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On the velocities of the barotropic perfect fluids Bartolomé Coll Département de Mécanique Relativiste, UA 766 CNRS, Université de Paris VI, Paris, France Joan Josep Ferrando Departament de Fisica Teórica, Universitat de Valencia, Burjassot, Valencia, Spain (Received 16 August 1989; accepted for publication 8 November 1989) The conditions for a unit vector field to be the velocity of a relativistic barotropic perfect fluid are given. These conditions induce an eightfold classification of such fluids; for every class, the admissible barotropic variables are found. Some special cases, in particular polytropic fluids, are analyzed separately. L INTRODUCTION In relativity, a perfect fluid is characterized by an energy tensor Tof the form T= ( p + p)ue u — pg, where pis the total energy density, p is the pressure, and + is the (unit) velocity of the fluid, and g is the space-time metric. The con- servation of T' leads to a system of equations in (u,p,p), open from the evolutive point of view, which is usually closed by the adjunction of a barotropic relation p =p(p). So com- pleted, this system is called the fundamental system of baro- tropic hydrodynamics. Thus in a given domain of the space-time, a barotropic perfect fluid is a solution s=(u,p(p),p) to the fundamental system. Let us denote by U the set of unit vector fields u, by R the set of functions ofa single variable p = p(p), and by F the set of functions p over the given domain of the space-time. In the total space UXRXF, the space of solutions (5) to the fundamental system defines, by circumscription, a parallel- epiped U, XR, XFs. The Cauchy problem for the fundamental system shows that R, = Ror, in other words, that locally, eny function of asingle variable p(p) is an element of a solution (u,p(p),p) to the fundamental system.! Nevertheless, it can be shown that U, is a proper subset of U, U, 4U, that is, there does not exist, in general, a barotropic perfect fluid having as the ve- locity field an arbitrary unit vector field of U. Thus it is natural to ask the following question: Is it possible to intrin- sically define U, or, more precisely, is it possible to express, solely in terms of u and its derivatives, the necessary and sufficient conditions for y to be the velocity field ofa barotro- pic perfect fluid? The answer, as we shall show, is affirmative. The search for the conditions on u leads to a classification of the unit vector fields in eight classes. For each class, we obtain the necessary and sufficient conditions on u and its differential concomitants for insuring that u is the velocity field of some barotropic perfect fiuid. Furthermore, we give the holonomy potentials that allow us to determine the corresponding bar- otropic relations. Similar problems to that of the intrinsic characteriza- tion of U,, but restricted to particular forms of the barotro- pic relation or to particular evolution laws, may be also con- sidered. As an illustration, here we obtain the intrinsic characterization of the unit vector fields u that are the veloc- ity fields of (i) perfect fluids with constant pressure, (ii) 1020 y. Math. Phys. 31 (4), April 1990 0022-2489/90/041020-08$03.00 perfect fluids with constant total energy density, (ii) baro- tropic perfect fiuids with -invariant (i.e., constant along the streamlines) pressure, and (iv) polytropic fluids. For these cases, the conditions on u are simpler than those correspond» ing to the generic barotropic case. From a formal point of view, the differential system in defining the set U, is nothing but the conditional system in the variable u associated to the fundamental system of baro- tropic hydrodynamics. In other very different contexts, such as thermodynamic perfect fluids,? electromagnetic fields,? and almost-product structures or Killing tensors,* we have already shown the conceptual interest of conditional sys- tems. Now, whats the interest of an intrinsic characterization ofthe barotropic velocities in hydrodynamics? We think that such a characterization may be of interest in many domains, as, for example, in the following. (1) Our conditional systems allow one to divide the task of integration of the fundamental (test) system into two clearly defined steps: a first step in which, after selecting the desired class of velocities from our eightfold classification of the unit vector fields, one looks for a solution x to the corre- sponding conditional system and, once it is obtained, a sec- ond step in which, with the aid of our results on the holon- omy potentials, one constructs the barotropic relations p=p(p) associated to this 4. (ii) In the usual approach to the integration of the Ein- stein equations for barotropic perfect fluid space-times, one considers directly the Einstein system and its first integrabi- lity conditions; the problems of compatibility that appear because of the relation p(p) are well known. Our characteri- zation of U,, guarantees the existence of such a relation and allows one to relegate to a last, third step its computation: In a first step, taking local charts adapted to u, one translates the chosen conditional system in 4 into a system in the com- ponents of the space-time metric g; in a second step, for the corresonding constrained form of g, one evaluates its Ricci tensor and imposes that u be an eigenvector; and finally, in a third step, one considers the remaining Einstein equations with respect to the barotropic relation(s) computed from the g obtained in the first step. (iii) One of the few known results on the restrictions that the Einstein equations impose to the space of solutions of the fundamental (test) system is the Treciokas—Ellis* con- jecture, recently reconsidered by Collins.* The conjecture (O 1990 American Institute af Physics 1020 Downloaded 23 May 2005 to 147.156.125.102. Redistribution subject to AIP license or copyright, see http://imp.aip.org/jmp/copyright.sp states that a distortion-free barotropic perfect fluid space- time is either vorticity-free or expansion-free.? Because of its purely kinematical character, our associated conditional systems in 1 are well adapted to the study of this conjecture. (iv) In given (vacuum, Robertson—Walker, etc.) space- times, it is sometimes interesting to know if some particular congruences may be interpreted as the streamlines of baro- tropic test perfect fuids (e.g., weak accretion in the neigh- bors of a star). The answer to this follows directly from our results by a simple, direct computation. (v) Whatever its barotropic equation p = p(p), a (test) barotropic perfect fluid may always evolve following any (static or stationary) Killing direction of any space-time. Nevertheless, the analog statement for conformally Killing directions is false: In fact, the only barotropic perfect fluid thai may evolve following any conformally Killing direction of any space-time is that of isotropic radiation p = 3p in equilibrium with dust of constant energy density. Properties such as these may be easily obtained from our characteriza- tion of the barotropic velocities. (vi) Every barotropic velocity may be endowed with a barotropic relation p(p) and, of course, also with other more general thermodynamic relations. We think that in the study of nonbarotropic perfect fluids or nonperfect fluids (anisot- Topy, viscosity, heat conduction), the hypothesis that their velocities are barotropic may be useful in the study of the behavior of such fluids. Either this hypothesis is incompati- ble (the actual motion of the Huid cannot be reproduced by any barotropic test fluid) or it is acceptable (one can com- pare the ideal barotropic variables to the actual thermody- namic ones). Both results constitute an interesting comple- ment of information; in particular, the latter result may help us to better understand the limitations involved in the Eckart and Landau thermodynamic schemes. (vii) For the taxonomy of the solutions of the funda- mental (test) system and the Einstein equations, the eight classes of velocity vector fields not only allow one to label the known solutions, but also to play an heuristic role in the search of new solutions. The paper is organized as follows. In order to make the proofs of the main result easier, in Sec, II the case p = const is separate from the generic one, for which the data are re- duced to a unit vector field and a holonomy potential. Sec- tion II contains the main results of this paper: the eightfold classification of the unit vector fields (Definition 1), the characterization of the barotropic velocities corresponding to each of these classes (Theorem 1), and the associated equations for the holonomy potentials (Theorem 2). Final- ly, in Sec. IV we characterize the velocities corresponding to some particular cases often found in the literature: constant pressure or density, u-invariant pressure, and polytropic fluids. A portion of the present results (those leading to Theorem 1) with a sketch of the proof has been published elsewhere.? 11. THE BAROTROPIC PERFECT FLUID Let (V,,g) be the space-time sig(g) = — 2. Vector and tensor fields and the expressions that relate them, unless oth- 1021 y. Math. Phys., Vol. 31, No. 4, April 1990 erwise stated, are given in their covariant form. The symbols 1(u), +, d, Y, and 6 denote, respectively, the interior product, Hodge dual, exterior derivative, covariant derivative, and divergence operators. In a domain of (V,,g), the conservation 97 = 0 of the energy tensor Tof a perfect fluid amounts to the system dp=(p+pla+pu, 0+pP/(p+p)=0, (1 where a = ¿(u)Vu is the acceleration vector, expansion, and f"=£(u)f for any function f. A barotropic relation is a functional relation between p and p of the form dp Adp=0, (2) “When such a relation takes place (1) is called the fundamen- tal system of barotropic hydrodynamics. = — Su is the In the particular case of constant pressure Pp =p = const, system (1) becomes 2=0, 0+P/(p+D)=0. (0) Given u (and consequently, 9), the second of Egs. (3) asso- ciates one solution p to every ¿and to every u-invariant func- tion f(f*=0). Although simple, we explicitly state this result for completeness in the following proposition. Proposition 1: Perfect fluids with constant pressure have geodesic velocities. Conversely, to every geodesic (unit) vec- tor field u one can associate a family of perfect fluids with arbitrary constant pressure p and energy density p=f0+(f-— 1)p, where po is a given solution to 89 +p"/(p +6) =0 and fis any t-invariant function. From here on, unless otherwise stated, we have consider dp0. Perfect fluids with a barotropic relation such that dpX0 with be called barotropic ffuids. Because of (2), there exists a (local) function 7 verifying dp=(p+phdr. (4) This function is called the holonomy potential? For a non- constant p one has p=p(m), pr) =p+p%0, (5) where p'=dp/dw. From (4) and (5), the first of Egs. (1) may be written as dr=a+1Puz0, (6) the scalar p%/( p + p) adopts the form Pp = mp +p mr = [pm — pm 1/p (m), and the second of Eqs. (1) becomes 0=24 mw, (49) where gt) =1— (ln pm) Y. (8) Conversely, let be a function verifying (6), let p(+r) be an arbitrary function of 7, and define p() by pr) =p (mr) —p(m. We have dp =p'(m)dr = (p+pldrr, p? = (p+p)m and the first of Egs. (1) follows. Ifin addition, p(1) is a solution to (8), where g(7) is determined by (7), the second of Egs. (1) also follows. Thus we have shown the following proposi- tion. B. Coll and y. . Ferrando 1021 Downloaded 23 May 2005 to 147.156.125.102. Redistribution subject to AIP license or copyright, see http://imp.aip.org/jmp/copyright.sp fuña=dfMa, f'uña= —dfAu, 23 df Auñha=0, df*AuMa=0, es so that if du = u Aa and de = au Aa, one has PA JU af (24) Moreover, because of (21) and (23), the result is that all the scalars in (22) verify relation (24). From relation (24) it follows that a necessary integrability condition for Eqs. (22) ls BB -B"-aB*=0, (25) which, according to (22), gives uB? +8 + y=0, 06) where p4=-0%, y=0"40* 00%, y=00' —a. Let u be such that it verifies the hypothesis of Proposi- tion 7 with ¿e? + y? =0. Equation (26) then says that y van- ishes also and (25) becomes an identity. In this case, there always exists at least one solution to Eqs. (22); a simple way to see the solution is to consider an evolution problem with the constraint equation L=P4* — B— a =0. Taking into account the second of Eqs. (22) and (25), one finds L' = [2801 08) +0 ]L +48?+xB +7, so that since 4 = y =y=0, £? vanishes with L. Conse- quently, Eqs. (22) are in involution: If Bis a solution of the second of Eqs. (22) in a neighborhood ofa given instant and verifies the first of Egs. (22) at that instant, then it is a solu- tion to Eg. (22) in the neighborhood. Since the correspond- ing initial constraint admits a one-parametric family of solu- tions, we may state the following result. Proposition 8: A unit vector field u such that —wW4+e 424 y =0 and 9:4%0 is the velocity of a barotropic fluid if and only if it verifies 4g Au Aa =0 and y = 0. Equations (22) admit a one-parametric family of so- lutions 4, =f, [u]: For each of them, the one-form b, = a +8, is closed and the holonomy potential 7, is de- termined, up to a constant, by dr, = bz. Suppose now that y verifies the hypothesis of Proposi- tion 7 with? + 1740. If ¡440 the result is that from (26) a necessary condition for (22) to admit a solution is A=p — 4uy>0. 44) One then has f = fis, where Bs= (120 (—y +A. (28) On the other hand, if 4 = 0 (and, therefore, y 40,), the re- sult is that £ = A,, where Bi= —Y/X- Consequently, we have the following proposition. Proposition 9: A unit vector field u such that — ud +2? =0, 0:00, and 1? + xy? 40 is the velocity of the barotropic fluid if and only if it verifies either 440, x>%uy, (18), and (22) for $=8s as given by (28) or 4 =0, (18), and (22) for $ =8, as given by (29). In each case, the corresponding one-form b,=4 + KB, u, (i= 4,5) is closed and the holonomy potential 7, is determined, up to a constant, by dr, =b,. (29) 1023 +. Math. Phys., Vol, 31, No. 4, April 1990 Let u besuch that wo = O and 0-50. In this case, taking into account that du =u Aa, the exterior product of Egs. (11) by «a implies that a«Ada+dBAuMa=0, d(0/B) Nu Ma =0 and since 9:8£ 0, it follows that Bd0AuMa= — Ba Ada. The one-form z= — *(d8 Au Aa) does not vanish and is orthogonal to u. Consequently, 770 and [ = f¿, where Bs=(0/PrM(2+(a Ada). (30) Therefore, we may state the following proposition. Proposition 10: A unit vector field y such that w = O and 0-:240 is the velocity of a barotropic fluid if and only ifit verifies Eqs. (11) for 8 = f, as given by (30). Then the one- form b¿=4 + fógu is closed and the holonomy potential 7, is determined, up to a constant, by dir; = by. Consider now unit vector fields with w40 and da =0. By differentiation and the exterior product by um of dr =a + ue, one obtains u Ada + Mu Adu =0, that is, 7 =0; thus on account of (7), 9=0. Conversely, since da =0, let 7 be such that dr = a; then if 9 =0, rris a solu- tion to (9). Therefore, we have the following proposition. Proposition 11: A unit vector field y such that w*0 and da =0 is the velocity of a barotropic fluid if and only if it verifies 4 = 0. Then the holonomy potential 7 is determined, up to a constant, by dr= a. Finally, let us consider « such that w0%0 and daH0. Since the hypothesis of Proposition 3 is verified, the result is that u Ada + fuAdu =0 and since vw is a nonvanishing spacelike vector field, one has 1? 40; consequently, $ =P, where Bs=— (Wwditw)»(u Ada), Thus we have the following result. Proposition 12: A unit vector field u such that we da 40 15 the velocity of a barotropic fluid if and only if it verifies Egs. (11) for f =P, as given by (31). Then the one-form b¿= a + figu is closed and the holonomy potential 7 is de- termined, up to a constant, by d7¿ = bg. In the above we have obtained conditional systems in u for the barotropic fluids. These systems depend on the non- vanishing of some differential quantities associated to u and do not admit a unique simple form valid for any unit field. On account of the above results, we are lead to introduce the following classification of unit vector fields. Definition: A unit vector field + is said to be of class C; (í= 1,...,8) if it verifies the relations given in Table 1, where we have written w==*(4Adw), a=i(10Vu, 6= —Ón, 8=1060, a=(1/aJi(aJi(u)da, p=-—0%, y=09% 4 a*-a0*, andf*= £(w)f f* = (1/0?) £(a)f for any scalar f. The results of this section may then be summarized in the following two theorems. Theorem 1 (of characterization of barotropic veloc- itles): A unit vector field w of class C, (í= 1,...,8) is the velocity of a barotropic perfect fluid if and only ¡fit verifies the differential system A, given in Table IL, where the scalar B, (¡=4,5,6,8) is defined by (79) B. Coll and y. J. Ferrando 1023 Downloaded 23 May 2005 to 147.156.125.102. Redistribution subject to AIP license or copyright, see http://imp.aip.org/jmp/copyright.sp TABLE L. The eight classes of unit vector fields. Class Definition relátions c 0,8=0 C; ,9%0,4=0 C, ),0%0,4%0,0Ada=0,4 + P=0 C, ,0%0,440,0Ada=0, + Y 40,1 =0 Cs ), 00, a%0,0Ada=0, 4 + 140, 40 Cs , 040,040, aAdaz0 Cc wx0, da =0 CG 10%0,daj0 Ba=(—a0%/x Bs=(VMM—x14'?), Bs=(0/2P)i(2)*x(0 Ada), fBj=— (WuP)i(w)*(u Ada) and we have written A=yf +4u(a 00%), 7= —+*(d8NuAMa). Theorem 2: The holonomy potential 7 associated to a barotropic velocity of class C, (í= 1,...,8) is determined by the relations P, given in Table III. Let g(7) be the function such that 9 =g(1)7" and take ptr) = f 0xp f 11 —- sm) 1er Jar, pl) =p (7) — p; the triple (u,p,p) is then a barotropic perfect fluid. IV. SOME SPECIAL BAROTROPIC MOTIONS: THE POLYTROPIC CASE In many cases one may be interested in disclosing a more restricted character than that of barotropy. In this section, we study the following types of particular barotropic perfect fluids: (1) constant pressure dp = 0; (ii) constant total ener- gy density dp =0; (iii) u-invariant pressure (and density) p' =p? =0; and (iv) polytropic fluid, p = (4 — 1)p,A 41. We shall see that the characterization of these cases is easier than the general barotropic case. Proposition 1 already characterized fluids of type (1); such fluids also belong to one of the types (1i)-(iv) if and only if 8 = O, so that (1) may be stated in form of the follow- ing proposition. Proposition 13: The necessary and sufficient condition TABLEJI. Differential systems characterizing the barotropic velocities of class Cy. Symbol Necessary and sufficient conditions B, ? B, d0Au—0 B, — déNuMa=0,08%-4=0 d0NuMa=0 Bo pr=Brro BIBI ON +Bla + O) » dONuña=0, 470 , Bi=Bs+a Br =B3(1-0%)+Bla4 0%) Be día + Bgu) =0,d(0/B,) Ata+fB,u) =0 B 0=0 Be día + Bad =0, (0/8) Ala+ 8,0) =0 1024 y. Math. Phys., Vol. 31, No. 4, April 1990 TABLE lIL Characterization of the holonomy potentials for a barotropic velocity. Symbol Characterization of 7 P, — r=-Inr+AD,(u=rd0) P T=00,(u=rd0 dr, =a + Bu, where 8, is the one-parametric P, family of solutions to the system Br=B40,B9= Bl 0%) + B(a4 0% P, dia +84 Ps dí, =4+Bsu Po di =4 + Bau Pb de Pa dr= a+ Bu for a unit vector u to be the velocity of a perfect fluid with constant pressure and verifying one of the conditions (1í)- (iv) is that u be geodesic and expansion-free, Now, let dp0. From Proposition 2, the barotropic re- lation p =p(p) depends on the function g(7) given by (7); indeed, Pp =p (mp (rm) = —glr(p)). Thus one has g(77) = constifand only if p is a linear function in p. Itis then easy to see that cases (ii) and (iv) are charac- terized as in the following proposition. Proposition 14: The necessary and sufficient condition for u to be the velocity of a barotropic ftuid with dp = 0 and dp40 is 8 =0 and dí =a + 74 for some function 7. Proposition 15: The necessary and sufficient condition for u to be the velocity of a polytropic fluid with index 4 is the existence of a function rr such that (4,77) is a solution to (9) with g(7) =(1—A4)7?. Tn case (ii), because of p? =p? =0, one has m=0, which by (9) leads to € = O and da = 0. Since the converse is also verified, one has the following proposition. Proposition 16: The necessary and sufficient condition for u to be the velocity of a barotropic fluid with p? =p? =0 is 6=0 and da =0. Then the holonomy index is deter- mined, up to a constant, by dr =4. “When the conditions $ =0, da = O are verified for every function p(7), the triple (4,0,p) withp = p'(1) — pisa bar- otropic fluid verifying p*=p?"=0. Consequently, every function p = p(p) is admissible as a barotropic relation. By additing suitable conditions to the systems B, of Ta- ble IL, one may associate barotropic relations of types (ii) (iv) to unit vector fields of class C,. According to Proposition 16, the velocities of the classes C, and C; are of type (iii) if they verify de = 0. Consequent- ly, these velocities admit any function p(p) as a barotropic relation and the velocities of class C, (with da 40) and class C, (with 6 = 0) are of constant energy density. The velocities of classes C¿ (resp., Cs) with the additional conditions 640 and 6/8,=const (resp., 6 /B4 = const) admit polytropic barotropic relations, The velocities of classes C,, C,, and Cs admit a polytro- pic barotropic relation if B=k-8 is a solution to the system (22), where k is a constant. One then has a/(0— 9%) = const. B. Coll and J. J. Ferrando 1024 Downloaded 23 May 2005 to 147.156.125.102. Redistribution subject to AIP license or copyright, see http://imp.aip.org/jmp/copyright.sp