Generating Solutions of Einstein s Equations

Transcription

1 WDS'09 Proceedings of Conribued Papers, Par III, 13 19, ISBN MATFYZPRESS Generaing Soluions of Einsein s Equaions J. Hruška Charles Universiy, Faculy of Mahemaics and Physics, Prague, Czech Republic. Absrac. In his aricle he mehod of conformal ransformaion is used o generae perfec fluid soluions of Einsein s equaions from vacuum spaceimes. The condiions on he conformal facor in order o generae a perfec fluid spaceime are presened. Wih Minkowski as a seed spaceime, hese condiions are applied and a paricular soluion is found. I belongs o he family of Sephani spaceimes and is physical properies are furher invesigaed. Inroducion Einsein s equaions governing general relaiviy as a heory of graviy form a sysem of 10 non-linear second order parial differenial equaions for 10 unknown elemens of he meric ensor. This sysem is very complicaed o solve in general, bu here are many known soluions wih symmeries, which allow us o carry ou he calculaions. We can hen use he symmeric soluion and ry o ransform i ino a new one, perhaps wih less symmeries. The generaing mehod chooses a class of such ransformaions. Conformal ransformaion In his paper he generaing mehod uses conformal ransformaion g µν = f 2 g µν, (1) where g µν is he original (or seed ) meric and f is a scalar funcion. I is well known ha he new meric g µν preserves null cones of he seed spaceime and herefore has he same causal srucure. Afer choosing he seed meric, we have one degree of freedom in f o creae he new spaceime. Tha is usually enough o break he symmeries of he original spaceime, bu here are also some limiaions. For example, wo Einsein spaces ha are properly conformally relaed (f is no consan) mus be boh vacuum pp-waves or one Minkowski and he oher de Sier, as was proved by Brinkmann [1925]. Dafardar-Gejji [1998] generalized his heorem o he case of wo properly conformally relaed spaceimes wih equal Einsein ensors and showed ha boh spaceimes are generalized (no necessarily vacuum) pp-waves. Afer a conformal ransformaion, he new Ricci ensor R µν can be expressed in erms of f and quaniies from he seed spaceime as follows R µν = R µν + 2 f f ;µν + ( 1 f f 3 ) f 2 df 2 g µν, (2) where he covarian derivaive ; is also aken in he seed spaceime, f := g µν f ;µν and df 2 := g µν f,µ f,ν. In our case, he seed spaceime is chosen o be vacuum, i.e. R µν = 0. Afer specifying he desired form of he new sress-energy ensor T µν, we will obain resricions on he conformal facor f. Fluid from vacuum The new sress-energy ensor should no be vacuum because of he Brinkmann heorem. We choose o generae perfec fluid spaceimes, as did for example Loranger, Lake [2008], who sared wih an unphysical seed meric and obained a family of saic fluid spheres. The sress-energy ensor of a perfec fluid has he form T µν = (ρ + p)u µ u ν pg µν, (3) 13

2 where ρ is he energy densiy, p pressure and u µ he 4-velociy of an elemen of he fluid. The meric signaure is (+,,, ) here. The energy and he pressure mus also obey some energy condiions. We may for example choose he Plebanski (aslo called dominan) energy condiion, which saes ρ < p ρ for perfec fluids. Coll, Ferrando [1989] gave necessary condiions on a ensor o represen he sress-energy ensor of a prefec fluid wih he above menioned energy condiions. In order o generae perfec fluids, we mus ensure ha he Einsein ensor G µν = R µν 1/2Rg µν fulfills hese condiions as well, specifically: G µ ν has one single eigenvalue A = 8πρ which belongs o a imelike eigenvecor and a riple eigenvalue B = 8πp. The energy condiion ranslaes ino A B < A. (4) We now use eq. (2) o compue G µ ν in erms of he conformal facor f and he seed meric: G µ ν = R µ ν 1 2 Rδµ ν = 2ff ;µ ;ν + ( 3 df 2 2f f ) δ µ ν. (5) This allows us o rewrie he above menioned condiions for he Einsein ensor in erms of he funcion f and is derivaives. The relaion beween G µ ν and f ;ν ;µ shows ha hese 2 linear operaors share he same eigenvecors. Le s denoe a he single eigenvalue of f ;ν ;µ and le b sand for he riple eigenvalue. Equaion (5) relaes he funcions a, b o A, B. The energy condiions now have he form Le s apply hese condiions o a rivial seed meric. Fluid from Minkowski A = 3 df 2 6bf, (6) B = 3 df 2 2af 4bf. (7) af > bf, (8) af + 5bf 3 df 2. (9) The simples seed meric is he Minkowski meric η µν = diag(1, 1, 1, 1) and all covarian derivaives reduce o parial derivaives in Caresian coordinaes (, x, y, z). Diagonal Einsein ensor. To fulfill he firs condiion, we assume f ;ν ;µ is diagonalized in Caresian coordinaes, i.e. f,ν,µ = diag(a, b, b, b). We have The mos general funcion ha saisfies hese equaions is f, = a, (10) f,xx = f,yy = f,zz = b, (11) f,ij = 0 for i j. (12) f = T () 1 2 b(x2 + y 2 + z 2 i ) + c }{{} i x }{{} r 2 0 where b has o be consan and a = T,. The linear erms can be se o zero wih a suiable choice of he origin, i.e. we can shif he spaial coordinaes x i x i + c i /b, which leaves only a consan behind and i can be absorbed ino he unknown funcion T (). We can see ha he assumpion of diagonalisaion in Caresian coordinaes leads o spherical symmery and hus o he Roberson-Walker cosmologies and furher if b=0 we ge he sandard FRW meric. (13) 14

3 Nondiagonal Einsein ensor. Le s now make a small generalizaion and le f ;ν ;µ be diagonal in a frame {eˆµ } 3ˆµ=0 (haed indices are used o label erad componens), which is conneced o he Caresian basis {, x, y, z } by a poin Lorenz ransformaion, specifically by a boos in he x plane, which can be paramerized as follows e 0 = cosh φ sinh φ x, (14) e 1 = sinh φ + cosh φ x, (15) e 2,3 = y,z, (16) where he rapidiy φ is a funcion on he spaceime. We can now ransform f ;ˆµ ;ˆν = diag(a, b, b, b) ino Caresian coordinaes. The ransformaion reads a cosh 2 φ b sinh 2 φ (a b) sinh φ cosh φ 0 0 f,ν,µ = (a b) sinh φ cosh φ b cosh 2 φ a sinh 2 φ b 0. (17) b This is a se of 10 equaions for second derivaives of he funcion f. Afer solving he rivial ones, he mos general soluion is f = F (, x) 1 2 b(y2 + z 2 ), (18) where b is a consan. This form of f requires boh a and φ o be funcions of only and x. The funcion F has o saisfy he oher 3 nonrivial equaions F, = a cosh 2 φ b sinh 2 φ, (19) F,x = (a b) sinh φ cosh φ, (20) F,xx = a sinh 2 φ b cosh 2 φ. (21) To make he equaions simpler, le s assume b = 0 and so f = F. The equaions reduce o f, = a cosh 2 φ, (22) f,x = a sinh φ cosh φ, (23) f,xx = a sinh 2 φ. (24) The funcions a and φ canno be arbirary, hey have o be chosen so ha he inegrabiliy condiions f,x = f,x and f,xx = f,xx hold. Afer some simplificaions, hese condiions read φ φ = anh φ, (25) φ = a ȧ sinh φ cosh φ a a sinh2 φ. (26) Where is derivaive wih respec o x and denoes he ime derivaive. The general soluion o he firs equaion is given by he implici formula x anh φ + + G(φ) = 0, (27) where he funcion G is arbirary. A convenien choice is o se G = 0, hen we can find he soluion explicily as φ = arcanh x 1 ( ) x 2 ln. (28) x + 15

4 Having he soluion for φ, we can now solve equaion (26) for a, a = 1 x. (29) Such forms of a and φ allow us o inegrae he equaions (22)-(24), he soluion is f = ( x)arcanh + ( + x) arcan x x. (30) Boh energy condiions hold in a domain D resriced by > 0, x > 0 and x. Figure 1 shows he behavior of f in D. Figure 1. Conformal facor: The funcion f behaves as f = c(k) x on lines = kx wih consan k. The facor c(k) is an increasing funcion and ranges from c(0) = 0 o c(1) = π/2. Le us now look ino some physical properies of he soluion we obained. The generaed perfec fluid meric is ( ) 2 ds 2 = ( x)arcanh + ( + x) arcan (d 2 dx 2 dy 2 dz 2 ) (31) x x and i belongs o he family of Sephani universes according o Theorem in Sephani e al. [2003]. The Energy densiy and he pressure are given by ρ = 3 2π arcanh x arcan x, (32) p = ρ f. (33) 4π x On he hypersurface = 0, boh he energy densiy and he pressure vanish and he energy condiions are violaed. When approaching he boundary = x, he energy densiy goes o infiniy and is posiive everywhere in D. On he oher hand, he pressure is always negaive and ends o minus infiniy near = x. Boh funcions ρ and p can be expressed solely in erms of k := /x, as a consequence, here exiss an equaion of sae in he form p = p(ρ), so he fluid is baroropic. To explore he naure of he null singulariy = x, where boh he energy 16

5 densiy and he pressure diverge, we calculae he Kreschmann scalar K := R µνρσ R µνρσ. I can be wrien in erms of f as K = 12 [ df 4 + ( df 2 af) 2] (34) and i is also infinie on he singulariy, as well as he Ricci scalar, which is proporional o ρ 3p. I is also possible o calculae he worldlines of he fluid. I flows along he vecor field so he following equaions mus hold for he worldline f e 0 = f cosh φ f sinh φ x, (35) d(τ) dτ dx(τ) dτ If we divide he second equaion by he firs, we ge where a do denoes d/dτ here. The las equaion implies = f cosh φ, (36) = f sinh φ. (37) ẋ ṫ = anh φ = x, (38) d dτ (x2 2 ) = 0, (39) so he worldlines of he fluid are hyperbolae in, x coordinaes. These worldlines and he behavior of physical quaniies are displayed in figure 2. Alhough he fluid worldlines resemble hose of consanly acceleraed paricles in Minkowski spaceime, he four-acceleraion is zero wih respec o he new meric, so he fluid worldlines are geodeics. This resul is confirmed by he Euler equaions. Figure 2. Physical quaniies: All physical quaniies vanish on = 0 and increase up o (he pressure decreases o, respecively) on = x and are consan on lines = kx wih consan k. Le s now invesigae he symmery of he generaed soluion. Barnes [1998] showed, ha a member of Sephani universe family wih non-vanishing expansion (which is he case here) 17

6 can have only 1, 3 or 6-dimensional isomery group and in he ulimae case, he spaceime is a FRW. Our fluid has rivially 3 Killing vecors corresponding o he yz plane symmery, i.e. wo ranslaions and one roaion, and because i is no isomeric o FRW (due o he differen naure of he singulariies), here are no more. To compacify he domain D, we can choose Penrose coordinaes ψ = arcan( + x) + arcan( x), (40) ξ = arcan( + x) arcan( x). (41) Wih fixed y and z coordinaes, he perfec fluid meric now reads ds 2 = 1 a 2 f 2 sin ψ sin ξ (dψ2 dξ 2 ). (42) Le s denoe F := (a 2 f 2 sin ψ sin ξ) 1. The compacificaion of he perfec fluid spaceime is depiced in figure 3. Figure 3. Compacificaion: Anoher singulariy which was hidden in infiniy in, x coordinaes appeared on he righ side of he diagram. All physical quaniies behave similarly as on he lef singulariy. The excepion is ha he fluid is flowing o he righ singulariy and ignores he lef. Noe ha F is finie on boh singulariies (bu ends o near = 0), bu is derivaives are singular. Conclusion We used he mehod of conformal ransformaion in order o generae perfec fluid spaceimes from vacuum and presened he condiions on he conformal facor. The condiions were applied o Minkowski as a seed spaceime. We found he conformal facor ha saisfies hese condiions and generaes a special case of Sephani universe. The ransformaion from conformally Minkowski coordinaes o hose given by Sephani has ye o be found. We furher sudied some physical feaures of his spaceime. Is unusual propery is he exisence of null singulariy wih a finie meric on i. As was indicaed, even soluions generaed from fla spaceime can have some ineresing properies. Our fuure goal is o use a nonrivial vacuum seed spaceime and ry o solve he equaions for he conformal facor o obain new soluions of he Einsein s equaions. Acknowledgmens. The presen work was suppored by he Czech Gran Agency under Conrac 205/09/H033 and by he Charles Universiy Gran Agency under Conrac

7 References HRUŠKA: GENERATING SOLUTIONS OF EINSTEIN S EQUATIONS Brinkmann, H.W. (1925): Einsein spaces which are mapped conformally on each oher. Mah. Ann. 18, 119. Dafardar-Gejji, V. (1998): A generalizaion of Brinkmanns heorem. GRG 30, 695. Loranger, J. and Lake, K. (2008): Generaing saic fluid spheres by conformal ransformaions. Phys. Rev. D 78, Coll, B. and Ferrando, J.J. (1989): Thermodynamic perfec fluid. Is Rainich heory. JMP 30, Barnes, A. (1998): Symmeries of he Sephani universes. Classical and Quanum Graviy 15, Sephani, H. e al. (2003): Exac Soluions of Einsein s Field Equaions (2nd ed.), 601. Cambridge Universiy Press. 19