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Thermodynamic perfect fluid. lts Rainich theory Bartolomé Coll Département de Mécanigue Relativiste, UA 766 CNRS - Université de Paris VI, Paris, France Joan Josep Ferrando Departament de Fisica Tebrica, Universitat de Valencia, Burjassot (Valencia), Spain (Received 14 April 1989; accepted for publication 12 July 1989) The conditions for a relativistic perfect fuid to admit a thermodynamic scheme are considered, and the necessary and sufficient requirements for a metric to define a thermodynamic perfect fluid space-time are given. L INTRODUCTION Let g be the metric tensor of (a region of) a space-time, $ its Einstein tensor, and let (M,T) be the pair of the definition equations M of a medium and ofits energy tensor T. We call here Rainich theory of the medium the set of necessary and sufficient conditions on g insuring the existence of the pair (M4,T) such that the Einstein equations S = T (Ref. 1) hold. It is clear that this definition is nothing but a direct ex- tension to other media of results developed by Rainich? for the regular electromagnetic field; in it, T is the Maxwell- Minkowski energy tensor and M is the set of the vacuum Maxwell equations. Rainich worked out his theory about seven years after the Einstein paper on the foundation of the general theory of relativity,? where both media, the perfect fluid and the elec- tromagnetic field, were explicitly considered. It seems rather paradoxical that the perfect fluid had not, up to now, been the object of a work analogous to Rainich's one on the elec- tromagnetic field.* We would like to comment here on four of the factors that have contributed to this situation. (1) The apparent simplicity of the barotropic case. A Rainich theory involves two sets of equations: a first, alge- braic, set ensuring that S has the same algebraic structure as T, and a second, generally differential set translating in terms of g (and its differential concomitants) the definition equations M. In the barotropic case, the second set reduces to the expression of the functional dependence of the two algebraically independent invariant scalars of $, so that to complete the Rainich theory of the barotropic perfect fluid one only needs to know the algebraic characterization of the perfect fluid energy tensor. It is true that to obtain it is an easy task. Nevertheless, because of the Lorentzian character of the metric, it is not so easy a task as it has been evoked in the literature;' in addition to imposing T' to have a triple eigenvalue and be of algebraic type 1, one must give the con- dition insuring that to the simple eigenvalue corresponds a timelike eigenvector. For symmetric tensors, the general problems of finding the causal character of the eigenspace associated to a given eigenvalue, and its application to the perfect fluid, have been solved only very recently; we will need here these results. (ii) The apparent multiplicity of fluid thermodynamics. Both the equations of electromagnetism and relativistic con- tinuous media have been largely analyzed, discussed, and 2918 J.Math.Phys.30 (12), December 1989 0022-2488/89/122918-05$02.50 eriticized from the beginning of relativity. But, meanwhile, the matter for the electromagnetic field has been, in general, to find for it a nonlinear system.? For thermodynamic con- tinuous media, the matter has been to establish the basic sys- tem of equations, playing the role analogous to the Maxwell ones. And, as it is well known, there are many proposed versions for this basic system. This situation would indicate that thermodymamics is not yet ripe to be incorporated in relativistic continuous media. Nevertheless, Marle's work? pointed out in the opposed sense: many of these versions” may be obtained from a unique relativistic kinetic theory, their differences corresponding essentially to the different methods used to approximate the Boltzmann equation.!'" Furthermore, any two arbitrary versions differ in at least one of the following three aspects: the form of the conserved quantities (stress energy, momentum), the thermodynamic closure (generalized Fourier law, entropy balance), and the physical definition of the variables appearing in the equa- tions. What is important here for us is that, generically,'' the proposed versions, when reduced to the thermodynamic per- fect fluid, differ at most in the third aspeot,!? that is to say: the thermodynamic perfect fluid is generically unique, up to an eventual redefinition of some of its variables. (iii) The apparent independence of the thermodynam- ics from the energy tensor. In the usual presentation of the thermodynamic perfect fluid, the thermodynamic scheme is obtained by adding to the standard energy tensor a con- served matter current, an entropy relation, and an equation of state. It would seem that the existence of these three ele- ments could not be deduced from the metric and the energy tensor itself, so that a Rainich theory would not be possible. Nevertheless, we shall see that a unique condition from the energy tensor guarantees the existence of such a thermody- namic scheme. (iv) The wideness of Rainich's work. The work devel- oped by Rainich? to geometricize the electromagnetic field was, fortunately, superabundant. In particular, he revealed the (weighted) (2 + 2) almost-produet structure associated to the electromagnetic field*? and obtained the necessary and sufficient equations that the volume element U of the struc- ture must verify in order to have a solution of the Maxwell equations. As similarly, a perfect fluid has an associated (weighted) (1 + 3) almost-product structure, the extension of the Rainich work to the perfect fluid would implicate cor- respondingly the obtainment of the necessary and sufficient O 1989 American Institute of Physics 2918 Downloaded 23 May 2005 to 147.156.125.102. Redistribution subject to AIP license or copyright, see http://|mp.aip.org/jmp/copyright.sp equations that the volume element u (Ref. 14) of the struc- ture must verify in order to have a solution of the hydrody- namic equations. '* Rainich considered also the uniqueness of the Maxwell field, which he solved, but globally, the cor- responding uniqueness of the thermodynamic scheme would need to introduce some rather artificial ad hoc conditions. It is to palliate these features that we have chosen our above definition of a Rainich theory, which includes only a part of Rainich's work, The above analysis shows that a Rainich theory of the thermodynamic perfect fluid may be boarded. But, is it worthwhile? We think there are, at least, four reasons to constructit: (i) A general medium may not admit a Rainich theory. What are the media admitting it? According to Misner and Wheeler's geometrical point of view,'* the exis- tence of a Rainich theory would be a necessary condition for such a medium to be realistic. In any case, these media admit such a particular physical characterization [see (iv) below] that the question about the existence of a Rainich theory is already an interesting question. (11) A Rainich theory offers an alternative method'” of integration of the Einstein equa- tions: the set of all unknowns being reduced to the metric coefficients, the completed system of equations (the Einstein Ones plus those corresponding to the set M) is now an ouer- determined system (unless M =D), and the corresponding methods of compatibility conditions may be applied. (iii) This last consideration may be of interest in the study of those conjectures about perfect fluids which do not restrict the space of solutions of the hydrodynamic (test) equations, but restrict the space-times with which they are coupled;'* due to this fact, it seems plausible that the Rainich theory may help their analysis. (iv) In the penultimate phase of a Rainich theory, the set Mis reduced to a system of equations on the energy tensor: a medium which admits a Rainich the- ory isa medium which may be completely described in terms of the sole energy tensor variables. This fact may be ofinter- est for practical purposes;'? it is certainly of interest for con- ceptual and epistemological analysis. ?” In Sec, II we find a simple, necessary, and sufficient condition for a perfect fluid to admit a thermodynamic scheme (Theorem 1), and in Sec. II] we give the equations of the Rainich theory for it (Theorem 4), The barotropic and polytropic particular cases are given explicitly (Corollaries 2 and 3). The results without proof of this paper were communi- cated to the Spanish relativistic meeting E.R.E. 87.2! MH. CHARACTERIZATION OF THE THERMODYNAMIC PERFECT FLUID A, Thermodynamic scheme The energy conservation equations 97 =0 (Ref. 22) for a perfect fiuid T= (p+p)su— pg (Ref. 23) may be written dp=(p+p)la + pu, (1) (2+p10 +p=0, 2) where a and 0 are, respectively, the acceleration and the ex- pansion of u: i, O= — Ón. 2919 yl. Math. Phys., Vol. 30, No. 12, December 1989 From the evolution point of view, the system (1), (2) is open. A usual algebraic closure is obtained by imposing a barotropic condition p = p(p); however acceptable in some cases, it is known that this condition is too restrictive in many other interesting physical situations.?* The standard general closure to the energy conservation system is the dif- ferential closure consisting of a ¿hermodynamic scheme. Let r be the (Eckart) matter density”? of the fuid; de- noting by E=p—r the internal energy density and by €=E fr the specific internal energy, one has p=rll+6). [EY] When an equation of state, depending only on the inter- nal structure of the fluid, is known, €=E(pn), (4) the one-form de + p dv is integrable, v = 1/r being the spe- cific volume. Then, the temperature O of the fluid may be identified, by a classical argument, with an integrant divisor, and the specific entropy sis given, up to an additive constant, by O ds=de + pde. (5) As far as creation or annihilation of baryons do not take place,? the equation of conservation of matter holds: $(ru) =0. (6) The relation (5) allows us to write Eq. (2) in the form $(ru) = [10/f]5, 1) where f=1 + € + pvis the enthalpy index of the fluid;”” Eg. (7) shows the intimate relation existing between the local adiabatic motion and matter conservation. Ttis interesting to note that, while in classical thermody- namics, because of the nonequivalence between mass and energy, the internal energy Ey of a given volume Vis deter- mined up to an additive constant; in relativistic thermody- namics this energy is univocally determined once the matter density is given. However, this fact does not imply that the zero Of the internal energy E, be fixed in relativity; because ofits noninertial character, the matter density is only deter- mined up to a constant factor and, as a consequence, there still exists indeterminacy of E, by an additive constant. Thus, if Mand M' = kM denote two mass balances” of the particles contained in Vone has r = M /V, r = M'/Vandit resultsin E”, = (1 — k)M + Ey. This observation is perti- nent, for example, in the study of reaction fronts, where it allows us to localize conveniently the binding specific energy of the reaction,” or in the study of those hot perfect gases for which the limit 9 >0 is meaningless.*" B. Characterization theorem Einstein equations for the thermodynamic perfect fluid being not easy to solve, one often, in a first step, looks for a solution to the general perfect fluid and, once obtained, in a B. Coll and, J. Ferrando 2919 Downloaded 23 May 2005 to 147.156.125.102. Redistribution subject to AIP license or copyright, see http://|mp.aip.org/jmp/copyright.sp e2é(x)T— y+ be positive for any timelike vector x, where e denotes the sign of the quantity t* — 6ts + 8 tr T?, We are assuming that the perfect fluids considered here correspond to a macroscopic level of description. For this reason it is plausible to submit them to the Plebañski energy conditions, which state that, for any observer, the energy density is positive definite and the Poynting vector is non- spacelike.** In terms of, '£ and p, the Plebaíski conditions for the perfect fluid are equivalent to the inequalities —p
0. Taking into account the above two lemmas, one ob- tains the following theorem.* Theorem 2: In a space-time of signature — 2, a second rank symmetric tensor T' defines algebraically a perfect fluid submitted to the Plebañski energy conditions if, and only if, Q(7?— yT) =0, 4s>t, t<2y20, (18) 2) T>x, where t=tr 7, s=tr 7?,Q=I-— (1/4)g tr, and x is any timelike unit vector. The intrinsic decomposition of T may then be obtained according to the following result.* Theorem 3: The total energy p, the pression p, and the direction of the unit velocity u ofa perfect fluid energy tensor Tare given by p=12GBr-0, p=1/(y—0, (19) uxilx)T + px, where x=1/2(t +2) 2=[(4s-)/31'”, (20) and x is any timelike vector. B. General case Let us write R=Ric(g), r=trR, and s=tr R?; from Einstein equations, we have (Ref. 1) R= T'— 1/2tg so that F= —t= —tr Tahd s = tr 7?. Taking into account these values in definitions (20) and the expressions (19), the Jaco- bian of r and s with respect p and p is given by Jsp. p) = — 6(2y +r), which does not vanish under the third of the assumptions (18). Thus according to Coroilary 1, the perfect fluid admits a thermodynamic if (17) holds forA=randu=s. Iff =0, (17) holds trivially; ifF40, (17) is equivalent to d(S/t) AdrAds=0, Q) and we have to evaluate the scalar $/t in terms of the con- comitants R, r, and s of the space-time metric g. To doit, let us observe that the direction of the unit velocity +, as given by the third of the relations (19), is the image of the endo- morphism U given by U=ST +pg=R+1/4(2—1)g, (22) so that u = Ai(») U, where pis any vector field not belonging 2921 y). Math. Phys., Vol. 30, No, 12, December 1989 to the kernel of U: (y) U 40. Thus, for any function f we have f= ¡(u)df= (df)ju = Adf)i(y) U, in particular, tak- ing.f =r and y = dr, we have + = AF (dr) U, which vanishes only if dr belongs to the kernel of U. Also, for f =$ we have 3 = Ai(dr)i(ds) U and, consequently, 8/t=i(dedi(ds)U/P (dr) U. (23) On the other hand, let us note that the three inequalities expressed by the second and the third of the relations (18) are equivalent to 4s>r? and z>r, which are nothing but — 2811? r, and Pd) U=0 or dli(dr)i(ds) U /P(dr)U] AdrAds=0, where R=Ric(g), r=trR, s=trR?, r=1/4r + [(4s —12)/3]2), U=R + (r— 1/2)g and x is an arbitrary unit timelike vector field. As a corollary of Theorem 3, the total energy density p, the pression p, and the direction of the unit velocity u of the perfect fluid are then give by p=3r=r, p=w, uci()R+Y(7—r/2)x. C. Barotropic case Let us note that in the barotropic case, since the Jacobi- an J(r,s;p,p) does not vanish, the condition dp A dp =0 is equivalent to dr A ds = 0, Thus we have the following corol- lary. Corollary 2: A metric g is a barotropic perfect fluid space-time with the Plebañski energy conditions if, and only if, it verifies the algebraic relations of Theorem 4 and the differential equation dr A ds =0. Also, in the case of a polytropic fuid of index y, p= (y — Dp, itis easy to show the following result, Corollary 3: A metric g defines a polytropic perfect fluid space-time with the Plebañski energy conditions if, and only if, it verifies the algebraic relations of Theorem 4 and the equation d(s/r?) = 0. Then, ifs/r? =c, the polytropic index is given by y=1l4c 14 [(40— 1)/319/(3c — 1). ACKNOWLEDGMENTS One of the authors (J.J.F.) would like to express his thanks to the “Conselleria de Cultura, Educació i Ciéncia de la Generalitat Valenciana” for the partial support of this work. B. Coll and J. J. Ferrando 2921 Downloaded 23 May 2005 to 147.156.125.102. Redistribution subject to AIP license or copyright, see http://|mp.aip.org/jmp/copyright.sp *n pertinent units. 2G. Y. Raínich, Trans. Am. Math, Soc, 27, 106 (1925). “A. Einstein, Ann. Phys. 49, 769 (1916). “At least, we have not been able to find it. “The Taub conditions for a (1,1) tensor to be the energy tensor for a perfect fluid [A.H. Taub, “Relativistic Hydrodynamics,” in Lectures in Applied Mathematics (Am. Math. Soc., Providence, RI), 1967, Vol. 8, p. 170] though not explicitly stated, apply only in absence of a given metric, see our Sec. MIA. $J. A. Morales, Ph.D. thesis, Valéncia, 1988; see also C. Bona, B. Coll, and J. A. Morales, “Caracterización algebraica de un 2-tensor simétrico,” in Actas de los E.R.E.86 (Pub. Univ. Valéncia, Valencia, to be published). "Including the Maxwell equations as linear approximation, $C, Marle, Ann. Inst. H. Poincaré 10, 67 (1969); 10, 127 (1969). *The Eckart and the Landau-Lifchitz ones, among others. "Marle considers the relativistic versions of the Chapman—Enskog and the Grad classical, methods. "Here, “generically” means “for almost al! the versions that have been pro- posed in the literature.” Of course, there are always some exceptions; for. example, the Arzeliés fuids [H. Arzeliés, Fluides Relativistes (Masson, Paris, 197131. *For example, the Catteneo fluids [C. Catteneo, Rend. Accad. Naz. dei Lincei 46, Sér. VIII, 699 (1969) ]. "Rainich called it the skeleton of the electromagnetic field. "Here, this volume element is nothing but the unit velocity of the fuid. "This task is not easy. Restricted to the barotropic fiuid, ¡t induces an eight- fold classification of the unit velocity (see B, Coll and J. J. Ferrando, “On the velocities of the barotropic perfect fluids,” to be published). 16€, W. Misner and J. A, Wheeler, Ann. Phys. 2, 525 (1957). “Usually, the corresponding differential equations are presented in the form of Cauchy or underdetermined systems for the metric coefficients and some other additional unknowns (pression, electromagnetic field, ete.) "This ís the case for Lichnerowiez's conjecture on spherical symmetry un- der appropriate asymptotic conditions [see H. P. Kunzle, Commun. Math. Phys. 20, 85 (1971) and references therein], or the Treciokas-Ellis conjecture on vorticity-free or expansion-free consequences under distor- tion-free conditions [see R. Treciokas and G. F. R. Ellis, Commun. Math. Phys. 23, 1 (1971), or the more recent analysis by C. B. Collins, J. Math Phys. 26, 2009 (1985)]. 2922 y. Math. Phys., Vol. 30, No. 12, December 1989 "As an application to the Maxwell case, see, for example, B, Coll, F. Fayos, and]. J. Ferrando, J. Math. Phys. 28, 1075 (1987). "Fortunately, the development of field theory began, historically, with force field variables and not with energy ficld variables. Otherwise Max- well equations should remain undiscovered; to think so, a glance on the nonlinear Raínich complexion equations is largely sufficient. 21B. Coll and J.J. Ferrando, “Fluido perfecto termodinamtico. Su teoria '4 la Rainich',” in Actas de los E.R.E.87 (Pub. Inst. Astrof. de Canarias, La Laguna, Spain, 1988). 28, ¡(1), L(w), Y, d, *, denote, respectively, the divergence, interior prod- uct, normal projection, covariant derivative, exterior differentiation, and Hodge dual operators. Newton's notation is used for timelike derivatives: á=¡(u)V, x being any tensorial quantity. Of course, u is the proper unit velocity of the fluid, p the presion, and p the total energy density. 24€, B, Collins, J. Math. Phys. 26, 2009 (1985). %Also called rest mass density, proper mass density, baryonic (average) mass density or, simply density of the fui, “The definition of r as a mass balance of the baryonic number allows us to include in this scheme the study of the propagation of chemical reactions fronts; see B, Coll, Ann. Inst. H. Poincaré 25, 363 (1976). "See A. Lichnerowicz, Relativistic Hydrodynamics and Magnetohydrodyn- amics (Benjamin, New York, 1967). “Molecular, atomic, or baryonic mass balances. See the paper quoted in Ref. 26. 3"Tn such a case, one does not have necessarily € = C,8—0, and every y- law, p= (y — 1)p, may be interpreted as a polytropic perfect gas [see B. Call, C. R. Acad. Sci, Paris A 273, 1185 (1971)]. 3!The symbol X denotes the cross product; contraction of the adjacent spaces of the tensor product. Of course, the operator * setects the antisym- metric part of VRXR. 32]. A. Morales, Ref. 6. See J. Plebaúski, Acta Phys. Pol. 26, 963 (1964), especially his prudent analysis (pages 1011 and 1012) on the validity of his two conditions. la The large scale structure of space-time (Cambridge U.P., Cambridge, 1973),S. W. Hawking and G. F. R. Elliscall them the weak and dominant energy conditions, seeming to be unaware of Plebaúski's work. B. Coll and y. J. Ferrando 2922 Downloaded 23 May 2005 to 147.156.125.102. Redistribution subject to AIP license or copyright, see http://|mp.aip.org/jmp/copyright.sp 
