Time-Periodic Solutions of the Einstein s Field Equations

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1 Tie-Periodic Solutions of the Einstein s Field Equations De-Xing Kong 1 and Kefeng Liu 1 Departent of Matheatics Zhejiang University Hangzhou China Departent of Matheatics University of California at Los Angeles CA USA In this paper we develop a new ethod to find the exact solutions of the Einstein s field equations by using which we construct tie-periodic solutions. The singularities of the tie-periodic solutions are investigated and soe new physical phenoena such as the tie-periodic event horizon are found. The applications of these solutions in odern cosology and general relativity are expected. PACS nubers: 04.0.Jb; 04.0.Dw; Jk; 0.30.Jr 1. Introduction. The Einstein s field equations are the fundaental equations in general relativity and cosology. The general version of the gravitational field equations or the Einstein s field equations read R µν 1 g µνr + Λg µν = 8πG c 4 T µν (1) where g µν (µ ν = 0 1 3) is the unknown Lorentzian etric R µν is the Ricci curvature tensor R = g µν R µν is the scalar curvature where g µν is the inverse of g µν Λ is the cosological constant G stands for the Newton s gravitational constant c is the velocity of the light and T µν is the energy-oentu tensor. In a vacuu i.e. in regions of space-tie in which T µν = 0 the Einstein s field equations (1) reduce to or equivalently R µν 1 g µνr + Λg µν = 0 () R µν = Λg µν. (3) In particular if the cosological constant Λ vanishes i.e. Λ = 0 then the equation () becoes or equivalently R µν 1 g µνr = 0 (4) R µν = 0. (5) Each of the equations ()-(5) can be called the vacuu Einstein s field equations. The atheatical study on the Einstein s field equations includes roughly speaking the following two aspects: (i) establishing the well-posedness theory of solutions; (ii) finding exact solutions with physical background. Up to now very few results on the well-posedness for the Einstein s field equations have been established. In their classical onograph [4] Christodoulou and Klaineran proved the global nonlinear stability of the Minkowski space for the vacuu Einstein s field equations i.e. they showed the nonlinear stability of the trivial solution of the vacuu Einstein s field equations. Recently by using wave coordinates Lindblad and Rodnianski gave a sipler proof (see [1]). In her Ph.D. thesis [18] Zipser generalizes the result of Christodoulou and Klaineran [4] to the Einstein- Maxwell equations. As for finding exact solutions any works have been done and any interesting results have been obtained (see e.g. [3] [16] []). In what follows we will briefly recall soe basic facts about the exact solutions of the Einstein s field equations. The exact solutions are very helpful to understand the theory of general relativity and the universe. The typical exaples are the Schwarzschild solution and the Kerr solution (see [3]). These solutions provide two iportant physical space-ties: the Schwarzschild solution describes a stationary spherically syetric and asyptotically flat space-tie while the Kerr solution provides a stationary axisyetric and asyptotically flat spacetie. The study on exact solutions of the Einstein s field equations has a long history. In Deceber 1915 Karl Schwarzschild discovered the first non-trivial solution to the vacuu Einstein s field equations which is a static solution with zero angular oentu (see [15]). The unique two-paraeter faily of solutions which describes the space-tie around black holes is the Kerr faily discovered by Roy Partrick Kerr in July 1963 (see [10]). These solutions are very iportant in studying black holes in the Nature which is just the study of these solutions (see [3]). Various generalizations of the Kerr solution have been done (see e.g. [16] and []). Gowdy [6]-[7] constructed a new kind of solutions of the vacuu Einstein s equations these solutions provide a new type of cosological odel. This odel describes a closed inhoogeneous universe space sections of these universes have either the three-sphere topology S 3 or the worhole (hypertorus) topology S 1 S. Recently Ori [13] presented a class of curved-spacetie vacuu solutions which develop closed tielike curves at soe particular oent and used these vacuu solutions to construct a tie-achine odel. The Ori odel is regular asyptotically flat and topologically trivial. Fro the above discussions we see that the exact solutions play a crucial role in general relativity and cosology so it is always interesting to find new exact solutions for the Einstein s equations. Although any interesting and iportant solutions have been obtained there are still any fundaental but open probles. One inter-

2 esting open proble is if there exists a tie-periodic solution to the Einstein s field equations. One of the ain results in this paper is a solution of this proble. In this paper we focus on finding the exact solutions of the vacuu Einstein s field equations (3) and (5). We will present a new ethod to find exact solutions. Using this ethod we can construct interesting and iportant exact solutions for exaple the tie-periodic solution of the vacuu Einstein s field equations. We analyze the singularities of tie-periodic solutions and investigate soe new physical phenoena enjoyed by these new spaceties. We find that the new tie-periodic solutions have tie-periodic event horizon which is a new phenoenon in the space-tie geoetry. The applications of these solutions and their new properties in odern cosology and general relativity ay be expected. More precisely our tie-periodic solution to the vacuu Einstein s field equations in the spherical coordinates (t r θ ϕ) can be written in the following for ds = (dt dr dθ dϕ)(η µν )(dt dr dθ dϕ) T (6) where G G + QK 0 where u v p q w ρ and σ are sooth functions of the (η µν ) = G + G coordinates (t x y z). It is easy to verify that the deterinant of (g µν ) is given by QK 0 QK QK K K sin θ g = det(g µν ) = uwρ σ v ρ σ p wσ q wρ (1) (7) in which Throughout this paper we assue that G = 1 + Ω + sin θ cos (t r) K = r + ln + sin (t r) M = Ω + sin θ Q = 1 (1 + sin θ)ω. (8) In the above ( ) and R are two paraeters and Ω ± are defined by the Einstein s field equations one key point is to choose a suitable coordinate syste. A good coordinate syste can siplify the equations and ake the easier to solve. In the study on the Einstein s field equations there are three faous coordinate systes: haronic coordinates wave coordinates and the Gaussian coordinates. In this paper we consider the etric of the following for (g µν ) = g 00 g 01 g 0 g 03 g 10 g g 0 0 g 0 (10) g g 33 where g µν are sooth functions of the coordinates (x 0 x 1 x x 3 ) and satisfy g 0i = g i0. In the coordinates (x 0 x 1 x x 3 ) the line eleent reads ds = g µν dx µ dx ν. For siplicity of notations we denote the coordinates (x 0 x 1 x x 3 ) by (t x y z) and rewrite (10) as u v p q v w 0 0 (g µν ) = p 0 ρ 0 q 0 0 σ g < 0 (11) (H) On the other hand it is easy to see that at least two of the functions w ρ σ have the sae sign. Without loss of generality we ay suppose that ρ and σ keep the sae sign for exaple ρ < 0 and σ < 0. (13) Ω ± = tan θ/ 1 ± tan θ/ 1. (9) In Section 5 we analyze the singularity behaviors and find soe new physical phenoena. According to the authors knowledge (6) gives the first tie-periodic solution to the Einstein s field equations. Here we would like to point out that by using our ethod we can re-derive alost all known exact solutions to the vacuu Einstein s field equations for exaples Gödel s solution [5] Khan-Penrose s solution [11] Gowdy s solution [6]-[7] etc. Our ethod can also be used to find exact solutions of the Einstein s field equations in higher diensions which will be of interests in string theory.. Lorentzian etrics. The Einstein s field equations are a second order global hyperbolic syste of highly nonlinear partial differential equations with respect to the Lorentzian etric g µν (µ ν = 0 1 3). To solve We can easily prove the following theore. Theore 1 Under the assuptions (H) and (13) the etric (g µν ) is Lorentzian. We now consider the solutions of the Einstein s field equations. By Bianchi identities we believe that the general for of the solutions of the Einstein s field equations takes one of the following fors (η µν ) = (η µν ) = u v p q v p 0 ρ 0 (Type I) q 0 0 σ 0 v p q v w 0 0 p 0 ρ 0 q 0 0 σ (Type II)

3 3 or (η µν ) = u v p 0 v w 0 0 p 0 ρ σ (Type III) For type I the assuption (H) is equivalent to v 0. Therefore by Theore 1 we have Conclusion 1 If ρ < 0 σ < 0 and v 0 (14) then the etric (η µν ) is Lorentzian. For type II we have Conclusion If w ρ σ are all negative functions and v + p + q 0 then the hypotheses (H) and (13) are satisfied and the etric (η µν ) is Lorentzian. Siilarly for type III we have Conclusion 3 If u is positive and w ρ σ are negative then the hypotheses (H) and (13) are satisfied and the etric (η µν ) is Lorentzian. We are interested in finding exact solutions of the Einstein s field equations of the above types I-III. 3. New ethod to find exact solutions. In order to illustrate our ethod as an exaple we use the Lorentzian etric type I to construct soe interesting exact solutions in particular tie-periodic solution for the vacuu Einstein s field equations (). For type I etrics our ethod can be described by the following algorith: G 11 = 0 G 1 = 0 G 13 = 0 G = Λρ G 0 = Λp G 3 = 0 G 01 = Λv 7 G 00 = Λu G 33 = Λσ G 03 = Λq FIG. 1: The algorith to construct the exact solutions In fact the equations () can be rewritten as G µν = Rµν 1 η µνr = Λη µν (15) where G µν is the Einstein tensor. Noting the special for of type I we have ( vx ρx G 11 = 1 v [ (ρx ) 1 + ρ ρ + σ x σ ) + ] ( σx ) σ One of the equations (15) reads Solving the ODE (17) gives v = v 0 exp ( ρxx ρ + σ ) } xx. σ (16) G 11 = Λη 11. (17) [ ρ xx ρ + σ xx σ 1 ( ) ρx 1 ρ ] ( σx ) σ where v 0 = v 0 (t y z) is an integral function depending on t y and z provided that (ρσ) x 0. In particular by taking the ansatz ρ = ρ(t y z) expf(t x)} (18) becoes σ = σ(t y z) expf(t x)} (19) v = v 0 f x e f. (0) According to the algorith shown in Fig 1 in a siilar way we can solve other equations in () and then we can construct any exact solutions of (). Reark 1 Our ethod can be used to solve the Einstein s field equations with physically relevant energyoentu tensors e.g. the tensor for perfect fluid: T γδ = (µ + p)u γ u δ + pg γδ where µ > 0 is the density p is the pressure u stands for the space-tie velocity of the fluid with u γ u γ = 1. Reark The ethod presented in this paper can also be used to solve the Einstein s field equations in higher space-tie diensions. 4. Tie-periodic solutions. By the ethod presented in the last section we can construct any new exact solutions to the vacuu Einstein s field equations G µν = 0. (1) For exaple in the coordinates (τ r θ ϕ) taking the ansatz in (19) as follows ρ = 1 σ = sin θ f = ln [ Ω + r + τ + sin τ ] one can obtain an interesting solution of the for η µν = η 00 η 01 η 0 0 η η 0 0 η 0 () η 33 } ρσ dx (ρσ) x (18)

4 where η 00 = 1 + Ω + sin θ cos τ + Ω + sin θ Ω η 01 = Ω + sin θ Ω η 0 = 1 (1 + sin θ) Ω [ Ω + r + τ + sin τ ] ] η = [ Ω + r + τ + sin τ ] η 33 = [ Ω + r + τ + sin τ sin θ (3) Proof. Noting ( ) we have η 00 = G = 1 + Ω + sin θ cos (t r) = sin θ cos θ + cos θ sin θ sin θ cos θ cos(t r) ( 1 4 θ sin3 cos θ + θ cos3 sin θ ) 4 in which is a paraeter 0 is a constant and ( ) r + τ Ω = exp + Ω± = tan θ/ 1 ± tan θ/ 1. (4) The solution () belongs to the class of type I also belongs to the class of type III. Making the transforation τ + r t = τ + + exp } τ + r r = + exp θ = θ ϕ = ϕ. } G = 1 + Ω + sin θ cos (t r) K = r + ln + sin (t r) M = Ω + sin θ Q = 1 (1 + sin θ)ω. In (8) Ω ± are defined by (8) Ω ± = tan θ/ 1 ± tan θ/ 1. (9) An iportant property of the space-tie described by (6) is given by the following theore. Theore When takes its value in the interval ( ) i.e. ( ) the solution (6) to the vacuu Einstein s field equations is tie-periodic. cos(t r) 1 8 > = 0. (30) On the other hand by calculations we have G G + G + G = () () < 0 (31) (5) for r 0 and θ 0 π G G + QK Then the solution to (1) becoes in the coordinates (t r θ ϕ) G + G QK = K () () > 0 ds = (dt dr dθ dϕ)(η µν )(dt dr dθ dϕ) T (6) QK QK K (3) where and G G + QK 0 (η µν ) = G + G G G + QK 0 QK 0 G + G QK 0 = QK QK K 0 QK QK K K sin θ K sin θ (7) in which K 4 sin θ () () < 0 (33) for r 0 θ 0 π and K 0. In next section we will show that θ = 0 π r = 0 and K = 0 are not essential singularities in other words they are not physical singularities. The above discussion iplies that the variable t is a tie coordinate. Therefore it follows fro (7) and (8) that the Lorentzian etric (6) is indeed a tie-periodic solution of the vacuu Einstein s field equations. This proves Theore. Reark 3 The tie-periodic solution (6) was first obtained by us in early 007. The authors have presented this solution in several conferences for the past year. As entioned above according to the authors knowledge

5 this is the first tie-periodic solution to the Einstein s field equations. Reark 4 In July 006 the first author discussed with Dr Y.-Q. Gu about soe ideas presented in this paper. We noted that for the case of type I etric the author of [8] constructed soe exact solutions none of which is tie-periodic because the variable t in these solutions can not be taken as the tie coordinate. Direct coputations give us the following property of the tie-periodic solutions. Property 1 In the geoetry of the space-tie (6) it holds that K t = G 1 M K t = 1 G M t (34) Ω + θ = Ω sin θ Ω θ = Ω+ sin θ M θ G r = G t K t + K r = r Q θ G r t = G t K t r = K t 3Ω+ = Q cot θ 4 sin θ Q K t = 1 G θ = Q (35) G θ r = G θ t (36) K θ r = K t θ (37) G θ = K Q t θ. (38) Q cos θ Ω Q (Ω + ) 1 (Ω + ) = sin θ. (39) Ω + Q cot θ 3 4 sin θ 1 + Q = M. (40) M The relations given above will play an iportant role in the future study on the geoetry of the tie-periodic space-tie (6). There are several iportant questions which deserve further study: (a) what is the topological structure of the tie-periodic space-tie (6)? (b) does the space-tie (6) have a copact Cauchy surface? (c) an iportant point is to consider the structure of the axial globally hyperbolic part of the space-tie (6) the question is whether this also exhibits tie periodicity. Probles (b) and (c) were suggested by Andersson [1]. 5. Singularity behaviors and physical properties. This section is devoted to the analysis of singularities and the physical properties of the tie-periodic solution (6). It follows fro (5) that r >. Since the solution (6) can be extended to the case r 0 we always assue that r [0 ). In what follows we analyze the singularities of the solution (6) on the doain r 0. For the etric (7) by a direct calculation we have g = det(η µν ) = M K 4 r () sin θ. (41) Cobing (8) and (41) gives g = (Ω + ) sin 4 θ [r + ln + sin (t r)] 4 r (). Noting (4) we obtain fro (4) that S r=0 = (t r θ ϕ) r = 0 } 5 (4) S K=0 = (t r θ ϕ) r + ln + sin (t r) = 0} S θ=0π = (t r θ ϕ) θ = 0 π} (43) are singularities for the solution etric (6). By a direct calculation we have R αβγδ = 0 and R αβγδ = 0 (α β γ δ = 0 1 3). (44) This gives and R µν = 0 (µ ν = 0 1 3) (45) R = R αβγδ R αβγδ = 0. (46) (45) iplies that the Lorentzian etric (6) is indeed a solution of the vacuu Einstein s field equation (1) and (46) iplies that this solution does not have any essential singularity. According to the definition of event horizon (see e.g. Wald [17]) it is easy to show that S θ=0π and S K=0 are the event horizons of the space-tie (6). If (t r θ ϕ) are the polar coordinates then the S r=0 can be regarded as a degenerate event horizon. However S r= is a new kind of singularity which is neither event horizon nor black hole. This differs fro that in Schwartzschild space-tie in which r = 0 corresponds to the black hole and r = corresponds to the event horizon. In other words the solution (6) describes an essentially regular space-tie it does not contain any essential singularity like black hole. It is an interesting topic to see how to cancel these kinds of singularities by aking soe coordinate transforation. Therefore we have Property The Lorentzian etric (6) describes a regular space-tie this space-tie is Rieannian flat in the sense of (44) it does not contain any essential singularity. However it contains soe non-essential singularities which correspond to event horizons and soe other new physical phenoena. Property 3 The non-essential singularities of the spacetie (6) consist of three parts S r=0 S K=0 and S θ=0π. S θ=0π are two steady event horizons; and S K=0 is the tie-periodic event horizon; S r=0 are two new kinds of singularities which are neither event horizons nor black holes. In particular if (t r θ ϕ) are the polar coordinates then S r=0 can be regarded as a degenerate event horizon. According to the authors knowledge the degenerate event horizon and the tie-periodic event horizon are two new phenoena in the space-tie geoetry.

6 6 We next consider the tie-periodic event horizon in detail. Without loss of generality we ay assue that and are positive constants. We have two cases: 0 r and r >. Case I: 0 r. Let f(r; t) = r + ln r + sin (t r). (47) For any fixed t R it holds that and At r = 0 we consider i.e. f(r; t) as r (48) f(r; t) as r. (49) ln + sin t = 0 (50) sin t = ln. (51) It is obvious that (51) has a solution if and only if ln 1. (5) In what follows we always assue condition (5). Therefore it follows fro (51) that f(0; t k ) = 0 (53) and In this case Thus we shall take ln < 0 (58) arcsin ln } < 0. (59) t k = (k + 1)π + arcsin ln } (k = 0 1 ). (60) In both case I-1 and case I- by fixing k 0 1 } and noting (48) we see that there exists a axiu r [0 ) such that for any given r [0 r ] the equation for t f(r; t) = 0 (61) has solutions. When r = r we denote the solution by t k. It holds that f(r ; t k ) = 0. (6) Suarizing the above discussion we observe that for case I the tie-periodic event horizons are given in Fig.. where t k = kπ + arcsin ln } (54) t t k in which k Z. We now divide the discussion into two cases. Case I-1: 0 < 1. In this case we have and then Therefore t k = kπ + arcsin Case I-: 1 <. ln 0 (55) arcsin ln } 0. (56) ln } (k = 0 1 ). (57) t k t 0 t 0 0 r r FIG. : Tie-periodic event horizons for case I Case II: r >. Siilar to the discussion of case I in this case the tieperiodic event horizons are given in Fig. 3.

7 t t + k t k t + 0 t 0 0 r 0 r + r FIG. 3: Tie-periodic event horizons for case II In Fig. 3 r 0 and r + are defined in the following way: noting (48) and (49) we see that there exists a iniu r 0 ( ) and a axiu r + ( ) such that for any given r [r 0 r + ] the equation for t f(r; t) = 0 (63) has solutions. In particular when r = r 0 (resp. r = r + ) we denote the solution by t k (resp. by t + k ). That is to say it holds that f(r 0 ; t k ) = 0 and f(r + ; t + k ) = 0. (64) Therefore we have proved the following property. Property 4 The non-essential singularities of the spacetie (6) consists of three parts r = 0 r = and r + ln + sin (t r) = 0. r = 0 is a degenerate event horizon r = is a steady event horizon and r + ln + sin (t r) = 0 are the tieperiodic event horizons. Tie-periodic event horizons for and disappear in finite ties they propagate tieperiodically. On the other hand by soe eleentary atrix transforations the etric (η µν ) can be reduced to (ˆη µν ) = diag G M r } G() K K sin θ. Noting that (8) gives (65) (ˆη µν ) diag 1 + Ω + sin θ cos (t r) (Ω + ) sin } θ 1 + Ω + sin θ cos (t r) r r sin θ. (66) In (66) we have ade use of the fact that when r is large enough it holds that K r because of the second equation in (8). (66) iplies that the space-tie (6) is not hoogenous and not asyptotically flat ore precisely not asyptotically Minkowski because the first two coponents in (66) depend strongly on the angle θ. Therefore we have Property 5 The space-tie (6) is not hoogenous and not asyptotically flat. Reark 5 Property 5 perhaps has soe new applications in cosology due to the recent WMAP data since the recent WMAP data show that our Universe exists anisotropy (see [9]). This inhoogenous property of the new space-tie (6) ay provide a way to give an explanation of this phenoena. Suarizing the above discussion gives the following theore. Theore 3 The vacuu Einstein s field equations have a tie-periodic solution (6) this solution describes a regular space-tie which has vanishing Rieann curvature tensor but is not hoogenous and not asyptotically flat. This space-tie does not contain any essential singularity but contains soe non-essential singularities which correspond to steady event horizons tie-periodic event horizon and soe other new physical phenoena. 6. Suary and discussion. In this paper we describe a new ethod to find exact solutions to the Einstein s field equations (1). Using our ethod we can construct soe iportant exact solutions including the tie-periodic solutions of the vacuu Einstein s field equations. We also analyze the singularities of the tieperiodic solutions and investigate soe new physical phenoena enjoyed by these new space-ties. We reark that by using our ethod we can obtain alost all known solutions to the Einstein s field equations. Our ethod can also be used to find exact solutions of the higher diensional Einstein s field equations which play an iportant role in string theory. The structures of these new space-ties the behaviors of their singularities and soe new nonlinear phenoena appeared in the tie-periodic solutions are very interesting and iportant. We expect soe applications of these new phenoena and the tie-periodic solutions in odern cosology and general relativity. The authors thank Professors Lars Andersson Chong- Ming Xu Shing-Tung Yau Dr. Wen-Rong Dai Chun-Lei He and Fu-Wen Shu for helpful discussions and valuable suggestions. It was Professor Yau who first brought to us the open proble of finding tie-periodic solutions to the Einstein s field equations. The work of Kong was supported in part by the NSF of China (Grant No ) and the NCET of China (Grant No. NCET ); the work of Liu was supported by the NSF and NSF of China. 7

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