1239
Progress of Theoretical Physics, Vol. 86, No.6, December 1991
Spinning Test Particles in Spacetime with Torsion
Koichi NOMURA, Takeshi SHIRAFUJI and Kenji HAYASHI* Physics Department, Saitama University, Urawa 338 * Institute of Physics, Kitasato University, Sagamihara 228 Downloaded from https://academic.oup.com/ptp/article/86/6/1239/1918293 by guest on 27 September 2021 (Received July 18, 1991)
We use the Fock-Papapetrou method to derive equations of motion for a spinning test particle moving both in the background gravitational field and a torsion field. The resultant two equations, the equations of world line and spin can be written in a covariant manner (as is the same for Papapetrou's result in general relativity). The equation of spin reproduces all the known equations of spin precession due to torsion.
§ 1. Introduction In general relativity, the problem of motion of extended bodies has first been 1 studied by Fock ) and developed by Papapetrou,2) who derived the equations of world line and spin for a spinning test particle moving in a background gravitational field. By a test particle we mean that the field produced by this particle can be neglected. Consequently, the back-reaction due to emission of gravitational radiation is ignored. The test particle is assumed to be neutral and of sufficiently small size so that effects due to electromagnetic field and the coupling of its multipole moments to in 3 homogeneities of the gravitational field can be neglected. ) Accordingto the -Fock-Papapetrou method, all test particles move like a spinless mass point in the single-pole approximation. Namely, the world line is the familiar geodesics of the metric. The effects due to spin appear in the pole-dipole approxima tion. In effect, the world line of a spinning test particle deviates from the geodesics of the metric. In this paper we study the motion of an extended body such as a gyroscope or a wave packet moving both in the background gravitational field and a torsion field. At present we have two theoretical frameworks of gravitation constructed on the spacetime with torsion: Poincare gauge theory4)-7) and new general relativity.S) The former theory is based on the Riemann-Cartan spacetime characterized by curvature and torsion. The latter theory, on the other hand, is built on the Weizenbock spacetime which possesses absolute parallelism: The gravitation is then ascribed to the torsion of the spacetime, since the curvature is identically vanishing. These two theories give the same predictions as those of general relativity to all the experiments so far performed. The torsion is coupled to the intrinsic spin of fundamental particles (quarks and leptons), and it will give, if it exists, nontrivial effects on the motion of spin of these particles. Thus, observation of spin precession may possibly give us a clue to detect the torsion of the spacetime. In a previous paper,9) we applied the WKB method to the Dirac equation and the Proca equation, and derived the equations of world line and spin precession for massive particles with spin 1/2 and 1 in units of fz moving in 1240 K. Nomura, T. Shirajuji and K. Hayashi the background gravitational and torsion fields. In particular, we showed that the world line for these particles in free fall is the familiar geodesics of the metric, but that the equation of spin precession is not represented in a geometric manner. Namely, the spin vector. of these particles is neither Fermi-Walker transported relative to the asymmetric connection with torsion, nor parallel propagated with respect to the ordinary symmetric connection. 10 It has been suggested by Adamovicz and Trautmann ) using a spin-fluid model/I) on the other hand, that the spin vector of a spinning test particle is Fermi-Walker transported along the world line relative to the asymmetric connection. Our result Downloaded from https://academic.oup.com/ptp/article/86/6/1239/1918293 by guest on 27 September 2021 obtained by the WKB method does not agree with their suggestion. It seems highly desirable to apply the Fock-Papapetrou method to spinning test particles moving in the Riemann-Cartan spacetime. We shall arrange the content as follows: In § 2 a brief review is given of the spinor and tensor analyses in the spacetime with torsion in order to fix our notation. In § 3 the Fock-Papapetrou method is summarized, and the basic response equations are extended to those for a spinning body in the Riemann-Cartan spacetime. In §§ 4 and 5 the equations of motion are derived in the single-pole and pole-dipole approxima tions, respectively. In § 6 a possible way to get the equation of spin precession is suggested. The equation of world line is discussed in § 7. The final section is devoted to conclusion. Detailed proofs of basic relations are given in three appen dices.
§ 2. A brief review of the Riemann-Cartan spacetime
Spacetime has a locally Lorentzian metric field gpv(x) and a global tetrad field ekV(x) with gpvekPe/= 7Jkl. Here !}kl is the Minkowski metric, which we choose as 7Jkl =diag( -1, + 1, + 1, + 1).*) The Latin indices and the Greek ones refer to the Lorentz basis ek and the coordinate basis E p , respectively. The fundamental particles such as quarks and leptons are described by spinor fields if, on which the covariant derivative acts like
(2'1) where (Jmn is the generator of the local Lorentz group, and Amnv is the Lorentz (or the spin) connection. In this paper we shall assume that matter fields are minimally coupled to the gravitational field and a torsion field: Namely, the interaction of matter fields with torsion is introduced through the covariant derivative, which satisfies the tetrad condition and the metric condition, (2·2)
(2·3) with r;v being the affine connection coefficients. Here the r;v is not assumed to be
*) We shall mainly follow the notation and convention of Ref. 7). Spinning Test Particles in Spacetime with Torsion 1241
symmetric with respect to fl and v, and accordingly the torsion tensor is defined by TA pv = rtv - rtp. The affine connection can then be expressed by (2·4)
where {~v} is the Christoffel symbol and KApv denotes the contorsion tensor,
(2·5) Downloaded from https://academic.oup.com/ptp/article/86/6/1239/1918293 by guest on 27 September 2021 which is anti symmetric with respect to It and fl. Putting (2·4) in (2·2), we see that the Lorentz connection is represented as Amnv = L1mnv + Kmnv " (2·6) where L1mnv is the Ricci rotation coefficients for the spin connection of general relativ ity. We shall write by l7 p the covariant derivative with respect to gv} and L1mnv: It also satisfies the tetrad condition and the metric condition. The comml,ltator of the covariant derivative reads
(2·7)
. with the curvature tensor given by
(2·8) As for the covariant derivative l7 v, its commutator is
(2·9)
where Rmnpv is the Riemann-Christoffel curvature tensor which is obtained from Fmnpv of (2·8) by replacing Amnp and rtv with L1mnp and gv}, respectively. The curvature tensor Fp(JPv is thus related to the Rp(JPv by (2·10)
The torsion tensor has three irreducible components; the tensor part tAPV, *l the vector part Vp and the axial-vector part ap, in terms of which the contorsion tensor is expressed by
(2·11)
In general relativity, the starting point of the Fock-Papapetrou method is the response equation of the energy-momentum tensor density TPV of matter fields in the background gravitational field. The response equation takes the familiar form: 17vTPv =O. (2·12)
*l'The tAPV is symmetric with respect to A and p. and traceless. Moreover it satisfies the cyclic identity. tAPV+ f pvA + tVAP=O. See Refs. 7) and 8) for details. 1242 K. Nomura, T. Shirafuji and K. Hayashi
Here TI-'v is symmetric with respect to fL and v, and the covariant derivative acts on the tensor density Tl-'v=J=[iTl-'v like [7ATl-'v=j-g[7ATl-'v. In the theory of gravitation constructed on the spacetime with torsion, on the other hand, the starting point is the response equations of the energy-momentum tensor density and the spin tensor density of matter fieds in the background gravita tional and torsion fields. These equations can be derived from Noether's 14 theorem. ),15) Since the action of matter fields is invariant under general coordinate transformations and local Lorentz ones, there are two sets of Noether's identity which 5 Downloaded from https://academic.oup.com/ptp/article/86/6/1239/1918293 by guest on 27 September 2021 are obtained by the well-known procedure. ) Using the field equations of matter fields in those identities which follow, we can get the following response equations:*)
(2·13)
(2'14) where the energy-momentum and spin tensor densities of matter fields are defined by
smnv=_2 oLM (2·15) oAmnv'
Here LM=j-gLM is the Lagrangian density of matter fields. Derivation of (2'13) and (2·14) is given in Appendix A: It should be stressed that these response equations are valid irrespectively of whether matter fields are coupled to the gravitational field and the torsion field minimally or not. We note that TI-'v is not symmetric with respect to fL and v: It is reduced to the canonical energy-momentum tensor in special relativistic limit, when the interaction is minimal coupling.
§ 3. Papapetrou's method
2 3.1. A brief review of Papapetrou's method ) Let us consider an extended body, whose motion sweeps a world tube in spacetime. Inside the tube we shall take an arbitrarily chosen, timelike world line L which will 'represent' the motion of the body as a whole. The coordinates of the points on L will be denoted by XI-'; they may be considered as functions either of t = Xo or of the proper time r on L. Putting
with oxo=xo-Xo=O, we shall consider the following integrals,
*) The symbols, ( ) and [ l. denote symmetrization and antisymmetrization, respectively. For example, T 1mn)=(l/2)( T mn + T"m) and T/lnn i =(1/2)( T mn - T"'n). Spinning Test Particles in Spacetime with Torsion 1243
(3·1) and so on, where TJl.v is the energy-momentum tensor density of the body and the integration is carried out over the three-dimensional space with xO=const_ Here UJI. denotes the four-velocity, UJI.=dXJl.jdr. We note that M°Jl.v=O=MoAJl.v holds owing to oxo=O. Downloaded from https://academic.oup.com/ptp/article/86/6/1239/1918293 by guest on 27 September 2021 If the dimensions of the extended body are very small, all integrals with one or more factors of oxP are also very small. Then the single-pole approximation is defined by the condition, (3·2) and the pole-dipole approximation by (3·3) Starting from the response equation (2·12), Papapetrou derived the following results: (1) In the single-pole approximation, the body beh~ves like a spin less test particle with a constant mass, obeying the geodesic equation
(3·4)
(2) In the pole-dipole approximation, the body can be treated as a spinning test particle, and its equations of world line and spin are
:r ( mUJI.- Uv 17;;V)+ ~ Rp(JJf.Vjp(5Uv=O, (3·5)
v A A 17r -UJl.U 17r +uvU 17r =0 17r A17r A17r· (3·6)
Here rv and m denote the total spin and the mass, respectively, given by
r v = j(oxJl.TVO-oxVTJl.O)d 3 x, (3·7)
(3·8) where the four-momentum pJI. is defined by
(3·9)
In the pole-dipole approximation, the MJl.v and MAJl.V in (3 ·1) are not tensors, and the pJI. is not a vector, either. It has been shown that the former two quantities are transformed under general coordinate transformations, xJl. -> XfJl., as follows: 1244 K. Nomura, T. Shirajuji and K. Hayashi
(3-11)
P where X'P,v denote ax'p lax V at the point X on L. These transformation properties ensure that m is a scalar and that rIO is an antisymmetric tensor. In the single-pole approximation, on the other hand, owing to (3-2), (3-10) is Downloaded from https://academic.oup.com/ptp/article/86/6/1239/1918293 by guest on 27 September 2021 reduced to the following form:
(3-12) which means that M PV is a tensor. 3.2. Application oj the method to spacetime with torsion Our starting point is the response equations given by (2 -13) and (2 -14), which can be rewritten as
(3 -13)
(3-14) These equations give
(3-15) and (3-16)
PV Now introduce the M and MAPV by (3-1), and NPVA by
(3-17) which we shall refer to as the spin-current hereafter. It should be noted that the energy-momentum tensor density is not symmetric this PV time. Nevertheless, it can be shown that the M and MAPV change in the same manner as (3-10) and (3-11), respectively, under general coordinate transformations. Also, the single pole and pole-dipole approximations for T PV are defined in the same way as in § 3.1. As for the spin tensor density, we shall restrict ourselves to the single-pole approximation throughout this paper, because higher moments of the spin tensor density is expected to be much smaller than the spin-current of (3 -17). Thus, we shall assume that
(3 -18) Spinning Test Particles in Spacetime with Torsion 1245
It can then be shown that the spin-current NI-'UA changes like a tensor under general coordinate transformations.
§ 4. The equations of motion in the single-pole approximation
Let us first study the equations of motion in the single-pole approximation, assuming that (3·2) and (3·18) are satisfied. Integrating (3·13) and (3·14) over the three-dimensional space with xO=const, we obtain Downloaded from https://academic.oup.com/ptp/article/86/6/1239/1918293 by guest on 27 September 2021
d%: +{~6}MP(} - KpdM(P6]- ~ Fp(}l-'uNp(}u=O (4·1) and
(I-'U] =---.!L( _ Nl-'uO ) _ 1""'1-' NPU6 _ 1""'U NI-'P6 2M dr UO .1 p(} 1 p(} , (4·2) respectively, where PI-' is defined by (3·9), and we have used UO=dXo/dr=dt/dr and dpl-'/dr= UOdpl-'/dt. Here {~6}, Kpd, Fp(}l-'u and r::r are now considered as functions of point XI-'(r) on L. We shall use the same convention hereafter, except for (5·4). In the same way, integration of (3 ·15) and (3 ·16) over the three-dimensional space leads to (4·3) and NI-'UOUA UO 0, (4·4) respectively. Now let us define the intrinsic spin by (4·5) Remembering that the MI-'u and NfJ.UA are tensors in the present approximation, we see' that the PI-' and SI-'u are also tensors .. The relation (4 ·4) then allows us to rewrite (4·5) as
(4·6)
Using this in (4·4) gives (4 ·7) Substituting (4·6) and (4·7) in (4·2), we get 2M(fJ.U]= DSI-'u . Dr ' (4·8) where D denotes the covariant derivative with respect to the asymmetric connection r,Ju. Putting (4·8) into the antisymmetric part of (4·3), we have 1246 K. Nomura, T. Shirajuji and K. Hayashi.
DSf.lJl =pf.lUJI_PJlUf.l (4·9) Dr
Then multiplying both the sides with UJI, we obtain
Pf.l=(-PJlUJI)Uf.l-UJlD~;J1 , (4 ·10) where we have used U JI UJI = -1. Let us define the mass of the body as the coefficient
of Uf.l in the expression for pf.l; namely Downloaded from https://academic.oup.com/ptp/article/86/6/1239/1918293 by guest on 27 September 2021 m=-PJlUJI. (4 ·11) Since pJI and UJI are vectors, m is a scalar. The four-momentum pf.l now becomes
f.l= Uf.l-U DSf.lJl P m JI Dr . (4·12)
Using (4 ·12) in (4·9) finally gives the equation of intrinsic spin, JI DSf.lJl _Uf.lU;.DS ;' +UJlU;.DSf.l;' =0. (4·13) Dr Dr· Dr This is the same as the equation of spin in general relativity except that the covariant derivative [7 is here replaced by D [see (3·6)]. If the left-hand side of (4 ·13) is multiplied by UJI, it vanishes automatically. This means that the (4·13) involves only three independent equations. Therefore, we must impose three constraints on 5 f.lJl , such as Sf.lJl UJI=012) or Sf.lJlPJI=0,13) which reduce the number of independent components of Sf.lJl from six to three. In this paper, we shall take the former constraint, (4 ·14) The intrinsic spin vector (or the Pauli-Lubanskii vector) is defined by
f.l=~cf.lJlPO"U 5 S 2 " JI pO", (4·15) which is orthogonal to the four-velocity Uf.l, namely Sf.lUf.l=O. Here the totally anti symmetric tensor (;f.lJlpO" is normalized as (;0123 = 1/ J - g . The constraint of (4 ·14) allows us to solve (4 ·15) with respect to Sf.l Jl , giving (4·16) Equation (4 ·13) can be converted to that for the intrinsic spin vector,
DSf.l =(Uf.lDUJI _UJlDUf.l)SJI. (4·17) Dr Dr Dr This shows that the intrinsic spin vector of a single-pole particle is Fermi-Walker transported along the world line relative to the asymmetric connection r:J1 , and that the Sf.lSf.l is a constant of motion. Substituting (4·3), (4·7) and (4·8) into (4·1), we get the world-line equation, Spinning Test Particles in SPacetime with Torsion 1247
P [1 PI' -lK I' DS <1 +lFP<1P1IS U =0 (4 ·18) [1 r 2 P<1 Dr 2 p<1 11 •
Using (2·10), we can also rewrite this in the form,
[1/rP + ~ RP<1!'1ISP<1U1I+ ~ ([1PKP<111)Sp<1U1I=0, (4 ·19) where Downloaded from https://academic.oup.com/ptp/article/86/6/1239/1918293 by guest on 27 September 2021 (4 ·20)
Putting (4·3) in the second and third terms in (4:1), and using (4·7) in the last term, we can get an alternative form of the equation of world line,
DJ;: + T PP1IPpU1I + ~ FP<1P1ISp<1U1I=0, (4·21) from which follows that
Dm= dm=O Dr dr . (4·22)
This shows that the mass is a constant of motion. The above equation of (4·21) is to be compared with the corresponding one of general relativity, (3·5): The covariant derivative [1 is replaced by D, an additional torsion term appears, and finally the curvature tensor of the asymmetric connection is used in the curvature term. The results of the single-pole approximation are very interesting from an aes thetic point of view. Nevertheless, we have some doubt about its validity, because of the following reasons. First, the intrinsic spin vector of elementary particles does not obey (4·17), as has been shown using the WKB approximation. Second and more seriously, it turns out that the Dirac particle behaves like a spin less point particle in this approximation. In order to see this, let us consider a Dirac particle described by a wave packet. The spin tensor density is then totally antisymmetric, and so is the spin-current N P1IA according to the definition (3·17). Putting ).1=0 in (4·4), we have NPAO=_NPOA=O, because of NPoo=O. Therefore, (4·6) gives SP1I=0, and (4·18) is reduced to the geodesic equation defined by the metric. Thus, higher-order approximation is indispensable in order to treat the intrinsic spin of fundamental particles.
§ 5. The equations of motion in the pole-dipole approximation
Let us consider the equations of motion in the pole-dipole approximation,now assuming that (3·3) and (3 ·18) are satisfied. Integrating (3 ·13) and (3 ·14) over the three-dimensional space with xO~const then gives
d:: +{~<1}MP<1- oA{~<1}MA(P<1)- KpdM[P<1] +oAKpdMA[P<1 L ~ F P <1P1INP<111=0 (5.1) and (4·.2), respectively, where pP is defined by (3·9). 1248 K. Nomura, T. Shirajuji and K. Hayashi
In the same way, integration of (3,15) and (3·16) over the three-dimensional space leads to
(5,2) and
(5·3) Downloaded from https://academic.oup.com/ptp/article/86/6/1239/1918293 by guest on 27 September 2021 respectively. An additional relation, which is independent of these four relations, follows from the equation, *)
A V V r 06'(x x Tl'o")=x TI'A+ XATI'V +XAXV( -{~6'} TP<1 + K p dTP6'- ~ FP<1l' SP<1r). (5,4)
Integrating this over the three-dimensioNal space gives
(5·5)
The intrinsic spin SI'V is still defined by (4· 5) as in the single-pole approximation. We note that though the expression (4,6) for the intrinsic spin does not hold in the present approximation, the definition (4·5) is reduced to (4·6) in the rest frame of the body where Uv=( -1,0,0,0). Let us now define the total angular-momentum around Xl'by
(5,6) which we shall call the total sjJin of the body. It can be shown that this quantity indeed changes like a tensor under general coordinate transformations (see Appendix C). By virtue of (5,3) and (5·5) together with (5·6), MAI'V can be expressed in terms of ]pv, UI' and NI'VA, as is shown in Appendix B. In paIticular, we have
MA(I'V) = -(P(I'UV)+ NA(I'V») + g~ (f°(I'UV)+ N 0(I'V»). (5,7)
We are now ready to derive the equations of motion for the body. Substituting (4·2), (5·3) and (5,7) in the antisymmetric part of (5·2), we get
(5·8)
Multiplying both the sides by Uv, we obtain
P *) Equations with three or more factors of x , such as a"(xpx'x"TM)=,,., give no new relations. ! The same is' true also (or smnv, . Spinning Test Particles in Spacetime with Torsion 1249
(5 0 9) where we have used UvUv= -1. Let us define iii as the coefficient of UP in the expression for PP: Inspection of (5 0 9) then gives Downloaded from https://academic.oup.com/ptp/article/86/6/1239/1918293 by guest on 27 September 2021
=-pvUv (5°10) with pv being defined by
p c5 pc5 Pc5 U N° 1 N 0 pv- = pv - {Xc5 }-----vo-- r {Xc5 }---vo+'[Kpc5v---vo ' (5°11) which is reduced to pv of (4°20) in the single-pole approximation. We shall prove in Appendix C that iii of (5°10) is a scalar, so that pv is a vector. Putting iii of (5 °10) into (5 °9), and using the resultant expression for pP in (5 °11) give the covariant form of PP:
P V V Ppc5 Pc5V V pP = iii UP - Uv( [7/;v + K pc5N pc5 - K pc5N + ~ Kp ,/N - ~ Kp c5 Npc5p) . (5°12)
If we use (5 °11) in (5 °8), all the noncovariant terms are cancelled out. Then using (5 0 12), we obtain the covariant equation for ]pv, [7 ]pv [7r A [7]PA vr=UPUAvr-uvUAvr
(5 °13)
We note that if the torsion is vanishing, or if the body has no intrinsic spin with NPVA =0, then (5°13) is reduced to (3°6) of general relativity. It can easily be shown that if (5 °13) is multiplied with Uv on both sides, it is automatically satisfied. Therefore, (5°13) involves only three independent equations. This allows us to impose three constraints on ]pv, which we shall take as (5°14) Then the ]pv has only three degrees of freedom, and is equivalent with the spin vector (or the Pauli-Lubanskii vector) defined by
(5 °15) 1250 K. Nomura, T. Shirajuji and K. Hayashi which is orthogonal to the four-velocity Uf1.. By virtue of (5'14), r ll can be solved in terms of r, (5 '16) Thus, Eq. (5 '13) can be rewritten into that for r, II [7r =(Uf1.[7U _UII [7Uf1.)J _lcf1.IIP6U (2K NAT+K NAT) [7 r [7 r [7 r II 2 '- II pAr 6 ATP 6. (5 ·17)
This equation is much simpler than that for r ll , and is consistent with the or Downloaded from https://academic.oup.com/ptp/article/86/6/1239/1918293 by guest on 27 September 2021 thogonality condition of rUf1.=o. Let us derive the covariant equation of world line, which follows from (5 '1). To do this, it is convenient to rewrite (5' 2) using (5 '11) as follows:
Mf1.II =Pf1.UII + ; ( - M;;O)+{~6}(]IIPU6 + Nllp6)- ~ Kp,/'Np611 . (5'18)
Substituting (5·11), (5·18), (5'7), (4·2) and (5·3) in the first, second, third, fourth and fifth terms of (5 ·1), respectively, we obtain the covariant equation of world line,
[7 pf1. +lRP6f1.1IJ, U -l([7f1.KP611)N. =0 [7r 2 p6 II 2 p611. (5 ·19)
Substituting (5 ·18) in the second and. fourth terms of (5 '1), and then using (5 ·11), (5' 7) and (5·3) in the first, third and fifth terms, respectively, we obtain an alternative form of the equation,
P6 ntrf1. + T Pf1.IIPpUII+ ~ K P6f1. [7/;6+ ~ R f1.IIJp6UII- ~ (Df1.KP611)Np6I1=0. (5'20)
If the torsion vanishes identically, or if Nf1.IIA=O, then Eqs. (5·19) and (5'20) are reduced to (3'5) of general relativity. Equations (5'17), (5'19) and (5·20) are not satisfactory ones, however, since the spin-current with three indices, Nf1.I1\ appears in them. To proceed further, therefore, we must make some assumptions (or approximations) about the form of Nf1.IIA . For example, if Nf1.IIA is totally antisymmetric such as the Dirac field, we can rewrite (5 '17) using (2'11), (4'5) and (4·16) as follows: II [7r =(Uf1.[7 U _ U II [7Uf1.)J _lL f1. IIP6U S [7 r [7 r [7 r II 2 E pa6 II. (5·21)
Futhermore, if we can put r equal to Sf1. in (5' 21), we obtain the same equation of spin precession as that for the Dirac particle in the WKB approximation. Here it should be emphasized that in deriving (5' 21), we do not use the Dirac equation at all. We shall consider more general expressions for Nf1. IIA in the next section. Since p lI does not agree withthe four-momentum in the single-pole approxima tion as is seen from (4'20), it cannot be interpreted as the four-momentum vector. Accordingly, iii cannot be regarded as the mass of the body, either. We shall intro duce the mass and the four-momentum of the body in § 7, where the equation of world line (5 ·19) [or (5·20)] will be discussed in more detail based on the explicit representa- Spinning Test Particles in Spacetime with Torsion 1251 tion of the spin-current to be obtained in the next section.
§ 6. Towards the equation of spin precession
It is a characteristic feature of the pole-dipole approximation studied in the previous section that the equations of motion, (5 ·17) and (5·19) [or (5·20)], involve the spin-current with three indices coupled to the torsion of the spacetime. Let us first briefly summarize the expression for the spin-current N PIJA in the various models of Downloaded from https://academic.oup.com/ptp/article/86/6/1239/1918293 by guest on 27 September 2021 spinning bodies so far considered:
ll 1) Spin-fluid model ) In the theory of spin-fluid, the spin-current is assumed to be (6·1) This form is valid also in the single-pole approximation [see (4·7)]. 2) Wave packets of spinning particles in the WKB approximation. For wave packets of particles with spin 1/2, 1 and 3/2, it can be shown that NPIJA is given by*) -SPIJ U A-2U[PSIJJA for spin 1/2, N PIJA = -SPIJUA- U[PSIJJA for spin 1, j -swUA for spin 3/2. (6·2) Here it should be emphasized that in all these cases the intrinsic spin SPIJ is orthogonal to the four-velocity, (6·3) In view of these examples, let us suppose the spin-current to be of the form, (6·4) with the intrisic spin SPIJ being orthogonal to U IJ , where I is an unknown parameter. This expression is the most general one under the following assumptions: (i) N PIJA is expressed by SPIJ and UP only, and does not involve the four-acceleration. (ii) SPIJ appears linearly. (iii) The parity of NpuA is even, excluding the odd parity terms like EPIJPrJUpSi. Substituting (6·4) in (5·17), and using (2·11) and (4·16), we obtain
(6·5) where SP is the intrinsic spin vector defined by SPIJ using (4 ·15), and the parameters 11 and 12 are given by 1 4 11=-1-2 and 12=3(1-/). (6·6)
*) The spin precession of particles with higher spin (s 23/2) wi Ii be studied separately. 1252 K. Nomura, T. Shira/uji and K. Hayashi
Equation (6-5) is not an equation of spin precession for J". This is not surprising if we remember that the intrinsic spin is coupled with the torsion field, while the rotational sjJin*) is not. In other words, when the spinning body under consideration has both intrinsic spin and rotational spin, Eq. (6-5) is not sufficient to determine the behavior of the total spin and the intrinsic spin. This equation, consisting of only three independent equations, must be supplemented with three additional relations or constraints between the total spin and the intrinsic spin. If the body has no intrinsic spin, Eq. (6·5) shows that the rotational spin is Fermi-Walker propagated along the world line in the same manner as in general Downloaded from https://academic.oup.com/ptp/article/86/6/1239/1918293 by guest on 27 September 2021 relativity. On the other hand, when the rotational spin of the body is negligibly small compared to the intrinsic spin, Eq. (6'5) is reduced to the equation of spin precession,
(6'7) which can also be rewritten using the covariant derivative with respect to the asymmetric connection as
(6·8) with the help of (6·6) and (2'11). These equations show that the SP.Sp. is a constant of motion, and that for the spin-fluid with the spin-current (6 '1), the spin vector is Fermi-Walker transported along the world line relative to the asymmetric connection, as has been suggested by Adamowicz and Trautman. 10) However, it should be emphasized that the Fock-Papapetrou method leads to the equation of spin precession with an unknown parameter /. Except for the particular case with /=0, the spin vector is not Fermi-Walker propagated, as is shown in (6'8). It has been suggested that for the spin precession of particles with spin, the 9 parameters 11 and 12 are represented as )
iI=S-2, 12=~(s-~) (6'9) with s being the spin value in units of n. Comparison of (6·6) with (6'9) implies that s is related to I by 3 s=-I+Z ' (6·10) from which we immediately see that the I of (6·2) reproduce the correct spin values forspin-1/2, -1 and -3/2 particles. When the motion of intrinsic spin of a body is not influenced by its rotational motion, (6'7) and (6·8) will still be valid. Conversely, if the intrinsic spin is strongly
*) Here we call LP=jP-SP the rotational spin. See also (7·2). Spinning Test Particles in SPacetime with Torsion 1253 coupled to the rotational spin, and furthermore if the former is aligned with the latter, then Sp can be expressed by Sp=gr with g being a proportional constant: According ly, Eq. (6 0 5) is reduced. to the equation of spin precession for r of the form (6°7) with the parameters f1 and f2 being now given by
(6°11)
The magnitude rIp is then a constant of motion. Downloaded from https://academic.oup.com/ptp/article/86/6/1239/1918293 by guest on 27 September 2021
§ 7o The equation of world line
Let us study Eq. (5°19) [or (5°20)] of world line in more detail, using the spin current NpVA of (6°4). First of all we note that use of (6°4) in (5 0 9) gives
pp=( iii-/KlJPrJUvSp (7 °1) pV where L represents the rotational spin defined by (7 °2) We shall express by m the coefficient of UP in the above pI', and call it the mass of the body: 1 0 m-m--- J'.fKvPrJU v S PrJ-TKPrJvS prJ Uv· (7 3) This is a scalar since iii is a scalar, and furthermore it coincides with the m of (4°11) in the single-pole approximation.*) Let us next define a vector quantity PI' by discarding all the noncovariant terms in (7 °1): Namely, PP=mUP- Uv( [7i;v + Dg;V)_ f(KpPrJSp Then it can easily be seen that PI' is related to PI' of (5 °12) by (7 0 5) Comparing this with (4 ° 20) shows that PI' becomes the four-momentum in the single pole approximation. Accordingly it is natural to interpret PI' as the fQur-momentum vector. Multiplying (7 ° 4) with Up, we have (7 0 6) *) The single pole approximation is attained if one puts LP"=O and /=0. 1254 K. Nomura, T. Shirajuji and K. Hayashi Substituting (6·4) and (7·5) in (5·19), we obtain the equation of world line as follows: (7 ·7) This equation reminds us once again of the fact that the torsion field is coupled to S"u, Downloaded from https://academic.oup.com/ptp/article/86/6/1239/1918293 by guest on 27 September 2021 but not to L"u. If we substitute (6·4) and (7·5) in (5·20), we can get the alternative expression for the equation of world line: DP" + Tmp U +lK ,,[7U<1 +lRP<1"UL U Dr, P U 2 p<1 [7 r 2 p<1 U (7 ·8) If the rotational spin L"u is negligibly small, we cari show that the mass is a constant of motion. For this purpose, we multiply (7·6) by D/Dr on both sides, and employ the equations of world line and spin, (7·8) and (6·8), respectively. Then a little algebra leads to (7 ·9) If the total spin is aligned with the intrinsic spin, namely if S"=g]" with g being a constant in time, then we can show in the same way that the mass is a constant of motion. § 8. Conclusion Following the'Fock-Papapetrou method, we have studied the equations of world line and spin for a spinning body moving in the background gravitational field and a torsion field. Our starting point is the two response equations, (2 ·13) and (2 ·14): One for energy-momentum of the body and the other for intrinsic spin. Throughout this paper we have assumed that the lowest (i.e., the single-pole) approximation is valid for the spin tensor density. For the energy-momentum tensor density, however, we have first considered the single-pole approximation and then the pole-dipole approxi mation. Let us summarize the main results. (1) In the single-pole approximation, the equations of motion are similar in form to those in general relativity which are obtained for the pole-dipole approximation., We should mention the two differences, however. First, the intrinsic spin of a particle in free fall is Fermi-Walker transported along the world line relative to the asymmetric connection of the spacetime. Second, the world line of a spinning particle is affected by its intrinsic spin, and accordingly it deviates from the familiar geodesics of the metric. It is also shown that the mass and the magnitude of spin is kept constant along the world line. We find, however, that the Dirac particle behaves like Spinning Test Particles in Spacetime with Torsion 1255 a spin less particle in the single-pole approximation: This suggests that higher-order approximation is indispensable. (2) In the pole-dipole approximation, we have two tensor quantities,],"1.1 and S 1'1.1 , which can be interpreted as the total spin (Le., the sum of the rotational spih and the intrinsic spin) and the intrinsic spin, respectively. We have derived the two covar iant equations of motion: One for the world line and the other for the temporal change of ]'"1.1. It is the characteristic feature of the equations of motion in this approxima tion that they involve couplings between the torsion field and the spin-current NI'I.IA of Downloaded from https://academic.oup.com/ptp/article/86/6/1239/1918293 by guest on 27 September 2021 the body. (3) On the basis of the spin-current in various models of spinning bodies, we have supposed that the spin-current is represented as (S·l) where f is a parameter characterizing the spinning body, and the intrinsic spin 51'1.1 is orthogonal to UI.I. We then get much simpler equations of motion: Nevertheless, the number of the quantities (]'"I.I, 51'1.1 and XI') to be determined is larger than that of the equations. This suggests that if a torsion field does exist, then the motion of a spinning body with both rotational and intrinsic spin cannot unambiguously be described by the Fock-Papapetrou method. At present we have no idea to overcome this problem. (4) If the rotational spin of the body is negligibly small, then the equation of spin describes precession of the intrinsic spin. The resultant equation of spin precession has an unknown parameter f characterizing the spinning body, and reproduces all the known equations of spin precession due to torsion. As for the equation of world line, we have shown that the mass of a spinning body without rotational spin is a constant of motion. Similar results are valid when the intrinsic spin is aligned with the total spin. Acknowledgements Two of us (K. N. and T. S.) would like to thank Professor H. Shimodaira, Professor K. Mori, Professor K. Kobayashi, Professor T. Saso, Dr. Y. Tanii and other members of theoretical physics group at Saitama University for valuable discussions and encouragements. One of us (K. N.) is also grateful to Mr. S. Toriyama and Mr. K. Kobayashi at Saitama University for encouragement at the early stage of this work. Appendix A --Noether's Theorem and Response Equations-- For simplicity we shall assume that matter fields have no Greek indices: Thus we shall ignore the effects due to eletromagnetic field and other gauge fields of internal symmetry. Then the action of matter fields can be expressed by*) *) ovq is denoted by q,v. 1256 K. Nomura, T. Shirajuji and K. Hayashi (A·l) where q denotes collectively the matter fields, and the subscript M is to remind us that this is the action only for matter fields, with ekp and AuI' being taken as external fields. For arbitrary variation 0, we have (A·2) Downloaded from https://academic.oup.com/ptp/article/86/6/1239/1918293 by guest on 27 September 2021 where F denotes q, ekp and Azjp collectively, and S is the Lie derivative defined by SF =oF-F,1I0XII. Let us suppose that the action is invariant under general coordinate and local Lorentz transformations: Namely, the relation, 01=0, shall be valid for any Q under such transformations. We then have N oether's identity, (A·3) which holds without the help of field equations. For local Lorentz transformations, variation of the fields is given by ~ _ Z mn uq-zWmn(J q, with (Jmn b~ing the Lorentz generator. The identity (A·3) can be represented in the form, (A·5) mnll mnplI where the explicit form of A mn, B and c can be found if we use (A ·4) in (A·3). Since Wmn are arbitrary functions of xl', the coefficients of Wmn and their mnll mnplI derivatives, A mn, B and c , must vanish separately. It follows from these identities that (A·6) In a similar way the invariance under general coordinate transformations gives the identity rJlll_K· SUII_OLMD 17 lI.Lp zjp T[ijJ+~F··2 UplI oq pq-.=0 (A·7) In deriving (A· 7), use is made of (A . 6). Using the field equations for matter fields in (A ·6) and (A .7), we obtain the response equations (2·14) and (2·13), respectively. Spinning Test Particles in Spacetime with Torsion 1257 Appendix B --Derivation of the Expression for MAI-')) __ Let us show that MAl-')) can be expressed in terms of UI-', JI-')) and NAI-')). Two other relations of the form (5· 5) may be derived by taking the cyclic permutations of the indices il, fJ. and 21. Adding the first and second of these three relations, and subtract ing the third, we have M[AI-']O U)) M[A))]O UI-' M(I-')))O UA Downloaded from https://academic.oup.com/ptp/article/86/6/1239/1918293 by guest on 27 September 2021 UO + UO + UO (B'l) Combining (5·3) and (5'6), on the other hand, we note (B'2) Using this in (B'1), we get (B·3) Putting il=O in (B·3), and remembering that MOl-')) = 0, we obtain (B·4) Substituting this in (B'3), we have (5'7). Taking the sum of (5·3) and (5·7), we finally obtain Appendix C -- Transformation Properties of r)), iii and PI-' __ First, let us prove that r)) is a tensor. Putting 21=0 in (3'11), we have (C'l) Substituting (B·5) in (C'l), we get M'AI-'O=_ ~X'A"X'I-',pf'PU'o- ~ N'AI-'o+(terms symmetric under il<=!fJ.). (C·2) Thus we obtain (C·3) Remembering (M'AI-'O_ M'I-'AO) + N'AI-'O -U'O (C·4) we see that JI-'V is a tensor. 1258 K. Nomura, T. Shirajuji and K. Hayashi Next, let us show that iii of (5·10) is a scalar. Multiplying both sides of (3·10) by U'J1. and U'JI, we have U~U~M'J1.v= UpUr5M Pr5 - U~U~(X'J1.,pX'v,r5),AMA(Pr5) +U'U,~(_l_X'o X'J1. X'V M A(Pr5») J1. v dr U'O ,A ,P ,r5 • (C·5) Substituting (5·7) and (5·18) in (C·5), and using the transformation property of Christoffel symbol, Downloaded from https://academic.oup.com/ptp/article/86/6/1239/1918293 by guest on 27 September 2021 P {J1.p(f }'-X'J1.- ,aX ,p' Xl ,6' {aPr }+X'J1.. ,a xa ,p'(J' , (C·6) we obtain - P'J1.U~= - PPUP. (C·7) Thus we have proved that iii"'=' - PPUP is a scalar, ~o that pJ1. is a vector. References 1) V. A. Fock, J. Phys. U.S.S.R. 1 (1939), 81. 2) A. Papapetrou, Proc. Roy. Soc. London A209 (1951), 248. E. Corinaldesi and A. Papapetrou, Proc. Roy. Soc. London A209 (1951), 259. 3) c. W. Misner, Kip S. Thorne and J. A. Wheeler, Gravitation (Freeman, San Francisco, 1973), p. 1050. 4) T. W. B. Kibble, J. Math. Phys. 2 (1961), 212. 5) K. Hayashi and A. Bregman, Ann. of Phys. 75 (1973), 562. 6) F. W. Hehl, P. von der Heyde, G. D. Kerlick and J. M. Nester, Rev. Mod. Phys. 48 (1976), 393. F. W. Hehl, in Spin, Torsion and Supergravity, ed. P. G. Bergmann and V. Sabbata (Plenum, New York, 1980), p. 5. 7) K. Hayashi and T. Shirafuji, Prog. Theor. Phys. 64 (1980), 866, 883; 65 (1981), 525. 8) K. Hayashi and T. Shirafuji, Phys. Rev. D119 (1979), 3524; D24 (1981), 3321. 9) K. Hayashi, K. Nomura and T. Shirafuji, Prog. Theor. Phys. 84 (1990), 1085. 10) W. Adamowicz and A. Trautman, BulL Acad. Polon. Sci., Ser. Sci. Math., Astr. Phys. 20 (1975), 339. 11) J. Weyssenhoff and A. Raabe, Acta Phys. Polon. 9 (1947), 7. 12) F. A. E. Pjrani, Acta Phys.Polon. 15 (1956), 389. 13) W. Tulczyjew, Acta Phys. Polon. 18 (1959), 393. 14) E. Noethe~, Nachr. Ges. Wiss., Gtittingen (1918), p. 235. 15) T. Yamanouchi, R. Utiyama and T. Nakano, Theory of General Relativity and Gravitation (Sh6kab6, Tokyo, 1967), (in Japanese), Chap. V. W Note added in proof: For spin-3/2 particles, the N ' of (6·2) is obtained by assuming that the gravitational interacton is introduced by the covariant derivative (J,+(1/2)AiJvSiJ)epp, which is usually employed in the Pv supergravity theory. If the full covariant derivative (J,epp+(i/2)A iJvSiJepp- rJvep,) is used, however,the N , is expressed by (6'4) with / now being given by 1/3. This result together with (6'2) for spin 1/2 and 1 suggests that the parameter / is related to the spin value s by /=1/2s, instead of (6'10).