing, a tangential magneti c polarization M = B li( is present, and when the current has decreased t o zero, there remains an amount 4M = 4B in the shell which, together wit h E, gives rise to a linear momentum in the z-direction . COSTA DE BEAUREGAR D integrates the force component f3 = - eaA 3 /cat over time and over the volum e of the wire, and obtains f3 dVdt J 1 gab4112. = (5.7 ) c wire If we use MINKOWSKI ' S tensor to calculate the remaining momentum component in the z-direction when the current has left, we find f DBdV 4G3 = 1 4 c body = 1 c EAMdV = 1 gab4M. c bod y (5 .8) A corresponding calculation with ABRAHAM ' S tensor yield s f EHdV = O . 4W = 1 4 c (5 .9) body Since (5 .7) and (5 .8) are obviously in accordance with the balance of tota l momentum, COSTA DE BEAUREGARD concludes that MINKOWSKI ' S expressio n for the momentum density should be preferred . Let us, however, continue to consider ABRAHAM ' S tensor and write th e force density in the form f 4_ a fM + at(gM - gA). (5 .10 ) Hence, by integration over the total syste m ( f+fJ =f + f = f f'dVd t +14c f DBdV = 0 , /3 dVdt wire body f3 dVdt wire wire 4 bod y (g3 9'3 ) d V (5 .11) bod y in view of (5 .7) and (5 .8) . Eqs . (5 .11) and (5 .9) show how the momentum balance must be interpreted in terms of ABRAHAM ' S tensor : Although th e electromagnetic field represents a non-closed system, eq . (5 .9) shows that 37 Nr . 13 the electromagnetic momentum is conserved . (In the case of MINKOWSEI ' S tensor this was not so, cf. eq . (5 .8)) This conservation is carried into effec t by the fact that the action from the force on the wire is equal and opposit e to the action on the body, in accordance with (5 .11) . We note in passing that only ABRAHAM ' S tensor leads to a mechanical force on the test body in th e z-direction, due Lo the fact that the surface forces on the body, which ar e common for the two tensors, are directed in the radial direction . There ar e also surface forces at the two end surfaces of the body, but these force s compensate each other . With MINKOwsaI ' s tensor, the presence of electromagnetic momentum is due to a momentum flow into the body . Following COSTA DE BEAUREGARD we mention that the recent C . GorLLoT(27 ) experiment might be considered as a possible test of the theory . I n this experiment a translational motion of a nature similar to the one described above was detected . However, although the qualitative features are similar , COSTA DE BEAUREGARD reports that the GOILLOT effect is far too large to correspond to the effect deduced from the electromagnetic energy-momentu m tensors . The effect of the experiment is presumably a spin effect( 28 ) . Th e inapplicability of the above theory should be expected in this case, sinc e systems exhibiting remanent magnetization are very different from thos e described by the simple phenomenological theory we are considering . On the Cerenkov effec t The CERENKOV effect is a convenient means for a study of the variou s energy-momentum tensors . We have touched upon this effect before, i n connection with relativistic considerations in I, section 10, and we shall tak e it up again in the relativistic considerations later on in this paper, but her e we examine some of its implications when the medium is at rest . In thi s kind of problem it is most convenient to use MINxowsxi's tensor, and let u s also employ the phenomenological quantum theory (see, for instance, ref . 29 or ref . 30) according to which the four-momentum of the emitted photo n is hk~ = h(k,iw/c) . With MrNxowsKI's tensor the balance equations fo r energy and momentum for the photon plus its radiating electron with momentum p --> p ' , are cj/p2 + nt 2 c2 = p = t7w + c hk -i- p' , + m2 c2 (5 .12a) (5 .12b ) from which we obtain the well-known expression for the angle 0 M betwee n pandk : 38 Nr . 1 3 c kht_j 1 cosOM = - + - 1 -mu 2p ` n2 , (5 .13 ) Here u is the modulus of the velocity of the incoming electron, u = p/m(u). From the point of view of ABRAHAM ' S tensor the above argument is only slightly modified : The momentum of the emitted photon in this case i s lik/n 2 , while the force fA gives rise to a mechanical momentum (n 2 - 1. ) •lik/n2 which runs together with the field . These two contributions togethe r yield the result hk which was used in (5 .12b) . Concerning the literature on this subject we should first of all refer t o the clear discussion by G . GY6RGYi( 3l ) . He shows the equivalence betwee n MINKOWSKI ' S and ABRAHAM ' S tensors along similar lines as above . On th e other hand, there has recently appeared a paper by J . AGUDfN (32) on th e CERENKOV effect in which ABRAHA M ' S tensor, but not MINKOwSKI ' S tensor, is claimed to be in accordance with EINSTEIN ' S mass-energy relation . Let u s therefore trace out the reason for this result, when we transform the formalis m to our notation and simplify the argument, which consists in a study of th e conservation equations for total energy, momentum and centre of mass velocity . Imagine that the initial electron moves along the x-axis with th e velocity u and that it emits a photon with mass m ' in the direction Ø at th e time t = t1 . After the emission the electron moves with the velocity u ' p/m(u ' ) in the direction (p . The energy balance is written a s m(u) _ hw/c 2 + m(u ' ) . (5 .14a ) With ABRAHAM'S tensor the magnitude of the momentum of the emitte d photon is hk/n2 = hw/(nc), and the balance equation for the x-component o f momentum is written a s m(u) u = -40) cos0 + G1 e"h + m(u ' )u ' cos(p, (5 .14b ) nc where Gme"h is the momentum transferred to the medium . Finally, AGUDÎN introduces an equation expressing the centre of masstheorem . During the time period t = 0 to t = t 2, where 0 < t l < t 2, th e centre of mass of the total system is displaced by a distance m(u)c 2 ut2 /Ve tot and the relation given by AGUDfN is equivalent to writin g m(u)ut2 = c m ' utl + -cos 0 ( n + Gmeah (t2 - tl ) + m (u' ) [ut l + (u ' cos 97) (t 2 - tl ) 1 . 39 Nr . 13 By inserting eqs . (5 .14) into (5 .15), one finds that the latter relation i s fulfilled if m ' = ttw/c 2 , which is EINSTEIN ' S mass-energy relation . Considering MINKOWSKI ' s tensor, AGUDIN uses the same set of equation s as above with the single difference that the first term on the right hand sid e of (5 .14b) is multiplied by a factor n2 . The new value for m ' one now obtain s shows an involved geometrical dependence which must be regarded as un physical . From this he concludes that ABRAHAM ' S tensor is the one of the tw o tensors that should be preferred . Let us now examine the above argument from the point of view of ou r earlier interpretation . Since Gmech in (5 .15) refers to the small motion of th e medium induced by the photon, we must have Glee' = ((n2 - 1)fnc)lico cos O . This value is in accordance with the value for Gmech appearing in equatio n (5 .14b), which is constructed on the basis of ABRAHAM ' S tensor . However , with AGUDIN ' s construction of the momentum balance in the case of MINKowsai's tensor the right hand side of (5 .14b) is changed into (nhw/c)cos O + Gmech become different ; in Grin ' + m(u' )u' coscp . Thus the two values of (5 .15) Gmech remains unchanged while in the momentum balance Gmech = 0 This is the reason for the diverging result . It is instructive to recall that th e centre of mass-velocity for an arbitrary (limited) total system is given b y c2G tot,rrtot (cf. eq . (3 .13)), which is a constant in view of the conservatio n equations for energy and momentum . Applied to the present case this mean s that the centre of mass-theorem can yield no more information than what is contained in eqs . (5 .14) We are evidently free to assign a mass m' = hw/c 2 to the photon also in the case of MINxowsKI's tensor . Finally we note that the CERENKoV effect provides a convenient opportunity to examine also the radiation tensor (1 .6) . If we in this case construct the energy and momentum balance similar to (5 .12) the only differenc e is that the term hk in (5 .12b) has to be replaced by hk/n 2 ; the radiatio n tensor is divergence-free and there is no force present to give rise to a mechanical momentum . Thus we find the following expression determining th e angle O s between p and k in this case : cos O s ne kh/ 1 \ -- 1 n2l u 2p` (5 .16 ) Since kh « p this equation leads to unphysical values for O s . It seem s therefore that there are even formal difficulties for the application of th e radiation tensor to situations where both particles and fields are present . Nr . 1 3 40 Final remarks So far we have limited ourselves to a study of previous treatment s advocating the validity of either MINIwwsIu ' s or ABRAHAM ' S tensors . In thi s subsection we discuss briefly, without going into detail, some papers i n which diverging tensor expressions have been given preference . The tensor (1 .8) introduced long ago by EINSTEIN and LAUB was en countered already in section 2, in connection with electrostatic phenomena . We recall the important result that the excess pressure effect in a dielectri c liquid predicted by this tensor does not fit the HAKIM-HIGHAM experiment . Let us yet write down the complete force expression in the time-varyin g case : fE = OE+(P•p)E+(M•p)H+1(j x H) + 1 (Px H) + 1 (ExM) . c c c (5 .17 ) It should be noted that according to (5 .17) the magnetic force density acting on a stationary current distribution, for instance in the interior of a wire, i s equal to (1/c)(j x H), instead of the usual (1/c)(j x B) following fro m ABRAHAM ' S or MINKowsKI's tensors . Now, in order to support their forc e expression, EINSTEIN and LAUB analyse in their paper( 5 ) two examples involving the presence of stationary currents . The second example considere d is the following : An infinitely long, rectilinear wire carrying a stationary current J is assumed to prossess a magnetization M in a direction perpendicular to the wire . When no external field is present, it is clear that th e electromagnetic force on the wire vanishes . EINSTEIN and LAUB verify by a direct calculation that their tensor leads actually to a vanishing force Fi per unit length in a direction i perpendicular to the wire . We must point ou t however, that this result is not peculiar for the EINSTEIN-LAUB tensor and thereby does not represent any particular support for this tensor . In fact , any of the actual tensor expressions will lead to this result, as an immediat e consequence of the relation s r Fi f akSixdV = - I Sk~n k dS, unit length (5 .18 ) J where the value of the last integral goes to zero when the integration surfac e is taken sufficiently far away from the body . Concerning the remaining terms in (5 .17) we mention that the argumen t for introducing the term We) c) (P x H) was that there must be no distinctio n in principle between external currents j and polarization currents P (cf . Nr . 13 41 also our remarks in connection with eqs . (3 .5) and (3 .6)) . The magneti c terms in (5 .17) were introduced by analogy considerations . EINSTEIN and LAUB ' S paper was criticized by R . GANS (33 ) . He employe d the force expression corresponding to MINKOwsKI's tensor, at least for para and diamagnetic media, and made an explicit calculation of the transversa l force on a conductor which carries stationary current and is surrounded by an external magnetic field . Ferromagnetic media were considered separately . In all cases the force was found to vanish when the external field is zero, i n accordance with our statement above . One remark is called for, regarding GANS ' claim that the EINSTEIN-LAU B expression comes into conflict with the energy balance . In his argument h e uses assumptions that are valid for MINKOwsKI' s tensor only, viz . that th e energy flux vector is given as S = c(E x H) also when the velocity of th e medium is different from zero . The other tensor expressions will lead to a n explicit appearance of the velocity in the energy flux expression . The use of thermodynamic methods in the present problem represents a special kind of approach . We have already employed such a method in this paper, although in a very simple form, in section 2 . In this context we shoul d refer to the work by DE SA(34) and to two papers by KLUITENBERG an d DE GRooT (35) . KLUITENBERG and DE GROOT postulate a certain relativisti c Goss relation and assume the material energy-momentum tensor to b e symmetric ; they obtain from these assumptions a symmetrical electromagnetic tensor which in the rest system is in accordance with eqs . (1 .9) , apart from a difference in the energy density component . Further, the y claim that the formalism yields ABRAHAM ' S tensor as an equivalent result, i f appropriate new definitions for the hydrostatic pressure and the interna l energy are imposed . Concerning this latter statement, however, we must point out that the formalism must always be chosen so as to conform to th e observed effects, and the HAKIM-HIGHAM experiment does not seem to leav e the room for ambiguities in the definition of pressure in the electrostati c case (cf. section 2) . The papers by G . MARX, G . GYORGYI and K . NAGY (3 36 , 37, 38) (with further references) contain a series of arguments of different kinds, an d represent together one of the most extensive macroscopic treatments of th e problem that has been given . We are considering elements of their paper s at various places in our work, for instance in the examination of the radiatio n tensor . Their main conclusion is that ABRAHAM ' S tensor is the basic electromagnetic tensor, while the radiation tensor (instead of MINKOWSKI ' s tensor) is claimed to he the result of a combination with the excited matter induced 42 Nr . 1 3 by a propagating field . Since in this section we consider fields within matte r at rest, we should mention that the difficulty they claim to exist for MINKOWSKI ' S tensor in explaining the propagation of the centre of mass for a limited radiation field within an isotropic dielectric, is cleared up of on e observes that the time derivative of the quantity (SM /c2 - gm), integrate d over the total volume, is equal to zero . As we have noted, the absence of terms containing polarization an d magnetization entities in MINKOwsKI ' s force has represented an obstacle fo r the acceptance of this expression (cf . also the book by FANO, CHU, ADLER (39 )) . In a series of papers published recently( 40 ), P . PoINCELOT took the ful l consequence of the opposite point of view and proposed the introduction o f all kinds of polarization and magnetization terms in the force on an equa l footing with the free charge and current terms, viz . 1 f = (e -p•P)E + - (j+P +cVxM)x B c t4 = -c E . (j + +cp xM) . (5 .19a ) (5 .19b) The tensor corresponding to the force (5 .19) can be expressed in terms o f E and B in the same form as the electromagnetic tensor in the vacuum-field . However, although (5 .19) cannot be rejected on purely formal grounds, w e cannot find any argument of convenience or experimental evidence that sup ports this expression. 6 . Angular Momentum in Arbitrary Inertial System s In the remaining part of our work we shall be concerned with topic s connected with relativity. To some extent we shall have the opportunity t o return to a study of situations which were considered already in I, chapte r IV, in connection with MINxowsKI ' s tensor . From the preceding it should b e clear that in a relativistic theory the latter tensor is convenient to use, i n order to obtain information about the direction and velocity of the propagating field energy . But it is instructive to consider also the behaviour o f the alternative tensors (especially ABRAHAM ' S tensor) in arbitrary inertia l systems, since such an anlysis will exhibit characteristic differences betwee n the tensors . In this section we assume that the medium is homogeneous an d isotropic, and let as usual K denote the inertial system in which the res t system KO moves with the velocity v along the x-axis . 43 Nr . 13 Evaluation of torques within an infinite mediu m Let us image a finite radiation field within a large (infinite) dielectri c medium . The angular momentum quantities Mm, are in general defined b y the integral mw = f xvgtic)dV, (6 .1 ) taken over the whole field, in any frame K . Let us further imagine that fo r each of the electromagnetic tensors in question we insert the appropriat e expression for g p into the integral in (6 .1) and calculate M . In this contex t it should be emphasized that in each case q 1, is considered as a field quantity , M~v thus being considered as a field angular momentum . This definition is the natural one and we have used it throughout, in I as well as in the presen t paper, although we have repeatedly pointed out that in the MINKOwSKI cas e the momentum density gj in reality includes also a mechanical part whic h is responsible for the asymmetry of MINxowsKm ' s tensor . In other words , MINKOwsar's angular momentum MX contains in a strict sense also a contribution from the mechanical part of the total system . To call M a fiel d angular momentum is obviously just tantamount to calling G' a field linea r momentum . If on the other hand we take the distinction between the tw o parts of G[ explicitly into account and exclude the mechanical part o f from the expression for field angular momentum, we obtain instead ABRAHam's expression M ,,, since that part of which pertains to the electromagnetic field is just g . The different ways of dividing the total angula r momentum into a field part and a mechanical part obviously have no influence upon the conservation of total angular momentum, which is a consequence of the symmetry and the zero divergence of the total energymomentum tensor . Thus, in each case we obtain the mechanical angula r momentum by inserting that part of the total momentum density which i s not counted as a field entity . As regards MINKOWSKI ' S tensor it seems appropriate to recall from I , section 11 that the quantities M are equivalent to the angular momentu m quantities one can most simply construct on the basis of N0KTHER ' S theorem . This is in accordance with the general property of MINKOwsKI ' s tensor tha t it readily adjusts itself to the LAGRANGIAN procedures . We recall also that MI% is in general not a tensor . For a comparison between the various tensors it is however not th e angular momentum itself which is of primary interest in each case, bu t rather its time derivative, i .e . essentially the body torque . The torque is Nr . 1 3 44 defined as N = - cliff/di, and we shall in the present subsection start to perfor m a direct calculation of the torques corresponding to ABRAHAM ' S and MINIiowsKI's tensors . It will turn out that the two values so obtained in genera l are different from each other . This difference is what we should expect , since the momentum densities gA and gm are themselves essentially differen t in direction and magnitude . The last point requires some further explanation . In all electrostatic (o r magnetostatic) cases and also in all high-frequency electromagnetic case s considered up till now we have found that ABRAHAM ' S and MINaowSKr ' s tensors yield just identical expressions for the torque on a test body immersed either in a vacuum or in a dielectric fluid . The reason for this equality can be understood in a simple way by observing that in those cases th e torque could be evaluated as a function of the field stress tensor taken i n the domain just outside the surface of the body, wherein the equality S,, k = Sk is valid . (Cf . eqs . (2 .11) and (4 .8b) for the electrostatic and electromagneti c cases, respectively .) In the situations considered in the present section ther e is however no similar reason why the torque expressions should be th e same ; we have to lean directly upon the formula (6 .1) and evaluate it over the field region within the body . In the MINK0wSIïI case the torque can b e looked upon as a consequence of the asymmetry of the mechanical energy momentum tensor (this fact having represented as an objection to th e acceptance of MINaowsKI ' s tensor), while in the ABRAHAM case the torqu e arises because of the force density . In spite of this difference between the two torque expressions obtaine d within an infinite medium we shall nevertheless in the next subsection se e that the torques are relativistically equivalent from a physical point of view , since both of them are compatible with uniform motion of the physica l system in K . In this context we shall draw into consideration the analogou s situation encountered in relativistic mechanics of elastic media : An elastic body subjected to stresses in the rest frame will in general require a torqu e to maintain steady motion in another inertial frame . Let us now start with ABRAHAM ' S tensor and perform the calculation . From (6 .1) it appears that the torque NA = - dMA /dt in K is given b y NA = J (r x f A ) dV. (6 .2 ) At first sight it seems that one will meet a difficulty in the evaluation of thi s integral . This difficulty is connected with the non-invariance in four-spac e of the world lines corresponding to ABRAHAM ' S energy flux S A (cf . the next Nr . 13 45 section) . On the other hand we pointed out in I, section 9 that the ray velocit y u, which is the velocity of propagation of the wave energy and which ma y be written as u = SM /WM, transforms like a particle velocity . From this it follows that the world lines corresponding to MINKOwsIU ' s energy flux SM really have the property that they remain invariant in four-space upon a LORENTZ transformation . Now it is clear that in order to obtain a picture o f the wave propagation in K one has to transform the total wave, i .e . one mus t include the effect also from the produced mechanical momentum g mech o in K° . This feature resolves the apparent dilemma in connection with th e evaluation of the integral in (6 .2) : Even though SA is different from S M both in direction and magnitude we have to integrate over that part of spac e where the field is actually present, i .e . across the world lines correspondin g to SM . It is now convenient to assume that the field travels parallel to the xyplane in such a way that any wave vector k which is contained in the wave , makes an angle 9 with the x-axis in K . It can readily be verified that the onl y non-vanishing component in (6 .2) is the z-component, the other component s fluctuate away. We evaluate the integral in (6 .2) over the domain AB, i .e . over the hypersurface t = 0 (cf . I, Fig . 2) . We obtai n N3 o ° = y f ( x °fz -4fi + ß ct °fe n ) dlr. (6 .3 ) AB This integral is to be transformed into an integral taken at constant time i n K°, and similarly as in I, section 12, we choose the domain CD for whic h t o = 0 . The world lines determined by SM will each intersect AB and CD i n two points with coordinates (x2(AB), t°(AB)) and (x°(CD), 0) in K°, such tha t x° (CD) = x°(AB) - t°(AB) cos M = x?(AB) 1 + ß cos 00 n \ ßx0u ( CD) sino0 x2 (CD) _ .x2 (AB) - Ç t° (AB) sin 190 = x2 (AB) + n n +ßcOs7% n Ç x3(CD) = x3( 4 B), t°(AB) = - ß x°(AB) . c The calculation is carried out in a similar way as in section 12 of I, so w e abstain from a detailed exposition . The relation between the volume elemen t dV and the element dV°, taken at constant Lime in K°, is given by (I, 12 .9) . We find 46 Nr .1 3 NA a = f x1 (1 2 °/Y2 + (ß/n) sin4° fI °) - (1 + dV (1 + (ß/n) cosI9°)2 CD Since fA ° [(n 2 - 1)/nc]8VV 0 /åt 0 , transformed as follows : f x~t2 cos a representative term in (6 .5) can b e 2 1°dV° (6 .5 ) 2 nRC-~°sinz90dt° i°fx°l~iOdV °1 = n 112 1 ~P°sin9'°cos'0° . (6 .6 ) In the second term we have here used the fact that d/dt°[ ] = (c/n) cos 0 0 , the centre of mass-velocity in the x°-direction . By a similar treatment of th e other terms in (6 .5) we find ßr3 __ n2 - 1 ß R2 sin49°cos99 0 0 (1 + (ß/n)COSl9°)2[ (6 .7 ) Thus there results a non-vanishing torque also with the symmetrical ABRAtensor . So far we have considered only the case where the domains A B and CD are placed at t = 0 and t o = 0 respectively ; however, the same resul t applies also when AB and CD are placed at arbitrary constant times in K and K° due to the fact that the force density fluctuates away when integrate d over space . So the expression (6 .7) is constant in time . Let us now consider MINKowslu's tensor . From (6 .1) we fin d HAM N3 = J(S_S)dV . (6 .8 ) Now S21 - S 12 = [ 012 - 1)/n]ßyW0 sinv9°, and the integration in (6 .8) can be carried out in the same way as above . We get N AT = P n2 sin'M n 1 + ( ß /n)COsY9 0 ,)(te ° (6 .9 ) We see that the expressions (6 .7) and (6 .9) in general are different from eac h other, although they both vanish in the rest frame as they should . It i s therefore natural to ask whether it is possible to single out one of thes e these expressions as preferable . As we shall now see this is not so in the cas e of an infinite medium, since the torque expressions (6 .7) and (6 .9) may b e looked upon as representing relativistic effects of the same nature as the non observable effect encountered in ordinary relativistic mechanics of a n elastic body possessing stresses in its rest frame . Nr .13 47 A relativistic effec t Let us first recall the following situation from mechanics : If an elastic body is subjected to stresses in its rest frame it may in other frames exhibit a momentum component at right angle to the direction of motion . Consequently, the body will require a torque in order to maintain its unifor m motion . We find it desirable to go into some details . Let r°k be the mechanical stress tensor of the elastic body in K° . The mechanical torque in K i s N= J(r xf)dV. (6 .10 ) Then make the explicit requirement that the body remain in steady motio n in K . This means that we can put dg/dt = 0, where g i = iri4/ c and th e time derivative is taken along the volume elements dV which follow th e body . Thus, the body experiences a change of angular momentum equal t o dM dt /dr f\x)d V, (6 .11 ) since also (d/dt)dV = 0 . Inserting dr/dt = v we obtai n dM = dt J( v x g)dV xG . (6 .12 ) If the torque (6 .10) is equal to the amount (6 .12) which the body actually requires in order to preserve stationary motion, then the scheme is consistent, and we have an example of a situation where the existence of a torque is not followed by a rotation . We have to stress the difference betwee n the calculations that led to (6 .10) and (6 .12) : In the first case, the velocit y of the body was required to be equal to v, and we can imagine that thi s requirement is fulfilled at a certain time in K just after the LORENTZ trans formation from K° has been performed . But in the latter case, the bod y velocity was required to be the same at an arbitrary instant afterwards , corresponding to the fact that the directions of the world lines of the bod y were required to be unaltered . It appears from the text-books (M . VON LATE (41) , R . BECxER(42 >) that th e equivalence between N and dM/dt has been verified in certain specia l cases . But the equivalence can also be shown quite generally for an elasti c body, by thefollowing simple consideration . Nr . 1 3 48 Let us calculate a typical component of the torque in K, say the z component . We readily find by an insertion into (6 .10 ) N3 = f [y(x° + uto )f2o _ ,~oyfl ` o~ 2 y -idvo = AB o - z x of l 2 - ` 2f ? ) dV° . (6.13) CD Using now the fact that Na (y h.° = åkik, we can writ e = f (y-2 x°z2k - 4s40)14°dS° 2 f -I- ß CD z° 2 dV° = CD ß 2f °2 dV °, (6 .14) CD since the surface integral is performed over a surface outside the bod y where z°k vanishes . Further, by means of the relation r24 = ißyz° 2 we readily obtain by a n insertion into (6 .12) dM 3 _ dt ß2 f -c° 2 dV ° . (6 .15 ) CD Eqs . (6 .14) and (6 .15) show the consistency in the case of an elastic body : The body is acted upon by a torque which is equal to the change of momentum required in order to maintain steady motion . After this digression let us return to the radiation field . The torque on th e body is defined as Ni = J(xf-xkf+Sik-Ski)dV, (6 .16) where i,k,l are cyclic . (Actually, the expression (6.16) has been derive d indirectly as NI = - dMik /dt ; however, the coordinate dependent terms i n (6.16) appear similarly as in (6 .10), and the two last terms in (6 .16) mus t yield the appropriate torque contribution from the tensor asymmetry, cf . fo r instance the considerations in section 4 of I .) The expressions for Ni tha t we need here have been derived in (6 .7) and (6 .9) . Next, require explicitly steady motion in K . The necessary and sufficien t conditions are : (1) The body velocity v = (v,0,0) = constant ; (2) dr/dt = u = SM/WM along the moving wave elements dV. From these conditions it follows that Ul, is a four-vector and that the world lines remain invariant i n four-space . Moreover, it follows that dg/dt = 0 along the wave trajectories , since g i (for any field tensor) is proportional to the energy density W° , which is a function of the invariant wave phase 99, 93 being constant along the Nr . 13 49 trajectories . Thus, taking the time derivative of the field angular momentu m we obtain in the two cases dM/dt { (u x GA ) 3 , dMg /dt = (u x G M) 3 . (6 .17 ) If we here insert the appropriate values for u, GA and G M we will find th e expressions (6 .7) and (6 .9) respectively, with the opposite sign. If now ABRAHAM ' S or MINROwsKl ' s tensor is taken to describe the field, it follow s from the conservation of total angular momentum that the rate of change o f the mechanical angular momentum is given by the expression (6 .17), wit h the opposite sign . In both cases we therefore find that the scheme is consistent in the same way as it was found to be in the situation considere d previously (cf . (6 .14) and (6 .15)) : The body is acted on by a torque whic h is just equal to the rate of change of mechanical angular momentum bein g necessary in order to prevent rotation . At this place we should make a comment on an assertion put forward by VON LAUE in § 19 of his book (41) , concerning a verification of the principl e of conservation of total angular momentum if MINKOWSKI ' s tensor is use d for the field . This is actually one of the arguments voN LAUE presents i n favour of MINKOWSHI ' s tensor . He first writes the rate of change of fiel d angular momentum similarly as the last of eqs . (6 .17), by taking the tim e derivative along the moving wave elements . Thereafter, and this is the crucial point, the z-component of the torque on the body is claimed to be given b y f (xi Ok S 2k - x 2 a k S k)dV = f (Si2 SM)dV . (6 .18 ) Since it can be shown that (u x gM)3 = S2 Slz, vox LAUE concludes that the conservation of total angular momentum is verified in the present case . We cannot find, however, any reason why this torque component shoul d be given by the left hand side of eq . (6 .18) . Moreover, one cannot fin d expressions for the rate of change of the field angular momentum and th e body torque independently of each other, and thereafter check the angula r momentum balance . Instead, the torque is found by just requiring th e angular momentum balance to hold, such that N be given by the relatio n N = - dM/dt . Finite bodie s Hitherto we have restricted ourselves to a consideration of very large (or infinite) dielectric bodies . The case of finite bodies is important, however , since it reveals characteristic features of the angular momentum balance . Mat .Fys .Medd .Dan .Vid.Selsk . 37, no . 13 . 4 50 Nr .1 3 Let us therefore consider this case, and for definiteness assume that a n optical wave passes through an isotropic and homogeneous glass plate, fo r instance at BREWSTER ' S angle of incidence in K° . The electromagneti c forces are present only in the boundary layers, and we shall assume that a n external mechanical surface force Fexto just counterbalances the surfac e force F° caused by the field, in such a way that the field is not disturbed . The consequence of the last assumption is that the mechanical angula r - N ext o and tha t momentum of the body is conserved in K°, NA° = NM° the presence of extra mechanical stresses due to the external forces is avoided . We now consider the system in the frame K, and adopt ABRAHAM ' S tensor as the field tensor . From (6 .16) it is apparent that the torque is give n as r x fA , integrated over the internal volume, plus r x FA , integrate d over the surfaces . We readily find that the contribution from the first ter m is zero, and as the electromagnetic surface force FA transforms similarly a s the external force F ext , we can writ e N A = dMA /dt = - Next . (6 .19 ) Thus we obtain the satisfactory explanation that the net torque acting o n the body is still zero . If, however, MINKOwsKI ' s tensor is adopted for th e field, the situation is changed . We see thatf M = 0 in the interior domain an d that FM = FA so that the contribution from the forces is the same, but ther e appears an extra volume effect in the torque because of the asymmetry o f the stress tensor According to the theory the body is thus acted upon by a net torque in K, although the motion is uniform and although no accoun t has to be taken of the influence from elastic stresses in K° . We find thi s property to be rather inconvenient . It does not mean, however, that MINKOwsKI's torque expression should simply be rejected . For we may carr y through an analysis of the same kind as in the previous subsection, wher e now the time derivatives are to be taken along the moving body, elements, and will find that also now the MINKOwsKI torque is compatible with th e requirement of steady motion . The peculiar property of MINKOWSKI ' s torqu e is obviously a consequence of the fact that the momentum density gM ° contains both a pure field quantity g A ° and a mechanical quantity gmeeh°, cf . also the remarks in the beginning of this section . In conclusion, the study of the case of finite bodies reveals the characteristic effect that the mos t natural division of the total angular momentum into a field part and a mechanical part is made when one adopts ABRAHAM ' S expression for th e field . On the other hand, in the case of infinite bodies we saw in the previous 4. Nr . 13 51 subsection that no preference could be assigned to either of the two torqu e expressions . At this place a remark is in order, in connection with a comparison wit h the situation where an electromagnetic wave passes through a finite, anisotropic body at rest . Such a situation was considered in section 4, and w e recall that the equation NA° = NM° was found to hold in general . Now ou r present situation resembles the wave-crystal situation from section 4, sinc e an isotropic body in K° becomes anisotropic in K. We may note that th e total angular momentum in the vacuum field when the wave has left th e body is independent of which energy-momentum tensor is used for the field , since the direction of the wave propagation in either case is determine d from SM . Yet we have found that NA in general is different from NM whe n ß+O. To point out the difference between these two cases let us once again examine the torque balance (4 .1) : N = - d/dtM° ac - d/dtM i , (6 .20 ) where now the time derivatives are taken along the moving body . In addition to the assumption of the independence of Mvae we could, in the cas e considered in section 4, require on physical grounds that N be independen t of the interaction period T (assumed a stationary field during this period) , expecially in the small period when the field leaves the medium . The crucial point here is that this latter requirement can no longer be upheld when th e body moves . Consequently, dMi /dt is in general different from zero, i .e . th e torque depends in this case also on the internal field . We note that dMi /dt + 0 also with ABRAHAM ' S tensor . As mentioned above the purpose of assuming F° = - F ext ° was to obtain a situation in which no complication will arise because of extra mechanica l stresses set up in K° . Let us now briefly consider how the situation is change d if we let the same value of N ext ° be obtained by external surface forces whic h do not compensate the electromagnetic forces at each surface element . I n this case there will appear mechanical stresses in K°, described by th e mechanical stress tensor These stresses may lead to non-vanishin g momentum components at right angle to the velocity v in K, and thus b e 4. connected with the torque Ni tress =- (i/c)åu k f vitk4dV which follows fro m the requirement of steady motion . This amount is equal to the resultin g torque exerted by the forces, so that we obtain instead of eq . (6 .19) th e equation ext = Nstress NA + N (6 .21 ) 4* Nr . 1 3 52 So far we have considered only ABRAHAM ' S and MINK0wsKI's tensors . Let us finally for a moment consider the radiation tensor S,,, which i s symmetric and divergence-free within an isotropic medium . In the situatio n considered in the first subsection above it follows immediately that Ni = 0 , so that according to the radiation tensor the angular momenta of the fiel d and the body are conserved separately . If the body is finite, the radiatio n tensor behaves similarly as ABRAHAM ' S tensor in the sense that the torque i n K is determined by the surface forces only . It should however be borne i n mind that the radiation tensor yields already in the rest frame a surfac e force with another direction and magnitude than ABRAHAM'S surface force , although the torques are the same (cf . section 4) . 7 . Further Considerations on Relativit y In this section we continue the investigation of relativistic phenomena . Only effects involving special relativity will be considered . For the main part we shall be concerned with topics that were studied in chapter IV of I in connection with MINKowsiu's tensor, and shall relate the phenomena t o the other tensors . In the following two subsections we study two subjects that are closely related to each other, namely the velocity of the energy in a n electromagnetic wave and the behaviour of the relativistic centre of mass . Transformation of the energy velocity in a light wav e Consider a plane light wave within an isotropic and homogeneou s insulator moving with the uniform four-velocity V~ in the frame K . Similarly as in I, section 9, the ray velocity u is defined as the velocity of propagatio n of the light energy . The ray velocity is in general different both in magnitud e and direction from the phase velocity . We recall that it is shown in MØLLER ' s boo k (7) that the ray velocity transforms like the velocity of a material particle , and further that this transformation property is verified experimentally i n the FIZEAU experiment, at least to the first order in v/c . If now an energy-momentum tensor 5t,,, shall describe the whole travelling wave, it must be possible to relate the ray velocity u to the components o f this tensor by the equation u = S/W. For such a tensor the quantity S/ W must therefore transform like a particle velocity . To investigate whether Sp ,, behaves in this way is tantamount to examining whether the quantitie s ~ SJW S 2 /(c2 W 2 ) ic ' V1 \ - `S2 /( e2W2 )/ Nr . 13 53 constitute a four-vector . As stated already in I, MOLLER has shown that th e sufficient and necessary condition for U p being a four-vector is tha t Bp, = Sp„ + 1 -2 Spa Ua U„ = 0 c (7 .2 ) in some inertial system . We recall that by inserting MINKOwsKI ' s tensor one really finds Rpm; = 0 in the case of a most general plane wave . This circumstance thus provides a further support for our general assertion that MrNKowsxi's tensor describe s the whole travelling wave . In particular, if a ray travels parallel to th e direction of the medium velocity, one obtains immediately by means o f MINxowsKI ' s tensor the well known formula, to the first order in vie , v= SM Wm c n = -+v I \ 1--1 1/f) n2 (7 .3 ) This formula was verified in the FiZEAU experiment . After this summary of the results obtained in section 9 of I, we investigate how the situation looks from the point of view of ABRAHAM ' S tensor . In this case one readily finds that R AA;, $ 0 in general, so that the equatio n (7 .2) is not fulfilled and SA /WA does not transform like a particle velocity . Correspondingly, the last of eqs . (7 .3) is replaced b y SA WA 1 -+2v 1 - - , c n n2 (7 .4) which is essentially different from (7 .3) . This kind of behaviour is what w e should expect : ABRAHAM'S tensor leaves out of consideration the influenc e from the produced mechanical momentum gmech° in K°, and thus SA /WA cannot be expected to be equal to the ray velocity . The non-compatibility between the transformation criterion and the ABRAHAM tensor evidently does not represent a real difficulty for this tensor . Let us now follow a more general line of approach and try to find a set of reasonable conditions under which the quantity S/W, obtained from som e energy-momentum tensor 5p,,, actually obeys the transformation criterion . To this end it is advantageous first to recall the essential assumptions inherent in MØLLER'S proof (in § 24 of his book( 7 )) about the transformatio n character of the ray velocity u : In the first place, the equation for the wav e front of an elementary spherical wave in K° being emitted from the origin a t the time 1 0 = 0 is written as 54 Nr . 1 3 I,02 2 _ L102 n2 = 0. (7 .5 ) In the second place, the corresponding equation for the wave front in K i s obtained by means of point transformations of each term in (7 .5), so that th e world lines are assumed to remain invariant in four-space upon a LORENT Z transformation . By means of these conditions MOLLER derives that u transforms like a particle velocity . Our task is now to transform the above conditions into equivalen t conditions imposed on the tensor S NP, . In accordance with (7 .5) we shall firs t require that the magnitude of the velocities of propagation of energy an d momentum is equal to c/n, as expressed by the equation s o S4k o ek, ° -''44 Iz ° = - - o °k e n Szk 24 , (7 .6 ) S where e° is the wave normal for the plane wave . Note that these conditions actually mean also that the field is closed, i .e. a°Sm„ = o, since each frequency component of the plane wave depends on the wave phase (k° • r° - (.0°t°) so that ekak may be replaced by - na/(cat°) . If we now insert the condition s (7 .6) into the expression (7 .2) for Rev , we really find that Rev - 0 . So far we have only shown that the conditions (7 .6) are sufficient to satisfy the transformation criterion ; we have not verified that they ar e necessary . In fact, if we merely maintain the single restriction that S°k be proportional to S 4ek, we find that the relatio n so i1c S° t S° e°k c W ° i4 i I (7 .7) is necessary to yield = 0 . Evidently eq. (7 .7) becomes equal to the las t of eqs . (7 .6) when IS°I/W° = c/n . Note that the weak condition (7 .7) doe s not even imply that sl,°, be divergence-free . We think that this condition is o f minor physical interest, however, since it is preferable to construct th e theory so as to conform to the equation (7 .5) (or the wave equation) in a simple way, i .e . one should always take S°I/W° = c/n. It has been pointed out by G . MARX et al( 3) that the radiation tensor Sins , also obeys the transformation criterion . This feature can be explained on th e basis of eqs . (7 .6), since the radiation tensor satisfies these equations . On the contrary, both ABRAHAM ' S tensor and the DE GROOT-SUTTORP tensor (1 .9) are incompatible with the condition (7 .7) as well as the transformation criterion RI , = 0 . Nr . 13 55 Centre of mass Let us now assume that the interior domain of the radiation field can b e taken as a part of a monochromatic plane wave with wave vector k . Similarl y as in section 12 of I we further assume that the small boundary layer-i n which the usual plane wave relations between the fields do not holdcontains negligible field energy and momentum . The spatial coordinates Xi (K) of the centre of mass of the field in K ar e defined by ~J Xi (K) = xz1VdV, (7 .8 ) whatever energy-momentum tensor is employed . Similarly as in the previou s section it must however be borne in mind that in any case the localizatio n of the field in K is determined by MiNKowsKi's tensor, i .e . one integrates over the volume of the field by integrating across the world lines correspondin g to SI Let us first study the velocity of propagation of the centre of mass in K . From (7 .8) we readily find the relatio n dtXi(K) = -. ~S . dV - icX~K)f f4dV + ~w f xzf4 dV ( 7 .9) which in general shows a complicated behaviour for a non-closed system . Inserting ABRAHAM ' S tensor into the right hand side of eq . (7 .9)-and assuming that corresponding world points in K and K° are connected by th e invariant (MINKOwsxi) world lines-we find however that the two last term s in (7 .9) fluctuate away . Moreover, since the field is homogeneous, we fin d from (7 .9) the simple relation d XA (K) dt A = SA. W (7 .10 ) By taking into account the result obtained in the previous subsection, w e thus find that the velocity dXA (K)/dt is different from the velocity of propagation of the total field, i .e . the ray velocity u . This feature severely limits the validity of the centre of mass as a representative point if ABRAHAM' S tensor is used . With the radiation tensor we get immediatel y d -Xs(K) Sss = u, (7 .11) 56 Nr .1 3 in accordance with the general equivalence between the radiation tensor an d the MINKOWSIiI tensor with regard to wave propagation properties . So far having studied the velocity of propagation of the centre of mass we now turn our attention to its localization . From the sudy of MINxowsKI' s tensor in I, we recall that the different centres of mass we obtain by varyin g the reference frames K, do not in general coincide when considered simultaneously in one frame . In fact, we calculated the difference XM (K) - XM, where XM denoted the simultaneous position in K of the proper centre of mass . The proper centre was defined as the centre of mass in the rest frame K° , i .e . XM (K o ) = XM ° . Let us write down again the formula (I, 12 .21 ) 0 aM (K) = XM (K) - XM = k° nk f' ß .k (7 .12 ) where we now have added a superscript M . Just the same procedure can now be applied to calculate the positio n XA (K) when ABRAHAM ' S tensor is used for the field . In this context we stres s that corresponding field points in K and K° are required to be connected b y the MINKOWSKI world lines, i .e . we simply ignore for a moment the above result dXA (K)/dt u . Since the proper centres coincide in K°, XA° = XM°, we evidently have also XA = XM in K. We do not give the details of th e calculation since it is just similar to the calculation carried through in I , section 12 . The result is aA (K) = XA (K) XA = aM (K), (7 .13 ) showing that ABRAHAM'S tensor yields the same position for the centre o f mass as MINxowsxi's tensor, XA (K) = XM (K), if we integrate across the worl d lines determined by SM . The radiation tensor exhibits very simple features with respect to th e centre of mass . Since â 0 (x, 2S6 - x,,Sm6) = 0 it follows that the angula r momentum quantities Msw constitute a tensor, and by calculating M in K at t = 0 we readily find that XS (K) = icM4 = XiM (K ) as (K) = aM (K) . (7 .14a ) (7 .14b) The equivalence we now have established between the three energy momentum tensors with respect to the centre of mass is not accidental . It i s connected with the fact that in (I, 12 .12) we introduced the radiation tenso r 57 Nr . 13 as a formal remedy in order to extend certain volume integrals, taken ove r the internal, plane part of the field, into integrals taken over the whole field . In the case of the radiation tensor we could just take advantage of the tenso r property of Mav . It does not seem, however, that the equivalence could easil y be foreseen . The last point we shall dwell on in connection with the study of th e centre of mass is a comment concerning a result obtained in a basic paper b y C . MøLLER (43) . On the basis of some definite assumptions, MOLLER showe d that the concept of mass centre for a non-closed system in general is incompatible with the equations of motion . This result seems to run into conflic t with the result obtained in the present section, where we have defined th e centre of mass even for the ABRAHAM field . However, there is no rea l discrepancy between the results, since one of the assumptions inherent i n MØLLER'S proof does not apply to the present situation . Let us point out in detail the mathematical reason for this circumstance . At an arbitrary point of the world line of the proper centre (with prope r time r) MOLLER assumes that the following relation can be written : 1 j'Svdav = ModzX, (7 .15 ) where the integration is taken over a hyperplane a which is normal to th e world line . The surface pseudo four-vector do, is given by do„ - iåv Q dxu 8x64xe, 8 1234 = 1, where dxg , 8x6 and Axe are four-vectors lying on a . If a is orthogonal to the x4-axis, we choose the latter three vectors so that th e non-vanishing component of do., is do-4 = - idV, when the outward normal lies in the direction of the positive x 4 -axis . In (7 .15) Mo = Mo(r) is a proportionality constant . If we now insert ABRAHAM ' S tensor into (7 .15) in the frame K` wher e the wave is at rest, we find for tC = 4 the relation tea : = Mo c 2 , while for ,u = i we find that Mo becomes infinite . This discrepancy shows that an equatio n of the form (7 .15) does not apply here . Hence MøLLE R's proof does not com e into conflict with the above results in this section . Nor does MINKOWSKI ' s tensor satisfy the relation (7 .15) ,while the radiation tensor does satisfy it . The Cerenkov effect As we already have noted, a study of the CERENKOV effect is very instructive for a comparison between the various energy-momentum tensors . In section 5 of the present paper we studied the CERENKOV effect in the case Nr . 1 3 58 that the emitting particle moves within a medium at rest, and in section 1 0 of I we considered the emitting particle in its own rest system from the poin t of view of MINKOWSKI ' S tensor . The reason why we shall here consider th e CERENKOV effect once more, it that we wish to point out how the relativisti c theory looks if ABRAHAM ' S tensor is used for the field . This kind of analysi s is desirable, since I . TAMM in his famous paper( 44 ) on the CERENKOV effec t studied the balance of momentum in the rest frame of the particle and cam e to the conclusion that MINKOwSK[ ' s tensor, but not ABRAHAM ' S tensor, is abl e to give a satisfactory description . We shall thus discuss the momentu m balance in the ABRAHAM case, since according to our general interpretatio n MINKOWSKI ' S and ABRAHAM ' S tensors ought to be equivalent in such a case . Consider then the same situation as in I : An electron is moving alon g the x-axis with a uniform velocity which in K° is larger than c/n . The rest frame of the particle is denoted by K ; as shown by TAMM, H = 0 in K, s o that there is no MlNaowsKr energy current in this frame . We integrate th e differential conservation law for momentum over a volume which contain s the electron and which is enclosed by a cylindric surface S of small radius an d infinite length such that the axis of the cylinder coincides with the x-axis . Since the field is stationary in K, one can thus write, in the case of MINKOwsxr's tensor, f Skn k dS = f -OdV, (7 .16 ) which is the same as (I, 10 .3) . As TAMM points out, MINKOWSKI ' S force must in any case represent th e force acting on the electric charge, because the terms which are added t o MINKOWSKI ' S tensor in order to form ABRAHAM ' S tensor will correspond t o additional forces acting on the medium itself, and not on the electric charge . The total force on the electron as given by the right hand side of (7 .16) ca n thus be found by transforming the total force from K° using the usual transformation formulas . Now TAMM evaluates the integral on the left han d side of (7 .16) and verifies that the two sides of the equation are equal . Further, since Sk = Ski for i = 1 and k = 2, 3, he concludes that a symmetrical "Ansatz" for S,, would give a different result in disagreement with th e force expression on the right hand side of (7 .16) . Let us now apply ABRAHAM'S tensor to the present case . It is instructive t o write the momentum balance in the for m f Skn k dS + f (fi - fZ`'r)dV = - f fv1 dv, (7 .17) Nr . 13 59 and so it appears that the second integral on the left may represent a sourc e (or sink) of electromagnetic momentum which also has to be taken int o account . Since the force on the matter cannot make up an appreciabl e magnitude in a small volume element just enclosing the electron, we ca n exclude this element from the second integration in (7 .17) and thus obtai n JSnk dS+JfdV where f = - f ffdV, (7 .18 ) means integration over the remaining part of the volume . However , also the second term on the left in (7 .18) vanishes due to the rapid oscillation of the integrand, so that eqs . (7 .18) and (7 .16) become identical, i .e . Sn = S . In fact, the relation S N = S,,k, valid for all combinations of i, k that occur i n (7,18), can be checked directly by expressing S X and SL in terms of th e tensor components in K° . Note that it is just the latter relation that represent s the main reason why the (macroscopic) descriptions corresponding t o MINKOwsi{i's and ABRAHAM ' S tensors are identical in this case ; properties o f symmetry or asymmetry of the energy-momentum tensors are of no direc t importance . In the remainder of the present section we shall be concerned with a study of the so-called "principle of virtual power" . Before embarking upo n this subject, let us however pause to make the following brief remarks i n connection with the topics considered in I : In sections 4 and 5 in I we gav e two sets of conditions from which we showed that MrNÜOwsxi's tensor i s uniquely determined . It should be clear that both these sets of condition s automatically exclude from consideration the alternative tensor forms tha t we have been studying : The first set because eqs . (I, 4 .1) and (I, 4 .2) requir e the tensor to be asymmetric and divergence-free ; the second set essentially because eq . (I, 5 .1) requires the tensor not to contain the four-velocity V » explicitly (cf . (1 .5), (1 .7) and the fact that also S„ will contain VI, in a complicated way) . In section 10 of I we discussed the negative field energy which appears with the use of MiNxowsKT ' s tensor in a certain. class of inertial systems du e to the space-like character of the four-momentum G,um . This property i s peculiar to MINriowSKI ' s tensor and is not shared by the other tensor forms . We may check by direct calculation that WA > 0 and Wr > 0 in any K , while the result W8 > 0 follows immediately from the fact that the four momentum G8 is time-like . If a plane wave moves parallel to the x-axis w e may conveniently write the total energy density of matter and field as Nr . 1 3 60 W tot = Y2(] + 2nß + ß2)Wa° + y2wmech° (7 .19 ) where the contributions arising from SIti, gmech° and Siech° are collected i n the first term . Principle of virtual power Quite recently, P . PENFIELD and H . A . HAus published a book( 45) on the electrodynamics of moving media which is a synthesis of work they per formed with various collaborators ; especially the earlier article( 46 ) by Cxu , HAUS and PENFIELD is of particular interest to us . The authors adopt a phenomenological point of view . In addition to employing the usual formulation (I, 1 .1) of MAXWELL'S equations in a moving medium (the MINKOWSIiI formulation), which we also have employed throughout our work , they consider the so-called Cxu formulation introduced in the book b y FANO, CHU and ADLER (39) . It is outside the scope of our work to go into a study of the CHU formulation . What really is of interest to us, is that th e authors, within the frame of the MINxowsai formulation, derive an expression for the electromagnetic energy-momentum tensor which is equal to ABRAHAM ' S expression in an isotropic fluid, while MINxowsai's tensor is claimed not t o describe the electromagnetic system in a meaningful way . We find it there fore of interest to trace out the reason why the authors have arrived at thi s result . The keystone of the derivation presented is the "principle of virtua l power", invented by the authors, so let us first sketch how the principl e looks in the present case . An isotropic fluid is considered, where the fluid velocity u(r,t) may be a nonuniform function of the position at a certai n time . We simplify the formalism (thereby ignoring the dependence of the fiel d energy on the material density), and transform it to our notation . Consider an arbitrary space-time point and denote by K° the inertial frame in which the velocity of a fluid element around this point momentarily is zero . Thus u° = 0 for the element, but one assumes that virtual deformations can be applied to the material to produce arbitrary values of a k u° an d au?/at . Then let K denote the frame in which K° moves with a small velocity u . To the first order in u f c we have Si = S° + uk Sk,~ + u i W° w= w0 -I- 1 u g° + z u• S°, c (7 .20a ) (7 .20b ) Nr . 1 3 61 and these equations are introduced into the energy balanc e o•S+aW/at = icf4 . (7 .21 ) The authors then let K approach K° so that terns containing u (but not th e derivatives of u) vanish . The resulting equation i s 0 S° + 1 S° c2 au0 aW 0 + at at + W °p • u° icf4 - Sikaku° g° au0 at (7 .22 ) (note that the differential operators 9, are not transformed) . The essential point is now that a knowledge of the physical quantities appearin g on the left hand side of (7 .22), i .e . of S°, W° and f4, is claimed to be sufficient to provide a determination of the remaining tensor components S k and g ° appearing on the right hand side of (7 .22) . The following expressions ar e chosen : S° = c(E° x H°) (7 .23a) aW°fat = E° • aD°fat + H° . aB°fat (7 .23b ) W° = z (E° D° + H° • B°), f4 = 0 . (7 .23c) The authors now argue that it is convenient to express the fields E°, D°, H° , B° appearing in (7 .23) in terms of the fields pertaining to the inertial frame K before inserting (7 .23) into (7 .22) (note again that K° means the fram e where the fluid element momentarily is at rest) . By inserting (7 .23) into th e expression on the left hand side of (7 .22) they thus obtain cp•(Ex H) E + [Ei Dk + Hi Bk - aD at +H aB at au aik (E D + H• B)] a k ui - 1c (E x H) • a t (7 .24 ) The three first terns add up to zero because of MAXWELL ' S equations . B y letting K approach K°, identifying (7 .24) with the right hand side of (7 .22) and taking into account the arbitrariness of the derivatives of u°, the authors finally obtain Sik o = - E°Dk H°Bk + (E° D° + H° B°) 1 g° _ -(E° x H° ) • C (7 .25a ) (7 .25b ) 62 Nr . 1 3 This is ABRAHAM ' S expression . (Actually, the expression given in ref . 46 , containing the detailed derivation, was somewhat different from (7 .25) but, according to a private communication by the authors, this difference is du e to a printing error . ) If we now proceed to examine this principle of virtual power, we ough t first to note that one must distinguish between the derivatives of the relativ e velocity v between the frames K° and K, and of the fluid velocity u . Th e formulas (7 .20) relate the tensor components in the frame K to the tenso r components in the momentary rest frame K° moving with the constan t velocity v with respect to K ; although v = u at the space-time point considere d the corresponding equality between the derivatives is generally not true . Thus each of the factors å ß,u° in (7 .22) should properly be replaced by 01,, v°, which is zero . In fact, by performing the transformation K -/ K° th e only result one can obtain is the covariant properties of the conservatio n equations a„5t,,, = - ft, . By starting from the relation (7 .21), and assuming the velocity v to be small, one will thus end up with the same relation writte n in K° . If we really subtract the equation V •S° -L ÔW°/dt - icf = 0 from eq . (7 .22), we see that obvious inconsistencies will appear in the remaining equation if arbitrarily adjustable terms O u° are present . However, the above remark does not elucidate the essential reason wh y a definite form of the electromagnetic energy-momentum tensor was obtained . To this end let us in the following simply assume the validity of eq . (7 .22) as it stands . The essence of the principle of virtual power seems i n reality to be that one starts from the energy balance (7 .21) in K, then trans forms the field quantities to K° and inserts some physical information i n this frame, and finally transforms back to K . Within the frame of the physica l information inserted in K° the formalism can therefore, if it is carrie d through consistently, yield only a mathematical identity . The reason why the authors instead obtained Abraham's expression in (7 .25) is that they implicitly introduced into the formalism a physical assumption which i s compatible with Abraham's tensor, but not with Minkowski's tensor . Let us go into some detail at this point . It is then necessary first to focus our attention on the force component f4 in (7 .21) . In the conventional theory fF , transforms like a four-vector, so that, in the limiting case of small u, f4 = f4 . This equation was used by the authors in the construction of eq . (7 .22) . In particular, if f4 = 0, as assumed in (7 .23c), one should obtain T'4 = 0 also in K . However, if we use the covariant expression for SF, v and calculate f4 in K according to the basic equation f4 = - a„S 4,„ we may obtain Nr . 13 63 a different result . For example, both in the Abraham case and the Minkowski case we know that f°4 = 0, while the covariant expressions (1 .5 ) and (1 .1) yield f4 = 0, fM = - i(n9 - 1)(ExH)•ôu/ôt (7 .26 ) to the lowest order . In the Minkowski case there is thus a conflict ; it is incorrect to transform f as if it were a four-vector . Due to this peculiar transformation property of f4 (which evidently i s closely connected with the covariance problem of the conservation equations discussed above), it follows that f4 should properly not have been re placed by f4 in (7 .22) but should rather have been retained unchanged . Accordingly, it follows that eq . (7 .24) implies the relation f4 = O . This is a choice which, according to (7 .26), implicitly singles out Abraham's tensor . The appearance of Abraham's expression in (7 .25) is therefore what w e should expect . It is also possible to make Minkowski's tensor emerge fro m the formalism ; to this end we must insert the explicit expression for TT, given by (7 .26), into (7 .22) . Generally speaking, the introduction of a specific expression for f4 implicitly implies the acceptance of a specific tensor , the remaining formalism thus effectively expressing an identity . 8 . Analysis by Means of Curvilinear Coordinate s In connection with the study of the canonical procedure in section 8 of I we mentioned that it is possible, in the case of a closed field, to make th e canonical energy-momentum tensor complete by means of a symmetrizatio n procedure . Now it is well known that in the presence of a gravitationa l field one can obtain the complete energy-momentum tensor directly, withou t having to perform a symmetrization, by means of a variational metho d involving the variation of the metric tensor . Actually, and it is this cas e which is of interest to us, the variational method can be applied also in th e absence of a gravitational field . Then the transition to curvilinear coordinate s occurs formally as an intermediate step in the calculation . Curvilinear coordinates have been used rather extensively in earlie r studies of the electrodynamics of material media, although one here i s confronted with a non-closed field . Incorrect use of the variational metho d caused a great deal of confusion in the literature some years ago . The 64 Nr . 1 3 ambiguity inherent in the calculation seems first to have been pointed out b y J . I . HoxvATH(47 ) (see also ref . 48) . However, we think that it is still o f interest to give a careful analysis of the electromagnetic field in terms o f these coordinates, to point out the detailed reason why the power of th e variational method is restricted, and to supplement with remarks pertainin g to alternative variational methods . The main part of the present section i s devoted to this task . In the last subsection we shall study again the Sagnactype experiment from section 9 of I, in connection with an application of th e various tensor forms . The cavity frame in this experiment is evidentl y non-inertial . A variational metho d Let us now leave out the imaginary x4 coordinate and work with the rea l coordinates x 1 , x2 , x 3 , x° = cl . The square of the line element is ds2 = g,,„dx1`dxv (it, v running over the numbers 1, 2, 3, 0), whence in GALILEA N coordinates gll - g22 g 33 = 1, g00 = ^ g = detgm, = - 1, gm „ = 0 for ,u + v . Further, in GALILEAN coordinates , E = ( F1o, F20, F30), B = ( F23, F31, F12 ) D = ( Hlo, H20, H30), H = (H23, H31, H12) , and the connection between field and potentials is in genera l Ft,v = V Ati - v„Al, = O, A, - d.,„At,, (8 .3 ) as the covariant derivative V I, can be replaced by the ordinary derivativ e when F~v is antisymmetric . For a radiation field MAXWELL ' S equations take the for m VI 2 Ftw + V F,, A + p„ F,I,a = O 2 F,uv + VVHHv = V -- g 0,(V ô1 Fr), -gH') + 0, Fri, = 0 = 0. (8 .4a) (8 .4b ) We have here assumed arbitrary coordinates where the g 1,„ are given functions of the coordinates . Then proceed to determine the constitutive Nr .13 65 relations . We shall keep the formalism so general that it includes the case o f an anisotropie dielectric medium, but we shall assume magnetic isotrop y with ,u = 1 . (The procedure runs similarly, however, also if N I, is a tensor . ) Introducing in the small region around each point a local rest system o f inertia k with the metric tensor given by (8 .1), we may writ e H~ 0 = E k 2 F k0 . (8 .5 ) Moreover, in R we introduce the quantitie s ~o = e° = 0 (v = 1, 2, 3, 0) (8 .6) and let in the arbitrary coordinate system the symmetric tensor e l" b e defined in such a way that its mixed components in R coincide with eT given by (8 .5) and (8 .6) . The constitutive relations written in covariant for m are then H ," = F~y + C el`aFa)Vy c2(Fy eyaFa ) (8 .7 ) where Fa = FaßV ß , and Vß is the four-velocity of the medium . In isotropic media eq . (8 .7) can be writte n H Fc' = F ly - /1 V y x(F - FV V~` ), x = (e - 1)/c 2 . (8 .8 ) This relation between (8 .7) and (8 .8) can readily be verified, since for a n isotropic body in K gia Fa = 4Fa = ~ ga' Fa = (8 .9 ) Here gå is the metric tensor in GALILEAN form and f is the dielectric constant in K . Note that e(o) = 0 according to (8 .6) while go = 1 ; however, this does not matter, since F° = o . Writing (8 .9) covariantly as et''Fa = fF'2 , w e obtain (8 .8) from (8 .7) . Thus, while e l' in (8 .7) is a tensor, the trans formation (8 .9) in the case of isotropic media causes the dielectric constan t in (8 .8) to be treated as a four-dimensional scalar . It can be verified that an appropriate LAGRANGIAN is L= = - 4 FuyFF`y H~v i F cv 12 - 1 FN F~` + 2c' b1at.Fys .Medd .Dan.vid.Selsk . 37, no . 13 . 2c e~`y F ~F y = L° + L' + L" . 5 66 Nr . 1 3 V- Multiplying with the pseudo-invariant gdx = - g dx l dx 2 dx3 dx 0 and inte grating over a region E in four-space lying between two space-like surface s and extending to infinity in the space directions, we get the action integra l J = L(x)V- gdx = fr(x)dx . (8 .11 ) Since (8 .10) corresponds to the field and its interaction with the matter, a variation of (8 .11) with respect to the potentials will yield the field equation s (8 .4b) . However, we are primarily interested in the invariance property of J under coordinate transformations . Let an infinitesimal coordinate transformation be given by x 'µ = + åx ß` = x Å -I- 2 , where the are small, but arbitrary functions of the coordinates, so that terms quadratic in may be neglected . The correspondin g change of (8 .11) is e åJ = f (x' ) dx ' f y (x) dx . (8 .12 ) E' By transforming this expression and using the assumption that the boundary, we obtain( 8 ) åJ e Jå*(x)dx = 0, E vanish o n (8 .13 ) where 8"Y(x) = '(x) - ..T(x) is the local variation . Eq . (8 .13) has the for m of a variational principle even though L does not correspond to a close d system ; only it must be remembered that all variations are generated by th e infinitesimal coordinate transformations . We proceed then to calculate these variations . By a vector trans formation we find V'1`(x' ) = av x '~`V v (x) = ( av + av µ )VV (x), (8 .14) whence å''' V/` ( x) = V' (x ) av - v av V It (x) = V vp v r - rV VI` . (8 .15 ) Here we have for example V„ VF' = av V'` + Va, where I'v is the CHRISTOFFEL symbol . It appears that åVµ is a four-vector, as should be the case , since this variation is the difference of the values of two four-vectors at th e same point . Correspondingly for the potential s b''A l.,(x) = -A,,p/4 e - s~v OvA~ . (8 .16) 67 Nr .13 The gµv will also be affected by the coordinate transformation, and we hav e g 'uv (x ' ) = g «ß (x g µv (x) + g feaa a $v + g va aa ~` . )a ax 'vaßx 'v = Thus Ô*g/ v (x) = g'/ v (x') - g f`v (x) Similarly å :t: Efr,v = Efta Va ty Paaev (x) + EvaVaSf~ - = V f` e +Vv e . e VaE l v . (8 .17 ) (8 .18 ) (8 .19) The part Jo of the action integral corresponding to Lo in (8 .10) is to b e varied with respect to Aft and gf`v . This term is present also in the case of a n electromagnetic field in vacuum . One obtains after some calculation (fo r details, see Focx( 49) , §§ 47, 48 ) åJ0 2f( FuaFva- 9 uv Fc,fl F aß)S*gf`v JVvFå*4gdx . gdx (8 .20 ) Here use has been made of the relations å*l/- g = - - gg f~ v b ~g~v , f`V *gm, = - gåg/'. By virtue of (8 .18) the first term in (8 .20) can b e transformed, so tha t g bJo = f Vi ,( Ff4IX Fva - g,u FaßF xß)gdx - f v, Far PAIL I/ gdx . (8 .21 ) We shall now give the detailed calculation for the action term J ' corresponding to L ' in (8 .10) . Variations are here to be taken with respect t o Af` , g f` v and Vf` . Let us first calculate the contribution from the potentials an d writ e SA J' since _~ OIL f F~ VaS`"Ff~a~~ggdx =- ~ fFV fL å*A Œ a a b*Af4)v/- gdx, and 6'. commute . By partial integrations the n fi A J' = - z J V„(F f` c Vy - FyVf`)S*Af,~/- gdx, (8 .22 ) where we have exploited the antisymmetry property of the expression in th e parenthesis . The variation with respect to the metric tensor is handled in the sam e way, and we get by means of (8 .18 ) 5* 68 Nr . 1 3 1 89J' = f 2L2 : :g.µv FµF,v([l-gs : 'zV ggµvg"ßbr: gap) dx V - gdx J(FF - 2gµFßFß)V (8 .23 ) ~z f 1 ( l0 a[(F, F - D a( Fµ F" - - g7 Fp Fß ) g7 Fß Fß )4 1` }v- gdx . Since the first term in this expression involves the covariant derivative of th e product of a scalar and a four-vector, we can write this term a s 1 fa[v- g(Fµ,Fa c2 g~FpF ß ) F`]dx (8 .24 ) and transform into an integral over the boundary . Therefore this ter m vanishes . It remain s 12 8 9 J' = c2 f O v(Ff ,, Fv - gdx . (8 .25 ) Finally we consider the variations connected with the velocity . By mean s of (8 .15) we hav e SvJ' = -z c f Faµ F"(Vv O v e gdx . (8 .26 ) Performing a partial integration we obtain, apart from an integral similar t o (8 .24) åvJ' 12 =-c f [Ov(FµaF"Vv) + Fva Fa Dµ Vv ] e [/ gdx . (8 .27 ) Similarly we can evaluate the contributions from the term L " in (8 .10) . We give the results : åAJ = c f Vv[(el' Vv -rv "V/ ) F" ]b*: A µ V-gdx 89J" 2c SvJ " = - 1 cs f [p IV ,( E"ß Fa F,e) ev gdx v (s" ß F"µ FßVv) + saßFavFßOµV v]~ µ ~l- gdx (8 .28a ) (8 .28b ) (8 .28c) Nr . 13 69 8E J„ = - lZ 2c r [ 2 p v ( E Va 1 ; J F) + FvF"pev"]~~`1/- gdx . (8 .28d ) In the last equation we have made use of (8 .19) . Now we are able to write down the total variation 6J, wher e å = åA + + å + c . We obtai n 0 = SJ = ôJo + SJ' + SJ" 1 =-J pv l F'`v _ Ç(F,u - eµa Fa)Vv gdx + f[V V (F I Œ H6*AmV- v" - c O Vv c- Fva( F" - e"ßFß) cL 2c2FvI'apµeval gv12 Fa ßH "ß ) V- gd.x . In this relation 8*A it and $E` are not independent, but related through (8 .16) . However, we do not have to express S*A /L by VI in (8 .29) since we know that L is the LAGRANGIAN for the field in interaction with the medium . There fore the coefficient of S f,4fc must be equal to zero, as we also see by virtu e of (8 .7) and (8 .4b) . Now the $/` are arbitrary at each point . This means that during th e displacement period the dielectric in general will not move as a rigid body, but the bulk density will vary throughout the body . However, even unde r this deformation process the LAGRANGIAN (8 .10) is permitted, since MAx WELL ' s equations are assumed to be valid within the body also when i t becomes inhomogeneous, with the small velocity changes that appear because of the deformations . So MAXWELL ' S equations do not restrict th e variations and we obtain from (8 .29 ) e, M Ov Sv = c2 Fv«( F" e"ßFß) V µ Vv + 2c2FvFpYev ", (8 .30 ) where M { S cc = FuaHva _ ~ 9 Fa ß H"ß µ is MrNKowsxi's tensor . We now introduce GALILEAN coordinates and us e that O, Vv = 0 for the undisturbed . body, whence M ôv S' = 1 2c2 Fv Faev " . (8 .31) 70 Nr . 1 3 We should like to mention the possibility of requiring the body to mov e as a rigid body under the deformation period in some coordinate system . Then the variations of one world line can be chosen arbitrarily, but th e variations on the surface t = constant will now be determined by the metri c tensor . Because of the relativity of simultaneity however, deformations will in general occur in another coordinate system . Besides, this type of variatio n does not lead to the strong result (8 .30) . To see this, let us confine ourselve s to GALILEAN coordinates, in which the restriction reads = constant on a n arbitrary hypersurface t = constant in some inertial system . If we let x,u mean the difference between the left and the right sides of (8 .31), we ca n write (8 .29) as 0 o Jz 1 dx = fdx et' fd3 xxt , from which we can only conclude that the volume integral (8 .32 ) f d3x4 - 0 . Let us now return to the main result (8 .31) emerging from the formalism . It should be clear that this result is only a certain combination of MAXWELL ' S equations . We could equivalently write eq . (8 .31) in terms of ABRAHAM ' S tensor, or any other expression . Apart from the statement of the LAGRANGIA N (8 .10), the subsequent calculation is of merely mathematical nature . The present behaviour arises from the fact that the LAGRANGIAN (8 .10 ) does not describe the total physical system . If the LAGRANGIAN had bee n complete, then we could further have reduced the expression for th e variation of the action integral in view of the mechanical equations o f motion, and would have been left with the total energy-momentum tensor a s a result of the remaining variations . In some earlier treatments the electromagnetic energy-momentum tensor was claimed to be determined simply by the variation of the action integral with respect to the metric tensor . A s mentioned above, HoRvATH(47, 45) has emphasized the ambiguity of such a procedure . Further, H . G . ScHöPF(50) has objected against certain calculational inconsistencies in the earlier attempts . The works of HORVATH an d SCHÖPF contain references to the preceding literature . In the treatment up till now we have generated all variations from coordinate transformations, since this seems to be the simplest kind o f approach . However, one will commonly find another method used in orde r to calculate the variation of the velocity (3, 50, 51) . Namely to preserve the relation Vp VP = - c2 (8 .33) Nr . 13 71 also after the variation, one introduces LAGRANGE variables a~ (A = 1, 2, 3, 0 ) to describe the medium, where a° = p is an arbitrary invariant parameter o f the nature of a time . Then, writing v~ cax ." lap - V- g aß ax a Jap axß Jap (8 .34) , the relation (8 .33) is identically satisfied . But when evaluating the variatio n of given by (8 .34), the change in the gaß must also be taken into account . In this way theare considered as arbitrary . However, we see that thi s procedure is necessary only if the LAGRANGIAN obeys an action principl e with respect to the xi" . In the case of an electromagnetic field in vacuu m interacting with incoherent matter, as treated by Focx( 49 ) for example, th e given LAGRANGIAN corresponds to the total system and must therefore yiel d the equations of motion of matter when the arbitrary e-variations are take n in a fixed system of reference . Therefore one must take the restriction give n by (8 .33) into account, for instance by the parametrical representatio n (8 .34) . Another method has been given by L . INFELD(52, 53) ; the method consists in introducing a LAGRANGIAN multiplier A to take care of the degre e of freedom being lost by (8 .33) . In our case, the LAGRANGIAN L given by (8 .10) obeys an action principl e only with respect to the potentials ; the e,'-` - variations are consequences of coordinate transformations which preserve the condition (8 .33) automatically . Therefore no attention was paid to the restriction (8 .33) in the calculation above . But it is not incorrect to use the representation (8 .34) . We then obtain instead of (8 .15) the velocity variatio n = V v pv - eO,V F' + 2 V~`VQ V y p~, e °', (8 .35 ) when the change in the ga ß is taken into account . However when evaluating the å;-variations, we vary also the gap, in (8 .34), so that å;17/2 1 2c2Vf'VaV,d gva 1 c2V~V6V'Vr e' (8 .36 ) where (8 .18) has been inserted . We see that in the total velocity variatio n (6 '* + å9)V'` the expression (8 .36) compensates the last term in (8 .35), s o that we end up with a certain combination of MAXWELL ' S equations, as before . 72 Nr .1 3 Similarly, by using INFELD'S method, the multiplier 2 drops out of th e calculation . We mention that in the case of isotropic media (fluids) some attempts(54, 50) have been made to complete the LAGRANGIAN so as to make the system closed . In such a case the LAGRANGIAN has to obey a variationa l principle also with respect to coordinate variations, so that one may use th e representation (8 .34) . In this way the total energy-momentum tensor ha s been found to he given by ABRAHAM ' S tensor plus the hydrodynamical tensor . The consistency of such a procedure may be illustrated by the followin g consideration . We first tentatively write the LAGRANGIAN density for th e total system as x Ltot - F/ `t'F~`v + F, Fu _ Cm c2 , (8 .37 ) 4 2 ` where x = (n2 - 1)/c 2 , F~ = FIV V V , and 6 m is the invariant rest mass densit y of the fluid . If we now perform coordinate variations (for fixed metric) an d evaluate the contribution to the action integral which arises from the secon d term to the right in (8 .37), we find the expression f feV/- gdx due to th e velocity variations (8 .35) . Here f means ABRAHAM ' S force density written in general coordinates . Therefore the coordinate variations, which effec t only the two last terms in (8 .37), lead to the hydrodynamical equations o f motion with ABRAHAM ' S force as the external force . This result is compatibl e with the interpretation we found in section 3, and this is the crucial point , since it permits the adoption of (8 .37) as the correct LAGRANGIAN density for the total system . If we then perform an infinitesimal coordinate trans formation so that the action integral remains invariant, we see that th e coefficients in front of AIL and vanish in view of the field equations an d the equations of motion, so that we are left with a divergence-free tota l energy-momentum tensor in front of å gµv which is equal to the sum o f ABRAHAM ' S tensor and the hydrodynamical tensor . Note that the present direct connection between the variation of th e metric tensor and the energy-momentum tensor, and between the remainin g variations and the equations of motion, is lost if we employ our first metho d and generate all variations from coordinate transformations . Thus, if w e use the LAGRANGIAN (8 .10), a variation of the action integral (8 .11) with respect to the metric tensor leads to ABRAHAM'S tensor only if both (8 .18 ) and (8 .36) are taken into account . However, in order to analyse how th e conservation equations emerge from the formalism when (8 .10) is used, ou r first method is simpler . e 73 Nr . 13 Final remarks on the Sagnac-type experimen t The last task that we shall take up in our work is to give an extende d analysis of the recent Sagnac-type experiment due to HEER, LITTLE an d Bu p P( 55 ) which we considered in sect . 9 of I in connection with MINxows g I ' s tensor . We shall examine how this experiment is explained by the othe r tensors . Let us briefly recall the essential features of the experiment . Th e apparatus is a triangular ring laser giving rise to two travelling electromagnetic waves in the cavity, one circulating clockwise and the othe r counterclockwise . A dielectric medium is placed in the light path . When th e system is at rest the photon frequencies in the two wave modes are equal. If the cavity is set into rotation with an angular velocity Q, the photo n frequencies of the two beams become different from each other and th e beams interfere to produce beats which are counted . With MINKOWSKI ' s tensor the energy density W m for one of the modes in the noninertial cavit y frame is related to the energy density W° for this mode in an instantaneou s inertial rest frame by Wm = W° + 1 12 • [r x (E x H)], c (8 .38 ) where the fields refer to the mode considered, and are evaluated for Q = 0 since only effects to the first order in Q are investigated . Further, within thi s approximation the total field energy in the cavity frame is a conserve d quantity, so that we obtain the formula (I, 9 .6) for the relative frequenc y shift ~v ln1 4 2•f rx(ExH)dV (8 .39) "J c (E . D + H•B)dV In the plane wave approximation the agreement between (8 .39) and the observed data is excellent, and the authors conclude that their experimen t supports the asymmetric MINKOwsKI ' s tensor . As we shall see now, the above conclusion should be somewhat modified : The experiment represents a nice verification of the predictions of phenomenological electrodynamics, but it is not a critical test of the convenience o f MINaowsKI's tensor as compared to all other tensor forms . In fact, bot h ABRAHAM ' S tensor and the radiation tensor give an equivalent description o f the experiment . For we have in any case, to the first order in Q, the followin g formula for the energy density in the cavity frame : 74 Nr . 1 3 W = Wo + g4ks4 k (8 .40 ) g44 where g1v is the metric tensor in the cavity frame and the superscript zer o refers to the instantaneous rest inertial frame . Since the tensor components Sov are equal for MINKOWSKI ' S and ABRAHAM'S tensors and also for the radi ation tensor, we must obtain the same value for W . Therefore, in any o f these cases, we can put the conserved total field energy of each mode propor tional to the corresponding photon frequency, and obtain again the fundamental formula (8 .39) . Note that the equivalence of the above three tensors with respect to th e energy balance in the cavity frame holds for all participating terms . Th e energy balance reads in genera l pvS4v 1 = v-g a axv(V - gS4v) rv _ 4v`SPv 1,4 , (8 .41) but it can he verified that the term involving the CHRISTOFFEL symbol yields no contribution to the first order in Q . Moreover, by performing a coordinate transformation between the inertial frame and the cavity fram e we find that f4 = 0, even in the ABRAHAM case, and that the components S4 7' take on common values . In all the three cases considered we can thus writ e the energy balance as a v S4 v = 0, with common values for the tensor components . Finally we note that with the DE GROOT-SUTTORP tensor (1 .9), complications arise because the expression for W° is changed . In this case the forc e component f4 is different from zero, yet the total field energy is a conserve d quantity in the cavity frame since f4 fluctuates away when integrated over the volume . However, we do not now obtain the expression (8 .39) for th e relative frequency shift ; in fact, if we put the total energy proportiona l to the photon frequency for each mode we find the formula (4v/v) G = (< ~M °/. G°)(dv/v)M, in disagreement with experiment. This tensor seems i n general not to be suitable for the description of propagating waves, since i n an inertial rest frame the magnitude of the quantity S G°/W G ° is different from c/n . 75 Nr . 13 Acknowledgement s Similarly as for Part I of my work, I am much indebted to Professo r C . MOLLER, and in particular Professor L . ROSENFELD, for careful readin g of the manuscript and for valuable comments . Discussions with various colleagues at The Technical University of Norway are also gratefull y acknowledged . Appendi x The table below gives a summary of the behaviour of the various energy momentum tensors in those examined physical situations which are o f experimental interest . References are given to those sections of Part I o r Part II where the actual subject has been investigated . Cf . also the summarie s in the introductory sections of I and II . Nr . 76 Situation considered Dielectric isotropic or anisotropic bod y surrounded by a va cuum or isotropic liquid and acted upon by an electrostati c field : Measuremen t of force or torque . a) Minkowski b) Abraham Within an anisotroTorque always depic body the tenso r scribed in terms of asymmetry is of mai n the force . importance for the II, sect . 2 . torque . I, sect . 3 ; II , sect . 2. No experimental distinctio n possible . II, sect . 2 . Excess pressure produced in a dielectri c liquid by an electrostatic field : HakimHigham experiment . In this case the electrostrictive term s must be taken into account . Thereby on e obtains a tensor which yields Helmholtz ' force, and which is in agreement with th e second tensor form put forward by d e Groot and Suttorp . Good agreement wit h experiment . II, sect . 2 . Radiation pressure exerted by an e]ectromagnetic wave travelling through a dielectric liquid : Jones-Richards ex periment . Good agreement with experiment . Simpl e interpretation . I, sect . 6 ; II, sect . 3 . 13 e) d e Groot Suttorp (firs t version) c) Radiation tensor (Marx et al ; Beck) d) Einstein Laub Not defined in this case . Same experimenta l result as in the case s a) and b) . II, sect . 2 . Not defined in thi s Force density equal case . to Kelvin's force. Disagreement with experiment . II , sect . 2 . Equivalent to cas e Disagreement with a), when the appro- experiment . II , priate interpretatio n sect. 3 . is imposed. II, sect . 3. Inconvenient. Dielectric isotropi c or anisotropic bod y surrounded by a vacum and acted upon by a high-frequenc y field : Measurement of force or torqu e (Barlow experiment , Beth experiment, etc .) . No experimental distinction possible . II , sect . 4 . Defined for isotropi c Same experimental media only . Same result as in the case s experimental resul t a)-c) . as in the cases a) an d b), although th e direction and magni tude of the surface force in general ar e different . II, sect . 4 . Dielectric isotropi c or anisotropic body surrounded by a li quid and acted upo n by a high-frequenc y field : Measuremen t of force or torqu e (experiment not per formed) . No experimental distinction possible . II , sect. 4 . Experiment of th e Barlow type shoul d represent a critical test . II, sect . 4 . Experiment of the Barlow type shoul d also here be critical. The torque formul a is different from th e formulas correspond ing to the cases a)-c ) II, sect . 4. Nr . 1 3 Situation considered 77 a) Minkowski b) Abraham Low-frequency vari - Does not predict Predicts oscillations . ation of electric an d oscillations . magnetic fields : Mea- The equivalence between the tensors does surement of oscillanot apply to this case . An experimental fions of a suspende d distinction should be possible . II, sect . 4 . dielectric shell (expe riment not performed) . c) Radiation tenso r 'Marx et al Beck) d) Einstein Laub e) de GrootSuttor p (first version ) Same behaviour as in the case b) . II, sect . 4 . Gerenkov effect . Good agreement with the experiments . Simple interpretation . I, sect . 10 ; II, sect . 5 and 7 . Equivalent to cas e Leads to unphysical a), when the appro - value for the Geren priate interpretatio n kov angle . II, sect . 5 . is imposed . II, sect. 5 and 7 . Inconvenient . Velocity of the ever gy of an optical wav e in a uniformly moving body : Fizea u type experiments . Good agreement wit h the experiments . Th e von Laue-Moller transformation cri tenon is fulfilled . I, sect . 9 ; II . sect . 7 . Equivalent to cas e Same behaviour as a), when the appro in the case a) . priate interpretation is imposed, II, sect . 7 . Inconvenient, Sagnac-type experi ment performed b y Heer, Little, Bupp. Good agreement with experiment . I, sect . 9 ; II, sect . 8 . Inconvenient . II, sect. 8 . References 1. I . BREVIx, Mat . Fys . Medd . Dan. Vid . Selsk . 37, no . 11 (1970) . 2. H. HERTZ, Ges . Werke II, 280 . 3. G . MARx and G . GYÖRGYI, Ann . 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