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PHYSICAL REVIEW D VOLUME 40, NUMBER 4 15 AUGUST 1989 Method to obtain shear-free two-fluid solutions of Einstein's equations Joan Ferrando, Juan Antonio Morales, and Miquel Portilla Departament de Física Teorica, Universitat de Valencia, 46100, Burjassot, Valencia, Spain (Received 8 May 1989) We use the Einstein equations, stated as an initial-value problem (3+1 formalism), to present a method for obtaining a class of solutions which may be interpreted as the gravitational field pro- duced by a mixture of two perfect fluids. The four-velocity of one of ihe components is assumed to be a shear-free, irrotational, and geodesic vector field. The solutions are given up to a set of a hy- perbalio quasilinear system. I, INTRODUCTION In many subjects of astrophysics and cosmology the energy content is described by a single perfect fluid, when, in reality, one should distinguish two or more cora- ponents. So, for example, dealing with the supernova phenomena neutrinos interact with the leptonic part of the matter. In cosmology the coid-dark-matter models distinguish a weakly interacting component and a baryonic one, lu the standard cosmology, after decou- pling of matter and radiation, one has a perfect-fluid en- ergy tensor describing an isotropic radiation, and a per- fect pressureless fluid describing the baryonic matter. The reason why only a vector field is considered for representing the energy content is that one expects that, because of the interaction between both components, equilibrium is soon established. However, if the two components are weakly interacting, the time taken to reach equilibrium may be significant. This situation has been outlined by Bayin' for the early times of a neutron star, giving some analytic solutions of Binstein's equa- tions in the case of spherical symmetry. Letelier? presented a method for solving Einstein's equations in the case that each fluid component is irrotational. The aim of this paper is to construct solutions of Einstein's equations interpretable as a mixture of perfect fluids, using the initial-value formmulation of general rela- tivity (3+1 formalism).** This method is more con- venient in order to construct solutions numcrically, when sufficient information about the energy-momentum ten- sor is available. But it is also useful in order to simplify Einstein's equations when a family of three-dimensional slices with certain geometrical properties is assumed to exist. So, for example, Stephani and Wolf presented a method for finding perfect-fluid solutions with flat three- slices and an extrinsic curvature proportional to the metric tensor. General properties of solutions admitting a spacelike folíation of constant curvature are discussed by Bona and Coll. * From the point of view of Einstein's equations, a two- fluid source is equivalent to considering a nonperfect fluid, with an anisotropic pressure tensor and a four- vector of “heat propagation.” The latter is interpreted as the propagation of energy due to the second fluid, if the 40 relative velocity of both components is different from Zero. The main assumption of this paper is to require a sim- ple kinematics to one of the components: no rotation, no distortion, no acceleration, only the expansion is allowed to be different from zero. This is equivalent to requiring the existence of a slicing with extrinsic curvature propor- tional to the metríc. So, in a sense, the intention of this work is similar to the Stephani-Wolf paper. The main difference is that we are looking for non-perfect-fiuid solutions with slices not necessarily flat. This simplifies considerably the field equations but produces a problem. Which are the initial conditions in order to guarantee that one of the two components moves in the way prescribed? We do not know if we have solved the prob- lem in general, but at least we have the method to con- struct a large family of solutions. We call these solutions “frozen” solutions, for the relative velocity field between both components always keeps the same direction. 1, INTIAL-VALUE FORMULATION OF EINSTEIN'S EQUATIONS The following conventions will be used throughout the paper: units are such that 41G =c =1, the signature of the metric is taken to be (— +++), and greek and latin indices run from 0 to 3 and 1 to 3, respectively. Let us assume that the space-lime may be foliated. So, it is possible to use a coordinate system (%,x1,x?,x%) such that on the surfaces of the foliation we have x'=const. Let n* be the unitary vector orthogonal to these surfaces. The expression in coordinates is usually given ín the form 2 Lg ao a nt= Each tensor is split into orthogonal and parallel com- ponents with respect to the covariant unitary vector x. (Henceforth all the tensorial equations will be written in covariant form.) So for the energy-momentum tensor we have T=tmen+Ten+aeT+T, 1027 (81989 The American Physical Society 1028 FERRANDO, MORALES, AND PORTILLA where 8 represents a tensorial product, Fand 7 are a co- variant vector and a covariant tensor orthogonal to n. In the case where n is a temporal vector, f will be the energy density in the proper reference system. Einstein's equations are also split with respect to each hypersurface of thc foliation, given a set of boundary equations, in which only “spatial” coordinate derivatives appear, and a set of evolution equations with respect to the coordinate x%, The boundary conditions may be writ- ten as (trKP—trK?eR =41 , (1 divíK —trKy)=—e2T . 102) And the evolution equations are (d)—Lgy=-2aK , 16) (0p—Lg)K =20a [== = trTy |—caRicty) +o(trKK —2K 30 +eVda , (4) where e=5?; y and K are the first and second fundamen- tal form of a slice, namely, y =g —engn, K =—Va; Ricly) is the Ricci tensor corresponding to metric y and R =trRic(y), where tr denotes the trace operator; Y is the covariant derivative with respect to the metric y; and Lg is the Lie derivative along the shift vector f£'. In Eq. (4), m is the dimension of (he manifold. In this case m4, but below, in Sec. V, we will need to take m=3. It is well known that, as a consequence of the Bianchi identities, the divergence of the energy-momentum tensor must be zero for any solution of the field equations. So we can write the following consequences of the field equa- tions (0) Ly =at trK —21(Mda—a div —eatríK xD), (5) (0) —LgM=—adivi—i(daF+eda+tarKF, (6) where ¿(F), div, X, are the interior product by T, diver- gence operator, and the contracted tensorial product, re- spectively. Next we give all the quantities appearing in these equa- tions in indices notation: trK =y UK, K?=KXK=KPK, 7 rK0=K'"K,,, divK =ViKydx'!, T: dxledx!, mp (7) TIL, SHEAR-FREE AND GEODESIC SOLUTIONS In this section we assume that the unitary vector field normal to the foliation is temporal, geodesic and shear- free. Then we take a=1 and e=—1 in Egs. (3) and (4). We choose a zero shift vector f8'. The shear-free condi- tion allows us to write the extrinsic curvature in the form K=-Hy. (8) Then, the boundary conditions are CGP+R=4t, (9) dH= (10) Let us write the three-dimensional metric y in the form 7=0%7, an where Y is the metric on the initial hypersurface, and N a function defined on the space-time, taking the value 1 on the initial surface. Equation (3) can be written in the form 30=0H (12) and substituting Eq. (8) into (4) one gets 9H =—H LU +0), (13) Ric(y)-4Ry=2UE 4 trip). (14) From Eqs. (3) and (6) we get Ot =— divi B(tri+ 30, as) di = —3H1— divi. (16) The latter are consequences of Einstein's equations writ- ten above, but it is very useful to consider them. Let us write some of these equations in a more con- venient form. Using the Eqs. (10) and (13) one gets NIH 4d an, 0) where we have introduced the variable 7 defined by JAY, 18 which is the difference between the energy density and the “critical energy density.” Taking into account Eqs. (13) and (15) we obtain an evolution equation for y: dy =—divF—2Hx . (19) Combining Eqs. (12) and (20) we get IO )=— (divo, (20) And Eq. (13) may be rewritten as YH =P = quan. Q1 Equations (12), (21), (20), and (17) form a quasilinear sys- tem of differential equations for the six variables 2,B,7,1. One can verify directly that the solutions of the equations of evolution verify at each instant the boundary conditions (9) and (10). The trace of the tensor Tis not determined by the field equations: then it must be an in- put of the problem. A solution of the system is charac- terized by the initial values (values on the initial sheet) 0=1, Hoyt» To. The requirement 7 =dH, must be satisfied on account of the boundary condition (10). The initial metric, í.e., 7, is constrained by the condi- tions (9) and (14), which can be combined in a sole equa- tion: Ric?) R(P7=277— 44) 2H DP] - 22) This equation is just a threc-dimensional Einstein equa- 1030 ing into account this result, and considering again Eg. (12) we get H =H(x%,H¿). Substituting T=T%, into Eq. (21) one gets that y +trTis also a function of x% and Hy. Finally, from the relation d inf « dH,, we get Ina ax “0H, ad n0)= dHo, and considering dy(dInfM)=TT, one gets T=(91n0)/(3x%9H8,)=0H /0H,. Using this lemma, we can state a sufficient condition to guarantee the two- perfect-fluid condition. Theorem. A solution of Einsteio”s equations, verifying the condition F“ ET and taking into account the condition (ii), one gets Vd In£2=(A98+80T,8 7, +u0B7 (34) with df = EN . 65 By substituting Egs. (33) and (34) into the expression (26) we get the characteristic energy-momentum tensor of a FERRANDO, MORALES, AND PORTILLA 40 two-component ftuid [Eq. (29)], with a=L04rr-LRPT?, (36) pU=L20 + BAB, (37) The following remark will be very useful in order to solve the three-dimensional Einstein equations. Let us write To=x0, with v a unitaty spatial vector, v1=+1. It is easy to prove, as a direct consequence of the requirement (ii) above, that v is geodesic, irrotationa!, and shear-free, and its length x depends only on Hp. This means that the initial sheet x%=0 admits also a foliation with a unitary normal v which is also shear-free and geodesic. Therefore we can try to solve the field equations by the same procedure we used at the begin- ning of this paper, that is we shail split the three- dimensional Einstein equations with respect to the folia- tion defined by the vector field b (2+1 formalism). A resume of the results of the section is presented in Table IL V, THE INFTIAL METRIC FOR TWO-PERFECT FLUID SOLUTIONS To determine the initial metric 7, one must solve the tirce-dimensional Einstein equations (22). We will devel- op the idca stated at the end of the previous section, that is to solve the three-dimensional Einstein equations by applying the 2-1-1 formalism. Taking into account the condition (i) above, one can write the soutce tensor 7 in the form Tb TT, Hb EP (38) The basic variables are, as is well known, the metric in- duced on the sheets, 9=)—v80 (39) 11,0=-—%v, which, and its extrinsic curvature, K= TABLE il. Shear-free and geodesic two-perfeci-fluid solutions. Initial metric Initial conditions Three-dimensional fieid equations Ah PPP), E =dHo Mp such that —3diy?, +d49y=0 Rio(y)—+R Y =2ULo — 4907) Evolution Initial conditions Arbitrary function Evolution equations L0=1,% Ho,D0, To=dHo 2 0=0H a I= —Ug+en IO) = — MPdiv(TT,) T=H' Y=T% Energy-momentum tensor T=in8n +108n +1817+1, I=ay+bier Ma = 1077 or? PT 020 +8 A 40 METHOD TO OBTAIN SHEAR-FREE TWO-FLUID SOLUTIONS... 1031 taking into account the fact that it is geodesic and shear- free, will be of the form K=-h0. (40) Let us introduce coordinates x,p,z, adapted to the foli- ation, with lapse function equal 1 and the shift vector equal to zero. Tn these coordinates we can write 11.0 v=dz Y= 0 Sn , (41) where 4,B=1,2; x'=x, x2=y. The tensor 7 can be written = 7% = = 2 T1=1090+780+7, T=2bygl— Lap > (142) 7=0, += Hbnd+aé . The three-dimensional Einstein equations can be split us- ing Eqs. (1)-(6) with m=3, and e=-+1, in a set of boundary conditions trRic(9)=24?47, (43) d¿h=0 (44) and a set of evolution equations 9,0 =240, (45) a,£ =25+—trr0)-Ric(8) . (46) The Bianchi identities give us a EAN 27h +hó, (47) ditrroedz, (48) where we have written £iz)=trf. All of the two- dimensional metric is conformally flat, and its Ricci ten- sor is proportional to the metric, So, we introduce the conformal factor « by ó=w%8, (49) where 5 is the flat metric. Then Eq. (43) is equivalent to Ria(6)=(22-2 6 (50) or what is the same Asino=(27—hk%)w?, (51) _ where Aj is the Laplacian operator corresponding to the metric 6. Substituting (50) into Eg. (46) and taking into account that trr=-+tr*, we write the latter in the form DAS Htrr (52) From the boundary condition (44), we know that A only depends on z; therefore we can write ur A2I4x 1,2, with -(z) taken to be equal 1 on the initial surface =0. Equation (51) can now be written Asing=1(27—k ¿19? (53) and the evolution equations simplified to dr A, (54) dh y2 e =P (55) This system admits a constant of motion 3,1(4?2—27)0?]=0 9) from which we can get rá ar? T=(279—h3) +41. (57) For some purposes it will be useful to introduce the radial coordinate + =+ (2); then we can write Y ¡dx dx i=eN"dr + rg Ux 08 pdx tdx?, 2 (58) de Mri= e dr From Egg. (54) and (55), it is easy to get the equation for 20, de” dr and using (58) and (54) one determines A as a function of n A =2rÉ0), (59) =p, (60) Finally, from Eq. (42) we get 2 ” dar br IE, (610 r 2) rá m9 — (279 hi A) (62) ” that, with Egs. (57) and (38) determines the source r of the three-dimensional Finstein equations. VI THE EVOLUTION EQUATIONS FOR TWO+FLUID SOLUTIONS Here we study the evolution of the three-dimensional metric with the time x%. For that, we need to solve the quasilinear system given by Egs. (12), (21), and (20): IAH, YH =P HH, INN) =— Min TE) Substiruting in the last equation QOiv(T)= TE MAA pp) we get Da=a te AB. (63) The space-tíme function trf in the evolution equation is related to the sum of the pressures, which we denote by a, by Egs. (36) and (37), .e., 
