Exactly solvable strings in Minkowski spacetime
aa r X i v : . [ g r- q c ] M a r OCU-PHYS 316, AP-GR 68
Exactly solvable strings in Minkowski spacetime
Hiroshi Kozaki , Tatsuhiko Koike and Hideki Ishihara Department of General Education, Ishikawa National College of Technology,Tsubata, Ishikawa 929-0392, Japan Department of Physics, Keio University, Yokohama 223-8522, Japan Department of Mathematics and Physics, Graduate school of Science, Osaka CityUniversity, Osaka 558-8585, JapanE-mail: [email protected] , [email protected] , [email protected] Abstract.
We study the integrability of the equations of motion for the Nambu-Goto strings with a cohomogeneity-one symmetry in Minkowski spacetime. Acohomogeneity-one string has a world surface which is tangent to a Killing vector field.By virtue of the Killing vector, the equations of motion reduce to the geodesic equationin the orbit space. Cohomogeneity-one strings are classified into seven classes (TypesI to VII). We investigate the integrability of the geodesic equations for all the classesand find that the geodesic equations are integrable. For Types I to VI, the integrabilitycomes from the existence of Killing vectors on the orbit space which are the projectionsof Killing vectors on Minkowski spacetime. For Type VII, the integrability is relatedto a projected Killing vector and a nontrivial Killing tensor on the orbit space. Wealso find that the geodesic equations of all types are exactly solvable, and show thesolutions.PACS numbers: 11.25.-w, 11.27.+d, 98.80.Cq,
1. Introduction
Cosmic strings are topological defects which are produced when the U (1) symmetrybreaks down in the unified theories. Such a symmetry breaking is supposed to haveoccurred at the early stage of the universe. If the existence of the cosmic strings isconfirmed, it is a strong evidence of vacuum phase transition in the universe. Besidesthe interests from the unified theories, cosmic strings have been studied in the contextof cosmology since they were proposed as possible seeds of the structure formation inthe universe [1, 2]. However, this scenario was rejected due to the confliction with theprecise observational data of cosmic microwave backgrounds [3, 4].Recently, cosmic strings gather much attention in the context of the superstringtheories since fundamental strings and other string-like solitons such as D-strings couldexist in the universe as cosmic strings [5]. In the brane inflation models, these cosmicsuperstrings are produced at the end of inflation [6, 7] and stretched by the expansion xactly solvable strings in Minkowski spacetime x -axis and y -axis. This identification isgeneralized as follows: two strings are equivalent if their world surfaces, say Σ and Σ ,are mapped by an isometry ϕ ,Σ = ϕ (Σ ) . (1)In the case of the cohomogeneity-one strings, we can identify the strings if the isometry ϕ sends the tangent Killing vector ξ which defines the cohomogeneity-one property ofΣ to the Killing vector ξ which defines that of Σ : ξ = ϕ ∗ ξ . (2)In Minkowski spacetime, the Killing vectors are classified into seven families (TypesI to VII) under identification by isometries. Therefore, the cohomogeneity-one stringsfall into seven families [21]. The Type I family includes stationary rotating strings.The Nambu-Goto equations of motion for this class are exactly solved and variousconfigurations are found [9, 10, 20, 11, 16, 18]. By using the exact solutions, the xactly solvable strings in Minkowski spacetime
2. Integrability of cohomogeneity-one strings
A trajectory of the string is a two-dimensional surface, say Σ, embedded in the spacetime( M , g ). We denote the embedding as ζ a x µ = x µ ( ζ a ) , (3)where ζ a ( ζ = τ, ζ = σ ) are the coordinate on Σ and x µ ( µ = 0 , , ,
3) are thecoordinate in M . The Nambu-Goto action is written as S = Z Σ √− γd ζ , (4)where γ is the determinant of the metric γ ab induced on Σ which is given by γ ab = g µν ∂x µ ∂ζ a ∂x ν ∂ζ b . (5)Let us consider the case that the spacetime M admits a Killing vector field ξ . Wedenote the one-parameter isometry group generated by ξ as H . The group action of H xactly solvable strings in Minkowski spacetime M generates orbits of H , or the integral curves of ξ . If the string world surface Σ isfoliated by the orbits of H , the string is called cohomogeneity-one associated with theKilling vector ξ . It is obvious that Σ is tangent to ξ .When we identify the points in M which are connected by the action of H , wehave the orbit space O := M /H . Under this identification, the cohomogeneity-oneworld surface Σ becomes a curve in O . This curve is shown to be a spacelike geodesicin O with the norm-weighted metric˜ h µν = − f h µν , (6)where f is the squared norm of ξ and h µν is a naturally induced metric on O : h µν = g µν − ξ µ ξ ν /f. (7)Therefore, the equations of motion for the cohomogeneity-one string are reduced to thegeodesic equations on ( O , ˜ h ).Integrability of the geodesic equations is related to the existence of Killing vectorsand Killing tensors. Let K µ be a Killing vector field on ( O , ˜ h ), which satisfies the Killingequations ∇ ( µ K ν ) = 0 , (8)where ∇ µ denotes the covariant derivative with respect to ˜ h . For a tangent vector u µ ofa geodesic, K µ u µ is conserved along the geodesic. Let K µν be a Killing tensor field on( O , ˜ h ), which is symmetric and satisfies the Killing equations ∇ ( µ K νλ ) = 0 , (9) K µν u µ u ν is also conserved along the geodesic. If the geodesic has enough number ofsuch conserved quantities which commute with each other, the geodesic equations areintegrable. In the case of geodesics in ( O , ˜ h ), where dim O = 3, two conserved quantitiesare required for the integrability. Therefore, if ( O , ˜ h ) admits two or more Killing vectorsand Killing tensors, geodesic equations are integrableIn the orbit space ( O , ˜ h ), we can find such Killing vectors without solving the Killingequations. Let us consider a Killing vector X in ( M , g ) which commutes with ξ . Wecan easily find that the projection of X , say π ∗ X , where π : M → O is the projection,is a Killing vector in ( O , ˜ h ): L π ∗ X ˜ h µν = L X ( − f g µν + ξ µ ξ ν ) = L X { ( − g ρσ g µν + g µρ g νσ ) ξ ρ ξ σ } = ( − g ρσ g µν + g µρ g νσ ) { ( L X ξ ρ ) ξ σ + ξ ρ L X ξ σ } = ( − g ρσ g µν + g µρ g νσ ) { [ X, ξ ] ρ ξ σ + ξ ρ [ X, ξ ] σ } = 0 . (10)Killing vectors which commute with ξ constitute a Lie subalgebra, called centralizer of ξ which we denote C ( ξ ). Let X, Y ∈ C ( ξ ) commute with each other. We can show thatthe projections of them on O also commute;[ π ∗ X, π ∗ Y ] = π ∗ [ X, Y ] = 0 . (11) xactly solvable strings in Minkowski spacetime C ( ξ ) except for ξ itself, ( O , ˜ h ) inherits the same number of commuting Killing vectors,and then the geodesic equations in ( O , ˜ h ) are integrable.In Minkowski spacetime, all of the cohomogeneity-one strings are classified intoseven families (Types I to VII). For each type, we list the Killing vector ξ , basis of C ( ξ ) and the number of commuting basis of C ( ξ ) except for ξ in Table 1. For Types Ito VI, there are more than two commuting Killing vectors in C ( ξ ), hence, the geodesicequations in ( O , ˜ h ) are integrable. As shown in the next section, the equations of motionsfor these strings are not only integrable but also solved exactly. For the strings of TypeVII, there is only one Killing vector in C ( ξ ). Nevertheless, the geodesic equations aresolved exactly. This is due to the existence of a Killing tensor in ( O , ˜ h ). We also solvethe geodesic equation exactly. Table 1.
Inherited symmetry of ( O , ˜ h ). P µ ( µ = t, x, y, z ) is the generator oftranslation for µ -direction. L i ( i = x, y, z ) are the generators of rotation around i -axis. K i ( i = x, y, z ) are the generators of Lorentz boosts for i -directions. n is thenumber of commuting basis in C ( ξ ).Type tangential Killing vector ξ basis of C ( ξ ) n I P t + aL z ( a = 0) P t , P z , L z P t P t , P x , P y , P z , L x , L y , L z L z P t , P z , L z , K z P t + P z ) + aL z ( a = 0) P t , P z , L z P t + P z P t , P x , P y , P z , K y + L x , K x − L y , L z P z + aL z ( a = 0) P t , P z , L z P z P t , P x , P y , P z , L z , K x , K y P z + a ( K y + L z ) P t − P x , P z , P y + a ( K z − L y ) , K y + L z P z + aK y ( a = 0) P x , P z , K y K y P x , P z , L y , K y K y + L z + aP x ( a = 0) K y + L z + aP x , P t − P x , P z K z + aL z ( a = 0) L z , K z
3. Solutions of cohomogeneity-one strings in Minkowski spacetime
The tangent Killing vector of this class is given as ξ = P t + aL z , ( a : const.) (12)where P t is the Killing vector of time translation and L z is that of rotation around z -axis. In the conventional cylindrical coordinate (¯ t, ¯ ρ, ¯ φ, ¯ z ) of Minkowski spacetime, ξ is written as ξ = ∂ ¯ t + a∂ ¯ φ , (13) xactly solvable strings in Minkowski spacetime ξ is f = | ξ | = − (1 − a ¯ ρ ) . (14)Then, ξ is timelike in ¯ ρ < / | a | and spacelike in ¯ ρ > / | a | . The surface ¯ ρ = 1 / | a | iscalled light cylinder. Cohomogeneity-one strings of Type I inside the light cylinder arethe stationary rotating strings. The constant a represents the angular velocity of therotation.Here, we introduce a coordinate ( t, ρ, φ, z ) = (¯ t, ¯ ρ, ¯ φ − a ¯ t, ¯ z ) so that ξ = ∂ t , i.e., oneof the coordinates, say t , is a coordinate along the orbits of H which is generated by ξ .In the new coordinate, the spacetime metric is g = − (1 − a ρ ) dt + 2 aρ dtdφ + dρ + ρ dφ + dz (15)and the norm of ξ is f = − (1 − a ρ ) . (16)Then, the norm-weighted metric on the orbit space is calculated as˜ h = − f g + ξξ = (1 − a ρ )( dρ + dz ) + ρ dφ . (17)We solve the geodesic equations in ( O , ˜ h ) with the action S = Z ( L N + N ) dσ, (18) L = (1 − a ρ )( ρ ′ + z ′ ) + ρ φ ′ , (19)where σ is a parameter of the geodesic curve, N is a function of σ and the prime denotesthe derivative with respect to σ . The action (18) is invariant under the transformations: σ ˜ σ = ˜ σ ( σ ) , (20) N ˜ N = dσd ˜ σ N. (21)Therefore, the function N determines the parametrization of the geodesic curve. Weshould note that even though we fix the functional form of N , there remains residualfreedom of the parametrization: σ ˜ σ = ± σ + σ . (22)The variation with respect to φ leads a conserved quantity related to the φ -independence of ˜ h : ρ φ ′ N = L (const) . (23)We also have a conserved quantity related to the z -independence of ˜ h :1 − a ρ N z ′ = P (const) . (24)The other variations lead(1 − a ρ )( ρ ′ + z ′ ) + ρ φ ′ = N , (25) ( − a ρ N ρ ′ ) ′ − N n − a ρ ( ρ ′ + z ′ ) + ρφ ′ o = 0 . (26) xactly solvable strings in Minkowski spacetime N = 1 − a ρ , (27)we obtain ρ ′ = 1 − P + a L − a ρ − L ρ . (28)This equation is readily integrated as a ρ ( σ ) = α + β cos 2 a ( σ + σ ) , (29) α := 1 − P + a L ≥ , (30) β := √ α − a L , (31)where σ is an integration constant. We can set σ to zero by using the residualreparametrization freedom (22). Then, the solution is written as a ρ ( σ ) = α + β cos 2 aσ, (32)Using the solution, we can solve (23) and (24) as φ ( σ ) = − a Lσ + tan − " aLα + β tan aσ + φ , (33) z ( σ ) = P σ + z , (34)where φ and z are constants.The string solution, i.e., embedding of the world surface ( τ, σ ) ( t, ρ, φ, z ), is givenby (32), (33), (34) and t = τ . The solution has four integration constants: P, L, φ and z . P and L determine the shape of the string. However, φ and z have no physicalmeaning, because we can identify the solution of φ = 0 and z = 0 with that of φ = z = 0 by the isometries in ( M , g ) φ φ + φ , (35) z z + z , (36)where we should remember that the spacetime metric (15) does not depend on φ and z . For Types II to VI, we can reduce the equations of motion to the geodesic equationsin the orbit space and solve them in the same manner as in the case of type I. Wesummarize the results in the Table 2.
The tangent Killing vector of Type VII string is ξ = K z + aL z , (37)where K z is a Killing vector of the Lorentz boost along z -axis. As in the case of Type I,we introduce a coordinate suitable for the reduction. In order to find such a coordinate, xactly solvable strings in Minkowski spacetime Table 2.
We show tangent Killing vectors ξ , coordinates for the reduction, orbitspace metrics ˜ h and solutions of the geodesic equations. P , Q and L are the conservedquantities related to the Killing vectors of ˜ h , and C is that related to the Killing tensorof ˜ h . (¯ t, ¯ x, ¯ y, ¯ z ) and (¯ t, ¯ ρ, ¯ φ, ¯ z ) are the Cartesian coordinate and cylindrical coordinateof Minkowski spacetime, respectively.TypeI ξ = P t + aL z , (¯ t, ¯ ρ, ¯ φ, ¯ z ) = ( t, ρ, φ + at, z ) , ˜ h = (1 − a ρ )( dρ + dz ) + ρ dφ , z ( σ ) = P σ, a ρ ( σ ) = α + β cos 2 aσ, φ ( σ ) = − a Lσ + tan − h aLα + β tan aσ i α := (1 − P + a L ) / , β := √ α − a L II ξ = P t + P z + aL z , (¯ t, ¯ ρ, ¯ φ, ¯ z ) = ( u + v , ρ, φ + au, u − v ),˜ h = dv − aρ dvdφ − a ρ dρ , v ( σ ) = aP σ, a ρ ( σ ) = aP ( L + √ L − aσ ) ,φ ( σ ) = − aLσ + tan − (cid:2) ( L − √ L −
1) tan aσ (cid:3) III ξ = P z + aL z , (¯ t, ¯ ρ, ¯ φ, ¯ z ) = ( t, ρ, φ + az, z ) , ˜ h = (1 + a ρ )( dt − dρ ) − ρ dφ ,t ( σ ) = Qσ, a ρ ( σ ) = α + β cos 2 aσ, φ ( σ ) = a Lσ + tan − h aLα + β tan aσ i , α := ( − Q − a L ) / , β := √ α − a L IV ξ = P z + a ( K y + L z ) , (¯ t, ¯ x, ¯ y, ¯ z ) = ( a λ u + v, − ¯ t + u, aλu, λ + w )˜ h = (1 + a u )(2 dudv − du ) − a u dw u ( σ ) = P σ, v ( σ ) = P + Q P σ + a P σ − Q a P σ , w ( σ ) = Q (cid:0) σ − a P σ (cid:1) V ξ = P z + aK y ( K y : timelike)(¯ t, ¯ x, ¯ y, ¯ z ) = ( y sinh at, x, y cosh at, z + t ) , ˜ h = a y dz − (1 − a y )( dx + dy ) x ( σ ) = P σ, a y ( σ ) = α ± (cid:0) e aσ + β e − aσ (cid:1) z ( σ ) = Qσ + a ln (cid:12)(cid:12)(cid:12) e aσ ± ( α + Q ) e aσ ± ( α − Q ) (cid:12)(cid:12)(cid:12) , α := (1 + P + Q ) / , β := p α − Q . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ξ = P z + aK y ( K y : spacelike)(¯ t, ¯ x, ¯ y, ¯ z ) = ( t cosh ay, x, t sinh ay, z + y ) , ˜ h = − a t dz + (1 + a t )( dt − dx ) x ( σ ) = P σ, − a t ( σ ) = α − (cid:0) e aσ + β e − aσ (cid:1) z ( σ ) = Qσ + a ln (cid:12)(cid:12)(cid:12) e aσ ± ( α + Q ) e aσ ± ( α − Q ) (cid:12)(cid:12)(cid:12) , α := (1 + P + Q ) / , β := p α − Q VI ξ = P x + a ( K y + L z ) , (¯ t, ¯ x, ¯ y, ¯ z ) = ( a λ + auλ + v, − ¯ t + λ, a λ + u, w ) , ˜ h = (2 au − du + dw ) + dv au ( σ ) − Q − P + a (1 − P ) σ , v ( σ ) = P Q − P σ + a P (1 − P )3 σ , w ( σ ) = Qσ VII ξ = K z + aL z ( K z : timelike)(¯ t, ¯ ρ, ¯ φ, ¯ z ) = ( z sinh t, ρ, φ + at, z cosh t ) , ˜ h = ( z − a ρ )( dρ + dz ) + z ρ dφ z ( σ ) = C ± (cid:0) e σ + β e − σ (cid:1) , a ρ ( σ ) = C + β cos 2 a ( σ + σ ) φ ( σ ) = tan − n aLC + β tan a ( σ + σ ) o + a ln (cid:12)(cid:12)(cid:12) e σ ± ( C + aL ) e σ ± ( C − aL ) (cid:12)(cid:12)(cid:12) , β := √ C − a L . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ξ = K z + aL z ( K z : spacelike)(¯ t, ¯ ρ, ¯ φ, ¯ z ) = ( t cosh z, ρ, φ + az, t sinh z ) , ˜ h = ( t + a ρ )( dt − dρ ) − t ρ dφ t ( σ ) = − C + (cid:0) e σ + β e − σ (cid:1) , a ρ ( σ ) = C + β cos 2 a ( σ + σ ) ,φ ( σ ) = tan − n aLC + β tan a ( σ + σ ) o + a ln (cid:12)(cid:12)(cid:12) e σ − ( C + aL ) e σ − ( C − aL ) (cid:12)(cid:12)(cid:12) , β := √ C − a L xactly solvable strings in Minkowski spacetime t, ¯ ρ, ¯ φ, ¯ z ) = ( z sinh t, ρ, φ + at, z cosh t ) , (38)such that ξ is written as ξ = ∂ t . Since this coordinate covers only the part of Minkowskispacetime where K z is timelike, we use another coordinate(¯ t, ¯ ρ, ¯ φ, ¯ z ) = ( t cosh z, ρ, φ + az, t sinh z ) . (39)in the spacelike regions of K z . In this coordinate, ξ is written as ξ = ∂ z . K z We take the coordinate (38) in the timelike regions of K z . With respect to the coordinate, the spacetime metric is written as g = − ( z − a ρ ) dt + 2 aρ dφdt + dρ + ρ dφ + dz , (40)and the norm of the Killing vector ξ is f = − ( z − a ρ ) . (41)Then, the metric ˜ h on O is given as˜ h = ( z − a ρ )( dρ + dz ) + z ρ dφ . (42)This metric admits a manifest Killing vector ∂ φ and an irreducible Killing tensor K = a ρ ( z − a ρ ) dz + z ( z − a ρ ) dρ + z ρ ( z + a ρ ) dφ . (43)In order to solve the geodesic equations, we start from the action (18) with L = ( z − a ρ )( ρ ′ + z ′ ) + z ρ φ ′ . (44)The existence of the Killing vector and the Killing tensor ensures two conservedquantities, say L and C respectively; L = z ρ φ ′ N , (45)2 C = a ρ ( z − a ρ ) z ′ N ! + z ( z − a ρ ) ρ ′ N ! + z ρ ( z + a ρ ) φ ′ N ! . (46)Here, we should note that the prime does not represent the differentiation with an affineparameter. The geodesic tangent with an affine parameter is written as x a ′ /N . We solve(45), (46) and the constraint equation( z − a ρ )( ρ ′ + z ′ ) + z ρ φ ′ = N . (47)By fixing the parametrization freedom as N = z − a ρ , (48)we can separate the variables; ρ ′ = 2 C − L ρ − a ρ , (49) z ′ = − C + z + a L z , (50) xactly solvable strings in Minkowski spacetime z ( σ ) = C ± e σ + β e − σ , (51) a ρ ( σ ) = C + β cos 2 a ( σ + σ ) , (52) φ ( σ ) = tan − ( aLC + β tan a ( σ + σ ) ) + a (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) e σ ± ( C + aL ) e σ ± ( C − aL ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (53) β := √ C − a L , (54)where σ is an integration constant. K z With respect to the coordinate (39), the metric ˜ h onthe orbit space is written as˜ h = ( t + a ρ )( dt − dρ ) − t ρ dφ . (55)This metric also admits a Killing vector ∂ φ and a Killing vector K = a ρ ( t + a ρ ) dt + t ρ ( t − a ρ ) dφ + t ( t + a ρ ) dρ . (56)These Killing vector and Killing tensor ensure the existence of two conserved quantities,and then the geodesic equations are integrable. With a calculation similar to that usedin deriving solutions (51), (52), (53), we obtain the exact solutions t ( σ ) = − C + 12 (cid:16) e σ + β e − σ (cid:17) , (57) a ρ ( σ ) = C + β cos 2 a ( σ + σ ) , (58) φ ( σ ) = tan − ( aLC + β tan a ( σ + σ ) ) + a (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) e σ − ( C + aL ) e σ − ( C − aL ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (59) β := √ C − a L , (60)where σ is an integration constant, L is a conserved quantity related to the Killingvector ∂ φ and C is the one related to the Killing tensor K .
4. Conclusion
We have shown that the Nambu-Goto equations of motion for all of the cohomogeneity-one strings in Minkowski spacetime ( M , g ) are integrable. The cohomogeneity-one stringis a string whose world surface is tangent to a Killing vector field ξ . The Killing vector ξ generates an one-parameter isometry group, say H , which acts on the world surface.Then, the world surface has symmetry due to H . By virtue of the symmetry on the worldsurface, the equations of motion reduces to the geodesic equations on the orbit space O := M /H with a norm-weighted metric ˜ h µν := − ξ g µν + ξ µ ξ ν . We have investigatedthe integrability of these geodesic equations.The integrability of the geodesic equations is related to the existence of Killingvectors. In the case of ( O , ˜ h ), we have shown that the projections of the Killing vectorsin ( M , g ) which commute with ξ are also Killing vectors in ( O , ˜ h ), i.e., the Killing xactly solvable strings in Minkowski spacetime M , g ). We have focused on the number of these inheritedKilling vectors.For the cohomogeneity-one strings of Types I to VI, we have found that there aremore than two commuting Killing vectors in ( O , ˜ h ). Existence of two or more commutingKilling vectors guarantees the integrability of the geodesic equations in ( O , ˜ h ) becausedim O = 3. Then, the geodesic equations for Types I to VI are integrable. We have alsofound that the geodesic equations are solved exactly. The exact solutions are shown inTable 2.For the remaining cohomogeneity-one strings, i.e., Type VII, there is only oneinherited Killing vector. However, we have found a Killing tensor in ( O , ˜ h ). Existenceof the Killing vector and the Killing tensor leads two conserved quantities of the geodesic,and then the geodesic equations are integrable. We have also solved the geodesicequations exactly. Acknowledgments
This work is supported in part by Keio Gijuku Academic Development Funds (T.K.)and the Grant-in-Aid for Scientific Research No.19540305 (H.I.).
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