Volume 104A, number 5 PHYSICS LETTERS 3 September 1984

CHARGED SPINNING FLUIDS IN GENERAL RELATIVITY

Ricardo AMORlM Universidade Federal do Rio de Janeiro, Institute de Fisica, Cidade Universitciria, Ilha do Fundao, Rio de Janeiro, CEP. 21.944, Brazil

Received 18 May 1984

The treatment of neutral spinning fluids in general relativity due to Ray and Smalley is generalized, by considering the

fluid as charged and having a magnetic dipole moment proportional to the spin.

In this letter we treat the problem of a perfect clas- sical charged spinning fluid with a dipole moment proportional to the spin in general relativity (GR). The equations of motion are obtained by a variational principle developed by Halbwachs [l] , Ray [2] and

Ray and Smalley [3]. We have generalized the work of Ray and Smalley so as to include the electromagnetic (EM) interaction terms. The notation used throughout this paper follows that of ref. [3].

The spin equations of motion are generated with the help of an orthonormal tetradic field api( where /J = 1,2,3, 4 are local Lorentz indices and i = 0, 1,2, 3 are the usual holonomic tensor indices.

The angular velocity [ 1,4] is described by

&Jii = :(l+Q - U/$j) ) (1)

where the over-dot represents the covariant proper-time derivative.

The spin kinetic energy density is

Ts = :pWijSi’ ) (2)

where

,Sii = k(x)(al@ - &j&i) (3)

is the spin tensor of a fluid particle and p(x) is the fluid proper matter density.

We identify the a4i with the four-velocity ui of a fluid particle, consequently, from (2) and (3)

T, = ;p,!c(&~ _ ;“a?) = p@(;? . 2 2 (4)

Assuming that the charge and mass are simulta-

0.3759601/84/$ 03.00 0 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

neously conserved and not considering dissipative processes, we write [5] the current as

Ji = epui , (5)

where ep is the charge density. The density of dipole moment is assumed to be

proportional to the spin,

MU = pxLSii . (6)

We represent the lagrangian density for the fluid as

(7)

~~/~g=(1/16nG)R_F(p,S, Sii)+hl(gijb!iUi+ 1)

+ h2(pu’);i + h3X,iUz + A4S,iUz , (8)

1,/G= -T, t A’l(~lialjgij - 1)

+ h22(&,2jg.. _ 1) + 2~12 9

li 2j gija a

t2h14g..aliuj + 2h24g..a2iuj 11

L?,/&= -epAiui - ipxSijFij,

and

(9)

(10)

X,,/fi = -:FijFi’ . (11)

L, is a generalization of the Ray lagrangian [2], which describes a non-spinning fluid in GR. A,, A,, A3 and X4 are Lagrange multipliers associated to the fol- lowing constraints

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Volume 104A. number 5 PHYSICS LETTERS 3 September 1984

gijuw+ 1 =o, (pUi);i=o, (12a,b)

X,iU’=O and S,iU’=O. (12c,d)

These constraints are obtained by variation of the action with respect to the correspondent Lagrange multipliers. (8a) is the canonical condition on four- velocity, (8b) represents mass conservation, (8~) is the conservation of the “Lin particle identity” [2] neces- sary to allow rotational motions of the fluid and (8d) ensures that the entropy is conserved along the current lines. G is the gravitational constant.

ZH was first used by Halbwachs [l] in special rela- tivity and contains the spin kinetic energy term and the other necessary conditions to ensure the ortho- normality of the tetradic field. L, describes the interac- tion of the charge and the dipole moment of the fluid with the EM field [6]. JZEM is the lagrangian of the EM field.

The equations of motion are obtained by varying the total action with respect to all the field variables.

Variation with respect to #, B = 1,2, leads to the

equations

kPila2’ - pkrii2 + 2X”a! + 2A12a: t 2A14Ui =O (13)

and

- kPilali t pia! t pkit t 2h22af + 2h12at

+2x%.=0 1 ’ (14)

where

Pi, = 2 aF/aS” - pXFil . (15)

By contracting these two equations with ali, a2i

and ui, and observing the orthonormality relations

imposed by &, we see that

x12 = i = 0 , ~14 = ik(uiPil t pcir)a2’ , O&b)

X24 = -:k(u’Pil+ pril)alz , (16~)

AlI = x22 = ;T, - :pi,sil. (164

From (13), (14) and (16) we see that the final form of the equations of motion for aBi are

&! = -(T, - iPikSik)af t k(uiPjl + ph,)a%i

t kPiia ‘i (17a)

and

260

pk# = (T, - iqkbyik)a: t k(UjPil t pil)a2’Ui

t kPiia2i . (17b)

From (3) and (17) we get the precession equation .

(18)

Note that ifpk, were null, Sij would be Fermi- Walker transported along ui.

Variation with respect to p gives

Fp = T,/p + :XFijS’J t eAiU’ t h2,iU’ ,

where Fp = aF/a,.

(19)

Variation with respect to the four-velocity leads to

2X’ + T, + pX2,iU’ t epAiu’ = 0 , (20)

after a contraction with ui. From (19) and (20) we have

2h’ = -pFp + ipxFij,yij. (21)

To obtain the energy-momentum tensor, we have to do a very tedious calculation. After several cancela- tions, we arrive at the Einstein equation

(2/fi)6L/6gij = -(1/8nC)Gij + Tij = 0 , (22)

where Gij is the Einstein tensor and

Tii = (F- pF,)gij t p(;xF,,Sk’ - FJuiuj

t (dpkl - &)u (W - p(ilsj)l

+ [puG,si)k] ;k - FilF,i - $gijF,mFlm (23)

is the matter plus EM field energy-momentum tensor. Variation with respect to Ai leads to the Maxwell

equations

Fijii =Ji +Miiij, (24)

which have the correct form, including the polarization current term [7].

Note that if we assume the first law of thermody- namics [8] as

de=TdS-pdp-’ tfwijdSij (25)

and

F = -p(l + e) , (26)

where pe is the internal energy density, it foJlows that

Volume 104A, number 5 PHYSICS LETTERS 3 September 1984

-F,=l+~+p/p, F-pF,=p,

and

(270)

Pii = p (XFii - Wij) . (27~)

As a consequence, the spin-independent part of Tij takes the usual form of a charged perfect fluid energy- momentum tensor [5]. In the case of a null EM field, rii has the same form as the one found by Ray and Smalley [8] in the framework of a Riemann-Cartan space-time.

The differential system for a stream line is formed by the divergence of Tij, the equation of continuity (12b), the Maxwell equation (24) and the precession equation (18). From this system and the usual relations involving the Riemann and Ricci tensors, we obtain the Euler equation

p(D/Dr){(l+ e +p/p - &&F)u’

+ S” [Cl - (xFlk - olk)uk] }

= Vj (pwpjl) - V’p - tp R$k$4jSk1

+ Fik Jk + &$fjkqkji . (28)

Eq. (28) has the desirable limit if the spin vanishes [5] and it is extremely analogous to the corresponding equation of motion of a classical spinning particle also submitted to the condition [6,9]

S”Uj = 0 . (29)

By contracting (28) with Ui it is possible to show that

UiVj (pw$sQ) = 0 , (30)

which is an additional constraint and not generally satisfied. On the contrary, if the first law of thermo- dynamics is assumed in its usual form, Pij contains only the x term in (15) and consequently (30) becomes

an identity. Note that condition (29) is identically con- served, which can be seen from (18), for any choice of E.

We observe, however, that eq. (28) is second order

in the four-velocity, allowing helicoidal solutions even for vanishing pressure and null interactions. These solu- tions are characteristic of the theories submitted to condition (29) [9-l 11. We are trying to extend the general ideas presented in this letter to fluids submitted to alternative spin constraints, which could avoid those spurious solutions. We are also interested in considering the spin-torsion coupling, which can be done in more

than one way [8,10,12,13].

References

[l] F. Halbwachs, Theorie relativiste des fluides a spin

(Gauthier-Villars, Paris, 1960).

[2] J.R. Ray, J. Math. Phys. 13 (1972) 1451.

[ 31 J.R. Ray and L.L. Smalley, Phys. Rev. D26 (1982) 2619.

[4] F. Giirsey, Nuovo Cimento 5 (1957) 784.

[S] A. Lichnerowicz, Relativistic hydrodynamics and mag

neto hydrodynamics, Math. Phys. Monograph Series (Benjamin, New York, 1974).

[6] F. Halbwachs, Prog. Theor. Phys. 29 (1960) 291. [7] D.E. Soper, Classical field theory (Wiley, New York,

1976).

[8] J.R. Ray and L.L. Smalley, Phys. Rev. D27 (1983) 1383.

[9] R. Amorim and J. Tiomnb, Rev. Bras. Fis. 8 (1978)

350;

R. Amorim, J. Math. Phys. 25 (1984) 874. [lo] G. Cognola, R. Soldati, L. Vanzo and S. Zerbini, Phys.

Rev. D25 (1982) 3109; Nuovo Cimento 76B (1983) 109.

[ 111 S. Hojman and T. Regge, Studies in Mathematical

Physics, Essay in Honor of V. Bargman, eds. E.H. Lied

and A.S. Wightman (Princeton Univ. Press, Princeton, 1976).

[ 121 P.B. Yassikin and W.R. Stoeger, Phys. Rev. D21 (1980) 208.

[ 131 R. Amorim, The motion of classical spinning particle in a Riemann-Cartan space-time, IF-UFRJ 83-26 (1983),

unpublished.

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