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General. Relativity and Gravitation, Vol. 24, No. 11, 1989 Relativistic Holonomic: Flirids * > : : Bartolomé Coll' and. Joan Josep Ferrando? Received Jamuary 25, 1989 'The notion of holonomic fluid in relativily is reconsidered. An intrinsic charac- terization of holonomic fluids, involving onty the unit velocity, is given: 1howing: lhal in spite of its dynamical. appearance. the: notion. af holonomiczfyid. is. a * kinematical. notion.. The¡ relations. belween: holonomic: and: thermedynamic pericct Muids are studied, . e 1. INTRODUCTION (a). la classical mechanics, there is a large class of fluids whose equations of motion admit a relative (Poincaré) integral invariant—namely, the velocity. For them, the vorticity becomes an absolute integral invariant, and it is well-known the important róle played by these two-invariants in the development of hydrodynamics: integration of the equations of motion, study oC particular fluid flows, ( Helmholtz) properties of the: vorticity, etc. En relativity, the analogue of the above class of fluids was corisidered many ycars ago by Lichnerowicz [1], who called them holonomic"fhuids. Holonomic fluids also admita relative integral invariant, but:itis not, as it could seem, the unit velocity of the fluid: from the point of'view-of the theory of integral invariants, the relativistic analogue: of: the- classical velocity is the current which differs from the unit velocity by a scalar factor called the index function. The corresponding absolute integral invariant is lhen given by the exterior derivative of the current which is called. the dynamical vorticity tensor. For the particular case ol barotropic perfect fluids, these appelations arc due to Synge [2], who was the first to develop systematically the relalivistic theory of such Muids. ' Département de Mecanique, ULA, 766; CNRS-Université de Paris Yi. (T.66, E3). 4; Place Jussicn, 75252 Paris Cedex 05, France, - ? Departament de Visica Teórica, Universitat de Valéncia, 46100 Burjassot, Valéncia, Spain. 1159 DOUI-7701/49/1 100-1159306.00/0 3 1989 Plenum Publishing Corporation 1160 Colk:and Ferrando (b)- Extending a: result by Eisenhart [3], Lichnerowicz [1] showed an: important property el holonomic fuids: Iheir stream lines are the uxtremes of a distance conformal to (he space-time distance. ln other words: the (unit) velocity of a holonomic Ñuid is + geodesic vector field for a metric. conformally related to the space-time metric. The vector fields verilying this property (or Uhuir congruence of integral curvus) are called conformally geodesic.? Killing, conform-killing, or'irrotational vector fields, beside barotropic perfert fluid velocitics, are usual examples of'conformally geodesic fields. However, how does: one recognize: a: given. vector. field: as. belonging to this class? We shall give here: necessary and: sufficient condi- tions for a vector field to be conformally geodesic: * : Also, as a converse of the above Lichnerowicz result we shall show thas any, conformally geodesic velocity makes any, conservative fluid holonomic. a ln: many" knowenexamples.of-luid; lows, the:index. function has been expressed as a function of the (thermo- kdynamicak variables, showing_tal the relativistic relative” integrat: invariant: is" a="dynamicab quantity, in contrast with the classical one, which is clcarly kinematical: In spite of he fact that the existence ol such a dynamical invariant is the main feature of holonomie fluids, our above results show that holonomic (Tuids.admit a purely kinematical characterization. : Barotropic perfect fluids are holonomic fluids, but holonomic perfect Nlvids are not, in general, barotropic. We shall study the relations of both concepts. (c) Holonomic fuids are interesting in relutivity because of (i)' the conformally geodesic character of their stream lines, (11) the known [51 relations between their invariant integrals and.their first integrals, and (iii) the conservation laws for the. vorticity thal lake place for them [1]. For lhese reasons, we think that the holonomic character of a Muid.may be use- fully taken into account in the vbtainment of explicit solutions to the Einstein equations as well as in the study of particular classes of motions and the analysis of certain conjeciures on baratropic perfect Muid. spucc- times (such as the Lichnerowiczs conjecture““ón spherical symmetry under appropiate asymptotic conditions, or the Treciokas- Ellis conjecture? on vorticity-free or expansion-free consequences under distorsion-frec condi- tions). 3 Conformai geodesics have been considered recently in a diferent relativistic context (see [4)). See, cg, [6] and references therein. 3 See [7] or the more recent analysis hy Collins (8). 1162: Coll ande Ferrando: where m=xX%0.and. /I are, respectively, a scalar and. a second-order. tensor, and denote by J the specific divergence ol I1? óll=mf (2) The conservation of T, 4T'=0, is equivalent to the system: * ón) = mi) A, a= Liu)! (3) where a is the acceleration OÍ u, u= i(u) Vu. Definition 2.1. (Lichnerowiez).* A fuid is said. to be holonomic (with respect to the associate velocity w) if the corresponding specilic divergence Lis exact: J=dF. Then; F is called. the hotonomiy: potential and: m'the holonomy densitw; f= e* is called:the index function: For a holonomic fluid, the conservation equations (2) become: a=dE— Fu (4) B=F-M (5) where 0 is the expansion of u, U= —óu, and M the logarithmic holonamy density, M = In »1.* Their solutions will be noted (u, F, 1). (c) Lichnerowicz.[1] showed that. the current Ox fircis.a relative integral invariant for the equations of motion of the holonomic fluids and that, consequently, is exterior differential dC is. an absolute- integral invariant. That. is, if D and. 0' are two 2-chains. on the same worid-lines tube of y, one has: [, <= C, [40= de (6) EDS Hi is not difficult to prove the differential version of these properties, In fact, it is sufficient to show that (qu) dC! vanishes for any function ¿i this follows from the development L (uu) dC == dl pita) d( fu) = di fit) LdP a u +01) =d [E dF+ a) and relations (4). 76, la) LG), Y, and d denote, respectively, the divergence, contraction (interior product), orthogonal projection, covarian! derivative, and exterior diflerentiation operators. Y His definition in Ref. [1] diflers slightly from ours: see our comment below (Section db). ?Newtonian notation has been used for time-like derivalives: X= i(u) Vx, x being any tensorial quantity. 19 1.Y) denotes the Lie derivative with respecl to the vector field X. Relativistic Holonomic Fluids 1163 3." CONFORMAL GEODÉSICS: (a) When dealing with conformal structures, conformal geodesics are considered as the solutions to a system of ordinary differential equations lhát generalizes the geodesic one in such a way [4] thar ¡ls space of Solútivás is invariant under conformal transformations of the metric. Here, being directly concerned with fluid flows in given space-times, we shuil adopt another point of view: we will look for a metric-dependent churacteriaaiva ef Lhe congruences ol conformal geodesics by the differen- tiul system satisfied by their tangent unit vector field. (b) In the space-time (Y,, g), let u be a unit (time-like) vector field. The vector field"! u'=e*e is a unit vector field for the conformal metric g'=e*g, and its acceleration a' is related to the acceleration. a of y by: d=a—-da+du (7 When there exists 4 function a:such lhat u'=0, we say the vector field u for (he congruence- of its integral curves) is conformally geodesic. The Eisenhart theorem [37 follows: « unit vector field u is conformally geodesic Uf. there exist a function a, called the acceleration potential, such that de =u+du (8) (c) Unless its acceleration is already given in the form (8). it is not simple to know whether a field « admits an acceleration potential. To answer this problem one must find the conditional system? in 1 attached to the differential system (8); that is to say, the necessary and sufficient conditions that u must satisfy to insure that-the system (8) in a admits. a solution. : By Exterior diferentiation of (8), we have du+dé a u+ádu=0 so that, taking the exterior product by u, we obtain un da+ dun du=0 (2) Let w=x+(ua duj be the vorticity of x, + being the (Hodge) dual operator. When w docs not vanish, applying the operator ¿(w)*-lo (9) we find ¡(w) » (ua da) +4w*=0 *' Remember that according to our covariant convention, y und w' denole the corresponding g-assuciuled 1-(0rms. Some conditional systems for Maxwell equations and barotropic fluids may be found, respectively, in [10] and (117. Relativistic Holanomic Fiuids 1165 geódesic congruence of the metric g'=e*” g is such that its g-unit tangent vector veriñies Eq. (8). Thus, any function a is an acceleration potential of a family of conformal geodesics. 4, HOLONOMIC FLUIDS AND CONFORMALULY GEODESIC VELOCITIES l 3 (a) As we already suid, the solutions to thk conservation” équa- tions (4-5) far a holonomic fluid will be noted by (3, F. m). According to thé Eisenhurt theorem, Ale fist of these equations expresses the confor- mally geodesic character ol the' associated velocity, and shows that the ucceleration and holonomy potentiais difler at most by an additive con- stunt. Suppose now, given a conformally geodesic unit vector field y as solution of the system (13-14), we lot F be an acceleration: potential for it: the equation (5) in the unknown m always admits a solution, determined up-to a i“invariant factor, so that we have a solution (u, F, m) to the con- servalion equations (4-5). With this solution (u, F, 21), let us consider a conscrvativo encrgy lensor T und define T=T—muQ u; one then has: Sil = T 4 1ñu + m0 ni mUM +0) u—- 4l=mdFE thal is to say, 7 is a holonomic energy tensor with associated velocity 4, holonomy potential F. and holonomy density mm. Therefore, taking into account Proposition 3.2, we have: Proposition 4,1. To every conformally: geodesic velocity e can be associated a family of solutions (La, F, m)) to the conservation equations for holoftomic Nuids, such that: (1) the Fs are determined up to an udditive constant if w 0 und up to'a function of the potential of u if w=0, (1) For every pair a, F, the ms are determined up to a u-invariant factor, (ii) For every solution (4, F, 1), ny conservative energy tensor T is a holonomic. fMuid with associated velocity «, holonomy potential F, and: holonomy density sm. (b) In his definition of helonomic fuids (see footnote 8) Lichnerowiez considered the associalcd velocity as being an cigenvecior of the energy tensor 7, but he never used this fact in his development of 1he theory. For this reason, we have excluded it in vur definition 2.1; which we give respect to any associated velocity. In doing so, we arc able 1o separate the features which are necessary and sulficient for the existence of [lre integral invariants, [rom the lealures related to the particular character of the associated velacities, Here we are only consideririg the first ones, und it is to be understood that our results must be constrained by the definition O 1166 Coll and Ferrando equations of the specific Lagrangian- or Eulerian-associated velocities, when they are previously given. From this point of view, assertion (iii) of Proposition 4.1 says that a fluid is holonomic if, and only if, its associated velocity is conformally geodesic, For this reuson, and whenever hydrodynamics is concerned, the confermally geodesic velocities will be, cquivalently.- called holonemia ., velocities. The different characterizations of the holonomic. velocitics that we* have seen, are collected in the following proposition. : Proposition 4,2. The following statements are equivalent: i The unit vector field u is conformally gcodosic, li. There exists u function F such that a=dF- Fu. iii, There exists a function f such that C<= fu is. u relalive integral invariant. iv. There exists a a function f such that d(fu) is. an absolute integral. invariant. , v. The vector field u is either vorticity-free or a solution to our system (13-149). vi. The vector field 1 is the associated velocity of a holónomic Nuid. (c) It is. well-known, and cusy to sce, thal a hurotropic perfect fuid, P=(p+p uu pg. p=p1p), is a holonomic fluid with respect to 4. with holonomy potential x such that dp=(p-+ p) de. Thos, the variables 1, R, p+ pare a solution (w,x, p+p) to the conservation equations (4-5), and this solution is particulur in the sense that x and p + p are functionally dependent. This property can be gencralized: a solution (u, E, m)' lo thé conserva= tion equations (4-5) will be called harótrropie il ¿Fa din=0,- corre- spondingly, the velocities y giving rise to barotropic solutions (4, F,m1) will be called barotrupic velocities. Any trivial barotropicity (ie, F=F,, const.) corresponds to au gcodesic velocity and, conversely, every peodesic velocity 4 may be associated to a family of solutions ([(w, Fy,m)] with F, const. the holonomy density 1 being determined, up to a t-invariant factor, by M+0=0, When dF%0, by virtue of dFadm=0, we may wWrile F- M=FT[1— M'(F)], so that we have O= a) E (15) Conversely, if F is u holonomic-potential verifying (15) for some g(F), let M be a solution to lhe cquation M(4)=1-— g(F) then (a, Fm) is a 1168. Coll and: Ferrando. (6) In this paragraph: we will. consider vorticity-free: lows- of: ther- modynamic Muids, When a fluid admits a thermodynamic scheme,” pis decomposed in the form p=r(l +2), and the thermodynamic closure of the Nuid is obtuined by reyuiring that: (1) the I-form o=déi+pd(ifr) be integrable: a do =0; and (iij the current ru be conserved: ó(r4)=0. In terms of the variables w, p, p, a perfect Muid admits a thermodynamic scheme if! the differential equation in 4.41) h +8 =0-admits-u solution ef the form 4=4(p, p) [14] and, in the case p%0, it oucurs ie PIP isa function of state, thatis p/ó=(p, p) [157 We already know that, in the vorticily-frec case, 1 =t de, 4. is confor- mally geodesic (Proposition 3.1) with holonomy potential y= —In 1: a=dy — yu (19) Then, from (17-19) it. follows that cither the fluid is barotropic or the functions 1 and y lie-in the thermodynamical 2-plane (dp, dp). When 450, it follows that + and 1 are functionally independent; then we have x(dp, dp) = atdt, di) => ru, a) and, consequently, Hp, p;1,1)%0, where J denotes the Jacobian. Under this condition, ¡it is casy to sce Lhut the fluid admits a thermodynamic scheme iff. d(t/¿) a di a dr=0; but as u is a unit vector field, we have ¿=> *, so that, this condition may be written dia diadi=0, that is, dada di=djacaa=0. On lhe other hand, bucause ol (19), il resulls da+ di nu+p-du=0, se thal as e is vorticity-free, we have a a da=dj a u a a. Thus, we have shown: Proposition 5,2, A (nonbarotropic) perfect Muid with vorticity-free and nongeodesic proper velocity u admits a thermodynamic scheme if. ts acceleration « is an integrable form: qn du=0. Then, the velocity, acceleration, and holonomy potentials ure functions of state. When a=0, we have u=«t and, from (17, it follows dp =p dt, that is, 6= p'(1), Consequently, A/ó is u function of p and p il, and only il, $ does. Therefore, by virtue of cg. (4), and to the last characterization of a thermodynamic scheme, we have: Proposition 5.3, A (nonbarotropic) períecl Muid with vorticity-free and geodesic velocity u admits a ¡hermodynamic scheme if, and only if, dd A do a dp=0. (c) Now, let us consider 4 perfect fluid with a nonvanishing vorticity and holonomic (i.e., conformally geodesic) proper velocity, and let FF be Lhe holonomy potential. From (4) and (17), we can find a und y as combina- P It is Eckurt's scheme [12] (see also [13] and references thercin). Relativistic Holanomic Fleids. % 1169 ade : tions of dE and dp, and evaluate 1he elements-of imtegrability of the-2-plane: x(í a); thats; ira er de andtusa aa dix Where alu a) is not integrabie; one has Q6— F=0 and then: Qdp— HF =0: Thus, we have: Proposition 54. A perfect Muid with holonomic proper velocity for which the 2-plane rlu. a): is not inlegrable; 1 is barotropic:- On. the other hand, for: the: resulls. of: Section:3. we know tale the: holonomy potential F ia such that £= B[a]. where: ¿[1 is Lhe: function: of the velocily given: by: (14); taking into aceount (18), iu follows: : £= (1/0) ¿m) el da) = - (1/w* Yi (dQ on dp ac +06 From this expression, and the one obtained by solving in u. lhe system (4-17), we obtain: dE=Qdp—=uu, with p=(1/w) > (ao den dQ) (20) the converse being also casy to show, we: huve: Proposition 5.5. Let T=Q 'uQu-— pg be a: perfect fluid: wittr non- vanishing vorticity w, Then, « is a holonomic velocity if the 1-fosm O dp - pue is exa, where pu= (1/07) (16 a wa dp A dO). Applying the operator (4) +=d to the l-lorm Q dp e one obtains, by Proposition 5.4, w=(40%): «(ua dp a dph: a perfecto fluid: witb holonemic proper velocity has a. nonvanishing vorticity in so- far us. 1 separates from the 2-plane x(dp, dp). Finally, let us study the thermodynamical dependencé: ol: the holonomy potential. In (he vorticity-frec: case, we have shown (Proposi- tion 5.2 that lhe holonomy potential F isa function ol stale, F=F(p, p) say. Such u generic dependence is incompatible for nonvanishing. vorticity: from (20) it follows that e tics in the 2-plane z(dp, dp), Thus p=0 and, conseguently, F= Up): Proposition 5.6. la perfect Muid has holonomic proper velocity with nonvanishing vorticity, and if its holonomy potential is a function of state, then it. is barotropic. REFERENCES Lo Lichnerowacz, A. 11941). otom, Ec. Nori, 58, 285 304. 2. Synge, J. L. (1937) Proc. Londen Maih, Soe., 43, 316-416. 3, Eisenhart. L. P. (1924), Trans. American Math, Sor., 26, 205-220. 
