The 24 components of the relativistic spin tensor consist of 3+3 basic spin
fields and 9+9 constitutive fields. Empirically only 3 basic spin fields and 9
constitutive fields are known. This empirem can be expressed by two spin
axioms, one of them identifying 3 spin fields, and the other one 9 constitutive
fields to each other. This identification by the spin axioms is
material-independent and does not mix basic spin fields with constitutive
properties. The approaches to the Weyssenhoff fluid and the Dirac-electron
fluid found in literature are discussed with regard to these spin axioms. The
conjecture is formulated, that another reduction from 6 to 3 basic spin fields
which does not obey the spin axioms introduces special material properties by
not allowed mixing of constitutive and basic fields.Comment: 15 pages, dirac-electron example has been rewritte

Borzeszkowski, H. -H. v.

Herrmann, Heiko J.

Muschik, W.

Rueckner, G.

Theoretical and Applied Mechanics

English

arXiv

H.J. Herrmann

G. Ruckner

W. Muschik

H.H.v. Borzeszkowski

Crossref

Spin axioms in relativistic continuum physics

Heiko Herrmann

G. Rückner

H.-H. von Borzeszkowski

Spin Axioms in Different Geometries of Relativistic Continuum Physics

The 24 components of the relativistic spin tensor consist of 3 + 3 basic spin fields and 9 + 9 constitutive fields. Empirically only 3 basic spin fields and 9 constitutive fields are known. This empirem can be expressed by two spin axioms, one of them identifying 3 spin fields, and the other one 9 constitutive fields to each other. This identification by the spin axioms is materialindependent and does not mix basic spin fields with constitutive properties. The approaches to the Weyssenhoff fluid and the Dirac-electron fluid found in literature are discussed with regard to these spin axioms. The conjecture is formulated, that another reduction from 6 to 3 basic spin fields which does not obey the spin axioms introduces special material properties by not allowed mixing of constitutive and basic fields. 

Herrmann H.J.

Ruckner G.

Muschik W.

Borzeszkowski H.H.v.

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Spin axioms in relativisticcontinuum physicsH. J. Herrmann  G. R uckner W. MuschikH. H. v.BorzeszkowskiTHEORETICAL AND APPLIED MECHANICSvol. 28-29, pp. 169-183, Belgrade 2002Issues dedicated to memory of the Professor Rastko Stojanovi cThere is nothing so annnoyingas a good exampleMark TwainAbstractThe 24 components of the relativistic spin tensor consist of 3+3basic spin elds and 9 + 9 constitutive elds. Empirically only3 basic spin elds and 9 constitutive elds are known. Thisempirem can be expressed by two spin axioms, one of themidentifying 3 spin elds, and the other one 9 constitutive eldsto each other. This identication by the spin axioms is material-independent and does not mix basic spin elds with constitutiveproperties. The approaches to the Weyssenho uid and theDirac-electron uid found in literature are discussed with regardto these spin axioms. The conjecture is formulated, that anotherreduction from 6 to 3 basic spin elds which does not obeythe spin axioms introduces special material properties by notallowed mixing of constitutive and basic elds.Institut f ur Theoretische Physik, Technische Universit at Berlin, Germany (e-mail: hh@itp.physik.tu-berlin.de)169170 H. J. Herrmann, G. R uckner, W. Muschik, H. H. v.Borzeszkowski1 IntroductionWe investigate the constitutive theory of spin uids in a relativisticcontext. General Relativity Theory (GRT) using Riemann geometryfor geometrization of gravitation is currently assumed to be the mostappropriate geometrization.Here we consider in a relativistic framework the general spin balance.The systematic reduction from 6 to 3 basic spin elds is introducedby two spin axioms which prohibit the mixing between the basic spinelds and the constitutive elds. This is obvious, because no specialconstitutive assumptions should be introduced by this reduction. Asan example the Weyssenho uid and the Dirac-electron uid withregard to the general spin balance and the spin axioms are discussed.2 The Spin BalancesBalance equations are dierential equations for the wanted basic elds.Beside these elds constitutive quantities describing the material ap-pear in the balances. This structural distinction into basic elds andconstitutive properties is independent of writing down the balanceequations in relativistic or non-relativistic form.First of all we consider the non-relativistic spin balance for characteriz-ing, what are the basic spin elds and what the constitutive properties.2.1 Non-relativistic spin balanceStarting out with the denition of the angular momentum as a skew-symmetric 3-tensor of second rank (i;j; = 1;2;3)Mij = x[ivj] + sij (1)the balance of angular momentum is12@t(sij) + @tx[ivj] +@k 12sijvk   mijk + x[ivj]vk   x[itj]k=lij + x[ifj](2)Spin axioms in relativistic continuum physics 171Here the basic elds are (1) consisting of the orbital x[ivj] and thespin angular momentum densities sij. The constitutive properties aredescribed in (2) by the stress tensor tjk and the couple stress mijk. Anexternal angular momentum is given by lij.Because the specic spin sij , sk can be identied with the axialspin vector sk in R3 the balance of angular momentum can be writtendown as a vector equation. If from this balance equation the balanceof momentum multiplied by x is subtracted, we obtain the balanceof spin in vector formulation [1]@t(si) + rk vksi   wik+ ijktjk = gi (3)Here the distinction into the basic elds s and the constitutive quan-tities w and t is clear: There are 3 basic spin elds s, 9 elds of thecouple stress w, and 3 elds from the balance of momentum by theskew-symmetric part  : t of the stress tensor. A given external angu-lar momentum density is g.2.2 Relativistic spin balanceThe non-relativistic momentum ux density vj in (1) is replaced inrelativistic formulations by the energy-momentum tensor T . There-fore the tensor order one of the momentum ux density is replacedby the order two of the energy-momentum tensor in a relativistic for-mulation. Consequently in special relativity theory the denition ofskew-symmetric angular momentum becomes (;; = 1;:::;4)M := x[T] + S (4)and the balance results in@M = L + x[f] (5)Here S is the skew-symmetric spin tensor and L the externalangular momentum. Caused by the extension of the tensor order inrelativistic theories with respect to non-relativistic ones, we now have6  4 = 24 elds of the spin tensor, including the basic spin eldsand the constitutive quantities. Of how many basic and constitutive172 H. J. Herrmann, G. R uckner, W. Muschik, H. H. v.Borzeszkowskields these 24 elds are composed, can be seen, if the spin tensor isdecomposed into its spatial and time-like parts by a 3+1 split. By useof the 4-velocity u the 3+1 split of the spin tensor is S  = S  = s  +1c2~ su +1c2u[^ s] ++1c2 s[juj] +1c4~ s[u]u +1c4u[^ s]u (6)= s  +1c2~ su +1c2u[] +1c4u[]u (7)Here we have introduced the following abbreviationss  := S  hhh couple stress (8)~ s := S  hhu spin density (9)^ s := S  hhu (10) s := S  huh (11)~ s := S  huu (12)^ s := S  uhu (13) := ^ s    s spin stress (14) := ^ s   ~ s spin density vector (15)The spin balance equation determines the divergence of the spin tensorS  ; = T[] + L (16)Here T [] is the skew-symmetric part of the energy momentum tensorwhich couples the spin balance to the balance of momentum. Beyondthis source, there may be other external moments L from (5). Wenow decompose (16) by its 3+1 split which results in two parts, thehh-part and the hu-part. Because of the skew-symmetry of the spintensor the uu-part is zero. In more detail the hh-part of the 3+1 splitresults in 3 equationsSpin axioms in relativistic continuum physics 173hhS  ; = s  ;hh | {z }+1c2~ s;uhh +1c2~ shhu;| {z }}  1c2u[] hh;| {z }4 1c4u[]u  hh;| {z }4= T[]hh | {z }]+Lhh | {z }\(17)and the 3 equations of the hu-part becomeshuS  ; = s  ;hu  1c2~ shu;u| {z }4++121c2;uh +121c2hu;| {z }}+121c2u ;uh| {z }= T[]hu| {z }]+Lhu| {z }\(18)Now we compare (17) and (18) with the non-relativistic balance of spin(3)@t(12si) + rk(vk12si)| {z }}  wik|{z}) =  ikjtkj | {z }]+ gi|{z}\(19)The }-terms in (19), (17), and (18) are equivalent to the total timederivative, the -terms belong to the couple stress, the ]-terms to theskew-symmetric part of the energy-momentum tensor, and the \-termsto the external angular momentum. The 4-terms can be interpretedas coupling terms between the hh- and hu-part in which the other eldsappear respectively. Consequently we obtain balance equations for thespin density (hh-part) and for the spin density vector (hu-part) whichare coupled to each other by the 4-terms.From (7), (17), and from (18) we can see, how the 24 components of174 H. J. Herrmann, G. R uckner, W. Muschik, H. H. v.Borzeszkowskithe spin tensor are distributed on the basic elds and the constitutivequantities. The rst term in (7), the couple stress (8), is according to(19) a constitutive equation. Because the couple stress is space-like,it represents 9 elds. The second term in (7), the spin density (9), isaccording to (17) a basic spin eld having 3 components. According to(17) and (18) there are other spin elds, the spin stress (14) and thespin density vector (15). Comparison of (17) and (18) with (19) yields,that the spin density vector is an other basic spin eld and that thespin stress is a constitutive quantity. The spin density vector eld has 3components, whereas the spin stress has 9 components. Consequentlywe have 3+3 = 6 basic spin elds and 9+9 = 18 constitutive quantitieswhich all together result in the 24 components of the spin tensor.3 Spin AxiomsAs discussed in the previous section there are 6 basic spin elds inrelativistic theories. But up to now only 3 spin elds are known inphysics, a fact which is formulated in the followingEmpiremAccording to (19) in non-relativistic physics only 3 basicspin elds (the spin density) and 9 constitutive functions(the couple stress) are known and measurable.This empirem can be interpreted in two dierent ways: 3 of the 6 spinelds are caused by typically relativistic eects. They are very smalland not be measured up to now. Interestingly these 6 spin elds areconnected with 18 constitutive couple stresses so that also materialsshould have relativistic properties which cannot be detected in thenon-relativistic limit. If this possibility of interpretation seems to betoo articial, the other possibility remains:Spin Axiom 1Also in relativistic physics only 3 basic spin elds and 9couple stresses exist.Spin axioms in relativistic continuum physics 175A consequence of axiom 1 is to reduce the 6 spin elds to 3 ones inrelativistic theories. The question is how to reduce them.One rst possibility is to demand, that the spin tensor is totally skew-symmetricS! = S[] (20)In this case we obtain only 4 spin elds which may correspond to thewanted 3 basic spin elds and 1 additional constitutive eld. Thereforeby (20) the constitutive equations are severely restricted by setting 8of the 9 elds of the couple stress to zero.A second possibility of reducing elds is to cancel 3 spin elds and 9constitutitive quantities arbitrarily. But this possibility seems not tobe very systematic, because the basic spin elds and the constitutiveelds are properly separated from each other: There is the spin den-sity (9) and the spin density vector (15) as basic spin elds, and thecouple stress (8) and the spin stress (14) correspond to each other asconstitutive elds. Because the hh-part (17) and the hu-part (18) ofthe spin balance are coupled to each other, a third possibility = 0; ^  = 0 (21)or~ s = 0; ^ s  = 0 (22)results in dierential equations which are not of balance type any more.Inserting (21) into (17) and (18) we obtain from (17) a balance equationof the spin density, whereas (18) has to be interpreted as a constraintfor the constitutive quantity couple stress: The couple stress cannot bechosen arbitrarily, because its divergence is restricted by the constraints(18). The same happens to the spin stress and the spin density vector,if (22) is inserted into (17) and (18). Consequently by choice of (21)or (22) hidden material properties are introduced.An essential point is, that the reduction of the elds should not restrictthe free choice of the also reduced constitutive equations. Consequentlythe possibility of identifying spin density with spin density vector andcouple stress with spin stress remains.Spin Axiom 2Spin density and spin density vector eld are semi-dual to176 H. J. Herrmann, G. R uckner, W. Muschik, H. H. v.Borzeszkowskieach other, as couple stress and spin stress are, that meanswe identify~ s! =121c2u ,1c2u[] =12  ~ s (23)s! =121c2u ,1c2u[] =12  s  (24)Here  is the Levi-Civita symbol. From (23) we see that ~ s and are semi-dual to each other if ~ s and u are dual to each other.According to (24) the same is valid for s and .Inserting (23)2 and (24)2 into (7) we obtain for the spin tensorS  =[] +12 1c2~ su +[] +12 s  (25)and inserting (23)1 and (24)1 into (7) results inS  =[] +12 1c4uu +[] +12 1c2 u (26)By adopting the spin axiom 2 (25) and (26) are dierent but equivalentrepresentations of the spin tensor, (25) in spin density and couple stress,(26) in spin density vector and spin stress. From (25) and (26) weobtainS  =[] +12 1c2~ su + s  (27)=[] +12 1c4uu +1c2u(28)Spin axioms in relativistic continuum physics 177The common bracket in the representations of the spin tensor has theremarkable property12 [] +12 =[] +12 (29)Consequently we have proven theCorollaryBy the spin axioms the spin tensor is self dual with respectto the rst two indicesS  =12  S  (30)From the two representations (27) and (28) for the spin tensor weobtain two equivalent versions of the spin balanceS  ; =[] +12 1c2~ su + s  ;(31)=[] +12 1c4uu +1c2u;(32)The 3+1 decomposition of (31) giveshh:hhS  ; = hh[] +12 1c2~ su + s  ;= hhT[] + hhL[]= t[] + hhL[] (33)uh:uhS  ; = uh[] +12 1c2~ su + s  ;= uhT[] + uhL[]= (q   p) + uhL[] (34)178 H. J. Herrmann, G. R uckner, W. Muschik, H. H. v.BorzeszkowskiAnd the 3+1 decomposition of (32) results inhh:hhS  ; = hh[] +12 1c4uu +1c2u;= hhT[] + hhL[]= t[] + hhL[] (35)uh:uhS  ; = uh[] +12 1c4uu +1c2u;= uhT[] + uhL[]= (q   p) + uhL[] (36)Equations (33) and (35) are identical, as well as equations (34) and(36). The reduction from 6 to 3 equations (spin balances) requires,that also equations (33) and (34) are dependent on each other, as wellas equations (35) and (36). This leads to the requirement that also(q   p) and t are semi-dual to each other:t =121c2  u(q   p) (37)The particular meaning of the spin axioms (23) and (24) can be char-acterized by the following proposition which we will prove laterConjectureAll reductions from 6 + 18 to 3 + 9 basic and constitutivespin elds which do not use spin axiom 2 are introducing aspecially chosen material.If this conjecture is true, the reduction of the spin elds has to obeythe spin axiom 2, because a reduction has to be performed before con-stitutive properties are introduced into the theoretical considerations.With respect to the spin axioms we now will shortly discuss two exam-ples of spin uids well-known from the literature.Spin axioms in relativistic continuum physics 1794 ExamplesThere are two spin uids which are typically dierent from each other:The Weyssenho uid which is a classical one, and the Dirac-electronuid which represents the classical description of a quantum-uid. Wenow will discuss these uids with respect to the spin axioms introducedabove.4.1 Weyssenho uidAccording to [2, 3] the spin tensor of the Weyssenho uid reads:S  _ =1c2~ su (38)This choice of the spin tensor can be interpreted in two ways: Com-paring (38) with (25) we could state, that the couple stress s   , thespin density vector , and the spin stress  are chosen to zero andthat therefore the choice (38) of the spin tensor does not satisfy thespin axiom 2, because in (38) the -term of the rst bracket in (25)is missing. Consequently we obtain from (17) the balance of the spindensity1c2~ s;uhh +1c2~ shhu;| {z }}= T[]hh | {z }]+Lhh | {z }\(39)and from (18) the constraint for the spin density 1c2~ shu;u| {z }4= T[]hu| {z }]+Lhu| {z }\(40)which only appears, if the spin axiom 2 is not taken into account.If the spin axiom 2 is taken into account, (38) is compatible with (25),if12c2  ~ su +[] +12 s  = 0 (41)180 H. J. Herrmann, G. R uckner, W. Muschik, H. H. v.Borzeszkowskiors  =  12 1c2~ su + s (42)is valid. This is a constitutive equation of the couple stress which isnot zero, if the spin axiom 2 is accepted. From (42) follows by thePropositions  =1c2[]    ~ su (43)the couple stress as a function of the spin density.Without accepting spin axiom 2 (38) describes an ideal spin uid, buttaking spin axiom 2 into account the spin uid is not an ideal one,because the couple stress does not vanish according to (43).According to (25) the spin tensor of an ideal spin uid isS  = +12 1c2~ su (44)if the spin axiom 2 is accepted.4.2 Dirac-electron uidThe following ansatz is made by B auerle und Haneveld [4]:S  g =1c2[u] with  := S  u (45)From this ansatz follows:[u] = S[] =1c2~ s[]u +1c2~ s[]u +1c2~ s[]u (46)=1c2~ s[]u +1c2u[] (47)with ! = ~ s  ~ s[] (48)Here one can see that a special material function is assumed, equation(48) is a constitutive relation that determines the spin stress. The totalSpin axioms in relativistic continuum physics 181antisymmetrisation mixes basic elds and constitutive functions whichis rather strange from a constitutive point of view.The interpretation of this spin tensor by use of the spin axioms isdicult.In equation (45) the couple stress is set to zeros _ = 0 (49)by use of the spin axioms also the spin stress vanishes = 0 (50)According to the total antisymmetrisation (46) the spin density vectoris zero _ = 0 (51)this implies by use of the spin axioms that~ s = 0 (52)This would means that the spin tensor vanishes.As the total antisymmetrisation seems only to be necessary for thereduction from the 6 elds in (45) to 3, there is another way using thespin axioms. Starting out with (45)2S  = u (53)and then using spin axiom 2 one gets the following spin tensorS  = +12 1c2~ su (54)which is the same as (44), and represents an ideal uid.AcknowledgementsWe thank F.W. Hehl for valuable hints concerning spin uids and T.Chrobok for continuous discussions and co-working. Financial supportby the DFG (German Research Council) and the Vishay Company,D-95085 Selb, Germany, is gratefully acknowledged.182 H. J. Herrmann, G. R uckner, W. Muschik, H. H. v.BorzeszkowskiReferences[1] W. Muschik, C. Papenfu, and H. Ehrentraut. A sketch of contin-uum thermodynamics. J. Non-Newtonian Fluid Mech., 96:255{290,2001.[2] J. Weyssenho and A. Raabe. Relativistic dynamics of spin-uidsand spin-particles. Acta Phys. Polon., 9:7{53, 1947.[3] Yu.N. Obukhov and V.A. Korotky. The Weyssenho Fluid inEinstein-Cartan Theory. Class.Quantum Grav., 4:1633{1657, 1987.[4] B auerle and Haneveld. Physica, 121 A:541{551, 1983.[5] Yu.N. Obukhov and O.B. Piskareva. Spinning uid in general rel-ativity. Class.Quantum Grav., 6:L15{L19, 1989.[6] H. Herrmann, G. R uckner, and W. Muschik. Constitutive the-ory in general relativity: Spin-material in spaces with torsion. InProceedings of the 4th International Seminar on Geometry, Con-tinua and Microstructures 2000, Rendiconti del Seminario Matem-atico dell'Universita' e Politecnico di Torino, October 26 - 28, 2000.Torino.[7] G. R uckner, H. Herrmann, and W. Muschik. Constitutive theoryin general relativity: Basic elds, state spaces and the principle ofminimal coupling. In Proceedings of the 4th International Semi-nar on Geometry, Continua and Microstructures 2000, Rendicontidel Seminario Matematico dell'Universita' e Politecnico di Torino,October 26 - 28, 2000. Torino.Spin axioms in relativistic continuum physics 183Aksiome o spinu u relativisti ckoj zici kontinuumaUDK 530.12, 531.0124 komponente relativisti ckog tenzora spina se sastoje od 3+3 osnovnapolja spina i 9 + 9 konstitutivnih polja. Empiri cki samo 3 osnovnapolja spina i 9 konstitutivnih polja su poznata. Ovaj "empirem" mo zese izraziti sa dve aksiome spina pri  cemu jedna od njih identikuje 3polja spina dok druga identikuje ostalih 9 polja. Ova identikacijaje objektivna i ne me sa osnovna polja spina sa konstitutivnim osobi-nama. Pristupi Weyssenhoovom uidu i Dirac-ovom elektronskomuidu nadjeni u literaturi se diskutuju pomo cu ovih aksioma spina.Formuli semo, zatim, lemu da jedna druk cija redukcija sa 6 na 3 osnovnapolja spina koja ne zadovoljavaju aksiome spina uvodi specijalne ma-terijalne osobine ne dozvoljavaju ci me sanje konstitutivnih i osnovnihpolja.

Spin Axioms in Relativistic Continuum Physics

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