InhomIsotropia, Apuntes de Matemáticas

¡Descarga InhomIsotropia y más Apuntes en PDF de Matemáticas solo en Docsity!

PHYSICAL REVIEW D VOLUME 46, NUMBER 2 15 JULY 1992 Inhomogeneous space-times admitting isotropic radiation: Vorticity-free case Joan Josep Ferrando, Juan Antonio Morales, and Miquel Portilla Departament de Física Tebrica, Universitat de Valencia, 46100 Burjassot, Valencia, Spain (Received 5 December 1991) The energy-momentum tensor of space-times admitting a vorticity-free and a shear-free timelike congruence is obtained. This result is used to write Einstein equations in a convenient way in order to get inhomogeneous space-times admitting an isotropic distribution of photons satisfying the Liouville equation. Two special cases with anisotropic pressures in the energy fow direction are considered. PACS numberís): 04.20.Jb E INTRODUCTION The highly isotropic microwave background (interpret- ed as a relic of an early hot stage of the Universe) is con- sidered as proof that the space-time, on a sufficiently large scale, is close to a Robertson-Walker universe. This has been stated in a paper by Ehlers, Geren, and Sachs [1] (EGS) and summarized in the book by Hawking and Ellis (2], who also proved that space-time should be time- like or nuil geodesically incomplete before the decoupling of matter and radiation, supporting in this way the big- bang models. On the other hand, small deviations from homogeneity at decoupling time are necessary in order to understand the formation of the observed structures; and, after the work by Sachs and Wolfe [3], a pattern of aniso- tropies in the microwave background (MWB) tempera- ture is predicted. Let us dwell a little more on this basic subject, recal- ling the assumptions necessary to arrive at these con- clusions. The EGS theorem starts from previous results [4,1] characterizing the class of space-times allowing iso- tropic radiation: the metric should be conformally sta- tionary and the distribution function depends only on the first integral of the null geodesic equation, defined by the conforma! Killing vector. In fact this is all one can ob- tain from the existence of a collisionless isotropic distri- bution of photons. The result of EGS (see Corollary 2 below) for the relation of isotropic radiation to a Robertson-Walker universe arises when (i) the observer who sees an isotropic MWB is geodesic, and (ii) the ener- gy tensor is a perfect fluid comoving with this observer. However, conditions (i) and (ii) should be considered as good approximations to reality, not as exact prescrip- tions. So, we can modify one or both of them in order to get perturbed Robertson-Walker universes compatible with the isotropy of the MWB, All we have to do is to assume a conformally stationary metric from the begin- ning. Cosmology is not devoid of examples of these metrics. For instance, the metric used to obtain the Sachs-Wolfe effect belongs to this type. The question arising here is why Sachs and Wolfe established a well- defined relationship between inhomogeneities and aniso- tropies in the MWB temperature. We have considered this subject elsewhere [5]. Other examples may be found among the Stephani universes [6,7]. 46 In this paper we develop a framework suited to obtain more general inhomogeneous solutions of Einstein equa- tions compatible with isotropic radiation. For the sake of simplicity, we shall assume that the observer measuring isotropic radiation admits orthogonal hypersurfaces. Un- der these conditions, space-time allows a shear-free and vorticity-free timelike congruence, or equivalently, an umbilical synchronization. Conformally flat umbilical synchronizations exist in any space-time admitting natu- ral symmetric frames [8]. This occurs for any sphericaily symmetric space-time [9] and also for the Stephani universes [10] that include (he Robertson-Walker metrics, On the other hand, dealing with scalar perturba- tions of Robertson-Walker universes in the longitudinal gauge, one removes the background space-time sym- metrics bui not the existence of an umbilical synchroniza- tion [11]. Moreover, after a study made by Treciokas and Ellis [12], the shear-free and vorticity-free timelike congruences constitute a large class among the observers measuting an isotropic distribution function obeying the Boltzmann equation. The study of space-times allowing an umbilical synchronization is justified by this last result as well as by the existence of a lot of usual space-times with this property. The particular case where the ob- server is geodesic has been previously considered [13]. In Sec. TI, using the 3+1 formalism, we write in a suit- able form the Einstein equations for the space-times ad- mitting an umbilical synchronization. The four boundary equations and the six evolution equations become nine boundary equations (with the energy density, the energy flow, and the anisotropic pressures tensor on the initial slice as sources) and a unique second-order evolution equation (with the mean pressure as source). In Sec. 11] we summarize the known results of Ehlers, Geren, and Sachs in order to justify the canonic form of the metrics compatible with isotropic radiation. The re- sults obtained in previous sections are used to write Ein- stein equations when the observer measuring isotropic ra- diation is hypersurface orthogonal. In general, inhomogeneous metrics compatible with isotropic radiation will correspond to energy tensors with anisotropic pressures. Section IV is engaged in the study of two particular cases with the pressure tensor present- ing anísotropies only in the energy flow direction, as occurs in spherically symmetric space-times. Thus, in a 578 (21992 The American Physical Society 46 INHOMOGENEOUS SPACE-TIMES ADMITTING ISOTROPIC... 579 sense, solutions with such an energy distribution may be considered as a natural generalization of the solutions with spherical symmetry. Anisotropic pressures of this type have been used in galaxy models [14] and cannot be ruled out by observations of galaxy clusters [15]. Else- where [16] we have given the conditions which allow these energy distributions to be interpreted either as a mixture of two noncomoving perfect fluids or as a tilted perfect fluid. II. UMBILICAL FOLIATIONS ON SPACE-TIMES A, The Ricci tensor of an (N + 1)-dimensional metric Let £ be a metric on an (N+1)-dimensional manifold Vy +1- Let us consider a foliation with a unit normal vec- tor n, £ín,n)=e=41, and let g and K be, respectively, the first and second fundamental forms of a slice, namely, ¿=8-enon, K=—¿£,8, £, being the Lie derivative with respect to m. For every vector field transversal to the foliation, É=an +8, a%40, 818,n)=0, there exist lo- cal coordinates (x)=1x0x1),¿=1,...,N, such that E a =% $ L)=0, pg. ax Eo a B Po Then the line element of the metric £ may be written da? =(e0+B8B,Mdx P +28, dx 'dx + gy dx dx! . 50) Henceforth we follow the notation of [17] and all the ten- sorial equations will be written in covariant form. From the definition of K one has 08 =Lgg—20K . e) The Ricci tensor of a metric £ may be decomposed rela- tively to n as Ric(£)=0n8n +58 n +5, where o, s, and $ are, respectively, a scalar, a covariant vector, and a co- variant tensor, which are orthogonal to 1, and $ denotes the symmetrized tensorial product, a8b=a8b+b8a. From the Gauss-Codazzi relations it follows that the Ric- ci tensor of the metric (1) is given by Ln da= hvda—d Ina8d InQ+y(d Ina, d ny, OUR 74 tr0pR — LK cba), s=eldtrK=divK), a S=Riclg)+e(2K?*—trKK) + =leapK LD da], where D and Ricíg) are, respectively, the covariant derivative and the Ricci tensor of the metric g and where the following notation has been used: af Ox* K?=KxK=K!K,¡dxiedx!, dap=“Lax, Af=t0Ddf, eK=gWK, yo divK =g%DKydx*, S being any function and K any tensor orthogonal to ». Now we consider umbilical foliations (those for which the first and the second fundamental forms are propor- tional: K=—HIg). Taking the vector field £ orthogonal to the slices (8=0), from Eq. (2) it follows [18] that an (N+1)-dimensional space admits an umbilical foliation if and only if there exist local coordinates (x%,x') such that the element may be written di t=eadx YA 08y ¡dx dx! , e where o=0(x%,x0), (Mx1%x", and y ¿Sy ¿(«%. Let us consider the Ricci tensor of the metric (4). Tak- ing 8=0 and g =0?y, Egs. (3) become vo=-NH?—- Lina +eña), a s=e1—NJH , (0) S= Rictg)—€ NH?+ Lan 1 =—Dda, g a a where, on account of (2), H=2a,1m0 . 6) a On the other hand, taking into account the relation be- tween the connections Y and D of two conformal metrics y and g=0%y we have Ric(g)= Ric) HN—=20VA —[A InQ+(N —2)y(d ln0,d m0) ly , Mm RGI=0 UA RIY)IAN — DA mnO0+(N —2)y(d 1a0,d 1031), where R ( ) is the scalar curvature of the corresponding metric. Then, as 1 PUáf=Vdinf+d mfod Inf for every function f, substituting the two first relations (7) in (5) and considering Eq. (6), one obtains that in an (N +1)- dimensional space admitting an umbilical foliation the Ricci tensor is given by 46 INHOMOGENEOUS SPACE.TIMES ADMITTING ISOTROPIC ... congruence [4,1]. In both cases, if the observer is vorticity-free, it will define an umbilical synchronization in the space-time. C. The Ricci tensor of an induced metric admitting umbilical surfaces We now give a result which will be used later. In Sec. IV we study the boundary equations when the induced metric y on the initial hypersurface admits an umbilical foliation. In this case, the Ricci tensor follows from Eq. (8), taking N=2ande=-+1. Proposition 1. If a spacelike hypersurface (V, ¡y ) ad- mits an umbilical foliation, y=4Urx Udredr+Y Ur x “YY ¿gl dx tedx?, A=1,2 its Ricci tensor Ric(y)=Awev+18v+L, v=udr is given by A=-2 10+ 20, | staiconams ctm] , I=—dh, L=[-— Vd Inu—d Inpe d Inu +d InY8d Ing] (16) 1 —y? ¿RM Y 22+ lan] 4 —Axy In Y —Y(d Ing, d In Y) [Y where V y and Ay are, respectively, the covariant derivative and the Laplace operator of the metric Y, and 1 a h==3,InY, d=—dx?!. (17) B ax? IL SPACE-TIMES ADMITTING ISOTROPIC RADIATION A. Summary of previous results For an energy tensor of the form P=(p+pIMm8n+pg with a barotropic relation p=p1p), the conservation equation is equivalent to differential relations involving the unit vector x only [20]. As a particular case, one has the following result. a Lemma 1. The energy tensor T=(p+p)I8n+pÉ with P=3p is conservative if. and only if, the unit timelike vec- tor field n satisfies d o, (18) a—-£a 3 d being the exterior derivative of the space-time. Then, if a is such that dna=a 2, , the energy density and the pressure are 581 p=3p, p=pya*, with pg a positive constant. From the kinetic theory point of view, isotropic radia- tion is described by a distribution function f(x,k), a solution of the Liouville equation along the null geo- desics, and dependent on the moments k only through the energy with respect to the observer 1 measuring iso- tropic radiation. This function determines a macroscopic energy tensor P=(p+p)m0n +p8 with p=3p. Then, on account of Lemma 1, the unitary vector field n will verify Eg. (18). However, a microscopic description using the Liouville equation imposes additional conditions on the observer 1 which lead to the existence of conformal motions on the space-time. This question was analyzed in [1] and we summarize the results below. Lemma 2. The following statements are equivalent: (i) There exists an isotropic distribution function with respect to an observer n,f(x,k)=h(x,—n:k), which obeys the Liouville equation; (ii) there exists a unit timelike field n such that o=0, da —L6n)=0, o, 0, and a being, respec- tively, the shear, the expansion, and the acceleration of n; (iii) the space-time metric is conformally stationary, that is lo say, there exists a timelike field E such that LE =DE. Moreover, the unit vector n in (1) and the conformal Kil- ling vector in (tii) are collinear, EÉ=an, a being a function such that d na=a —l6n; in the case of particles withowt mass (k?=0), the distribution function depends only on the first integral defined by the field E over the null geo- desics [ f(x, k)=M(EX]. B. Einstein equations in the vorticity-free case Let us consider now space-times allowing isotropic ra- diation with respect to a vorticity-free observer (L.e., hy- persurface orthogonal, 1 Adn 0). On account of Lem- ma 2, there exists a hypersurface orthogonal timelike conformal Killing vector; that is to say, the space-time metric is conformal to a static metric. Then, the line ele- ment may be written a 01 NP y aid, where y is the initial spatial metric [0(0,x)=1]. On the other hand, a =£1W is an integral divisor of n, 2=—adx0; thus, taking into account Corollary 1, we have ¿ma=a—L(9,1m00n =a (3, 10m =a—L9n . a a 3 From this, and taking into account lemmas 1 and 2, we have the following result. Proposition 2. The space-times allowing isotropic radi- ation with respect to a vorticity-free observer are those for which there exists an umbilical synchronization de =— AN PA O y y la dla, such that 0/8 is independent of x". The observer field is n=(1/a)0) and the density and the pressure of radiation are given by p=3p, p =pya *, re- spectivelp, with py a positive constant. 582 FERRANDO, MORALES, AND PORTILLA 46 Then, we may write Einstein equations for a large fam- ily of space-times admitting isotropic radiation. In fact, taking a=£2WV(x') in Theorem 1, we have the following. Theorem 2. For a conformal static metric a WN Py ¡a dd, Einstein equations Rig) RDE=x [Mon +18n+mg +0], n= WO dx? may be written. Evolution equations: INGE, (19) dH=— vor y ye 30 HO 4 a mt yd In, d Inv) +24 04d In0,d In8%) +3y(d InY,d nO) -Zowr . (0) Boundary equations: «r=3 2407 En. 280 —r1a 10,16 , Qu xi=2dH , a» «U=[Ricty)—Vd InV—d Ye d Inv -2Vd ln +2d Infed n0] . (o) Here Y and Á are, respectively, the covariant derivative and the Laplace operator of y. Theorem 2 gives the general expression of Einstein equations in the 3+1 way for space-times allowing iso- tropic radiation with respect to a hypersurface orthogo- naj observer. If we want solutions modeling space-times with physical properties established beforehand, we have to impose conditions restricting the field variables. For instance, when the energy flow 1 is zero, the expansion H is homogeneous; thc nature of anisotropic pressures II will affect the spatial metric y and the spatial dependence of the functions Y and (2; an hypothesis about the mean pressure 7 will have an influence on the evolution of the metric. The important result of Elhers, Geren, and Sachs [1) quoted in the Introduction establishing a relation be- tween the isotropy of MWB and Robertson-Walker universes can now be obtained casily. In fact, when the observer n who sees isotropic radiation is geodesic and expanding, from (19) it follows that H=H(x%)*0, and (18) implies that » is vorticity-free. Thus the hypothesis of Corollary 1 (ii) is satisfied and we have the following. Corailary 2. Any space-time that admits an expanding and geodesic observer measuring isotropic radiation (solu- tion to Liouville equation), and that is a perfect fluid on an instant, is a Robertson-Walker universe. TV, SOLUTIONS WITH ANISOTROPIC PRESSURE IN THE ENERGY FLOW DIRECTION In this section we will study the Einstein equations for conformal static space-times when in the energy flow direction f there exists a pressure 7, Which is different from the transversal pressures 7, in the orthogonal two- plane. That is to say, we consider an energy tensor T=mmen+lilegn+r,ege +7 (gege) flejer=1. Q4 Tn this case, it is possible to substitute the mean pressure «7 and the anisotropic pressures IT by the radial and trans- verse pressures 7, and 7: 3r=w,+2m,, M=(m,—m, Meoe— lg). Qs) Thus, in the evolution equation (20) the freedom in the mean pressure 71 is replaced by a free election of the transverse pressure 7,, and (23) becomes a unique bound- ary equation. In order to obtain solutions with an energy tensor of type (24), we need to impose additional conditions to the metric. Let us consider two cases. Case TI One requires that functions (land Y depend on a unique spatial coordinate +: 0=Ma, rn, Vr). (6) Moreover one states that on the initial hypersurface x"=0 the surfaces 7= const are umbilical with geodesic normal and homothetic induced metrics. That is to say, the initial spatial metric y may be written y=dredr+YArYY nl dx tedx?, A=1,2. Q7 Conditions (26) and (27) lead to Vd Inb=(InVYdredr+(nv)YVdr, din=B dr, Vd in0=3,Bdredr+BVdr, 08) B=9,(100), d InV=(InvYdr, Vdr=hiy—dredri, h=inYY, where, for a function fr), f'=df'/dr. Then, taking into account Proposition 1 with p=1 and Y(r,x 1)=Y(»), the Ricci tensor of the metric (27) is —34?-2h'+ icty)=2 Ricty)= 3 y? GíY ( b E) 1 - . HS | arodr hy |. 09) GUY) being the Gauss curvature of the metric Y, G(Y)=2trRic(Y). Relative to n, the magnetic part of 584 FERRANDO, MORALES, AND PORTILLA 46 _20 Ta [ecanon— ¿0 , (49) where Y and A are, respectively, the covariant derivative and the Laplacian operator of the metric c, and £=d0 /dx0. The metrics (45) become the Robertson-Walker metrics when W=1, Thus taking V=1+4 with $<<1, we can study perturbed Robertson-Walker universes which ad- mit isotropic radiation. Y. CONCLUSIONS The main results of this paper are summarized in two theorems: we have written Einstein equations in the 3+1 way for space-times with a vorticity-free and shear-free observer (Theorem 1), and for space-times admitting iso- tropic radiation with respect to a vorticity-free observer (Theorem 2). As Corollary 2 i¡llustrates, if we want space-times compatible with isotropic radiation other than Robertson-Walker universes, we must allow for a nongeodesic observer or for sources other than a comov- ing perfect fluid. In Sec. TV we have considered a special type of aniso- tropic pressure that we think may have physically in- teresting applications. In the first case, the evolution equations become a strictly hyperbolic systerí and the numerical methods may be applied. Elsewhere [21] we have showed that in this case an evolution governed by an ordinary second-order differential equation is possible, and we have constructed a model of spherical cluster. A second case includes a family of perturbed Friedmann models compatible with an isotropic distribution of pho- tons, ACKNOWLEDGMENTS The authors acknowledge the financial support of the Spanish DGICyT, project No. PB90-0416. 1113. Ehlers, P. Geren, and R. K. Sachs, J. Math. Phys. 9, 1344 (1968). 2]5. W. Hawking and G. F. R. Ellis, The Large Scale Struc- ture of Space-Time (Cambridge University Press, Cam- bridge, England, 1973). B]R. K. Sachs and A. M. Wolfe, Astrophys. J. 147, 73 (1967). [4] G. E. Tauber and J. W. Weinberg, Phys Rev. 122, 1342 (1961). [53 J. J. Ferrando, J. A. Morales, and M. Portilla (unpub- lished). [6] H. Stephani, Commun. Math. Phys. d, 137 (1967). [7] C. Bona, in Proceedings of the Spanish Relativistic Meel- ing, Santander, Spain, 1984, edited by M. Portilla, J. L. Sanz, and R. Lapiedra (Universitat de Valencia, Valencia, Spain, 1985). [8] B. Coll and J. A. Morales, J. Math. Phys. 32, 2450 (1991). [9] H. Takeno, The Theory of Spherically Symmetric Space» Times, Science Reports uf (he Research Institute of Theoretical Physics No. 5 (Hiroshima University, Hiroshi- ma, Japan, 1966). [10] C. Bona and B. Coll, Gen. Relativ. Gravit. 20, 293 (1988). [1173 J. Ferrando, J. A. Morales, and M. Portilla (in prepara- tion). [12] R. Treciokas and G. F. R. Ellis, Commun. Math. Phys. 23, 1 (1974). [13]3. 3. Ferrando, 3. A. Morales, and M. Portilla, Phys. Rev. D 40, 1027 (1989). [14] A. M. Fridman and V. L. Polyachenco, Physics of Gravi- tating Systems (Springer-Verlag, New York, 1984). [15] P. Merrit, Astrophys. J. 313, 121 (1987). [16] J. J. Ferrando, J. A. Morales, and M. Portilla, Gen. Rela- tiv. Gravit, 22, 1021 (1990). [17] B. Coll, These Vétat, Paris, 1980. [18] L. P. Eisenharr, Riemannian Geometry (Princeton Univer- sity Press, Princeton, NJ, 1949). [19] M. Triimper, J. Math. Phys. 6, 585 (1965). [20] B. Coll and J. J, Ferrando, J. Math. Phys. 31, 1020 (1990). [213 3. J. Ferrando, J. A. Morales, and M. Portilla (in prepara- tion).