Integrability of the Einstein-nonlinear SU(2) σ-model in a nontrivial topological sector
aa r X i v : . [ h e p - t h ] J a n Integrability of the Einstein-nonlinear SU (2) σ -model in a nontrivial topological sector Andronikos Paliathanasis, ∗ Tim Taves, † and P.G.L. Leach
4, 5, ‡ Instituto de Ciencias F´ısicas y Matem´aticas, Universidad Austral de Chile, Valdivia, Chile Institute of Systems Science, Durban University of Technology,PO Box 1334, Durban 4000, Republic of South Africa Centro de Estudios Cient´ıficos (CECS), Arturo Prat 514, Valdivia, Chile Department of Mathematics and Institute of Systems Science,Research and Postgraduate Support, Durban University of Technology,PO Box 1334, Durban 4000, Republic of South Africa School of Mathematics, Statistics and Computer Science, University of KwaZulu-Natal,Private Bag X54001, Durban 4000, Republic of South Africa
The integrability of the Λ − Einstein-nonlinear SU (2) σ -model with nonvanishing cosmologicalcharge is studied. We apply the method of singularity analysis of differential equations and we showthat the equations for the gravitational field are integrable. The first few terms of the solution arepresented. PACS numbers: 98.80.-k, 95.35.+d, 95.36.+xKeywords: σ -model; Pions; Integrability; Singularity analysis Nonlinear σ − models are important theoretical models in Physics for the properties that they provide which are ofspecial interest [1, 2]. The Einstein nonlinear σ − models, in which the total Action Integral is the sum of the Einstein-Hilbert Action Integral and the Action Integral which corresponds to the nonlinear matter source, have provideddifferent kinds of solutions for the gravitational equations. Specifically it has been shown that there exist black holesolutions with a regular event horizon which asymptotically approach the Schwarzschild spacetime, in the context ofthe Einstein-Skyrme Model, which violates the “no hair” conjecture for black holes (see for instance [3] and referencestherein).The purpose of this work is to study the integrability of the field equations of the Einstein-nonlinear SU (2) σ − model, which have been studied previously in [4]. We do this by using the method of singularity analysis ofdifferential equations . The application of singularity analysis in gravitational theories has been applied by manyresearchers in the past, for instance in the case of the Mixmaster Universe (Bianchi IX) [5–7], in scalar field cosmology[8] and in modified theories of gravity [9–11].Consider a Riemannian manifold M with metric g µν of Lorentzian signature. The action integral of the fieldequations for the Einstein-nonlinear SU (2) σ − model in a four-dimensional manifold is given by S = S EH + S ( σ ) , (1)where S EH is the Einstein-Hilbert action with the cosmological constant, i.e., S EH = R dx ( R − S ( σ ) is theaction integral of the nonlinear sigma model [15] S ( σ ) = K Z dx √− g (cid:0)(cid:0) U − U ; µ (cid:1) g µν (cid:0) U − U ; ν (cid:1)(cid:1) , (2)where U ( x ν ) is the SU (2)-valued scalar and K is a positive constant. The physical implication of the action, (2), isthat it describes the dynamics of low energy pions. The gravitational field equations are derived by variation of theaction integral (1) with respect to the metric tensor g µν . This leads to the following set of equations, G µν + Λ g µν = T µν , (3)in which the left hand side of (3) corresponds to the Einstein-Hilbert action where G µν is the Einstein tensor andΛ is the cosmological constant. The right hand side of (3) is that of the nonlinear σ -model and provides the matter ∗ Electronic address: [email protected] † Electronic address: [email protected] ‡ Electronic address: [email protected] For a review on the singularity analysis see [12] and subsequent developments in [13, 14] source. The explicit form of the energy-momentum tensor is T µν = − K (cid:0) U − U ; µ (cid:1) (cid:0) U − U ; ν (cid:1) + K g µν (cid:0) U − U ; κ (cid:1) g κλ (cid:0) U − U ; λ (cid:1) . (4)Furthermore variation with respect to the scalar-valued U ( x ν ), in (1) leads to the constraint equation (cid:0) U − U ; µ (cid:1) ; ν g µν = 0 , while the latter can follow from the application of the Bianchi identity in (3), that is, T µν ; ν = 0.By following the Ansatz, which was proposed in [16–19] and its generalizations [20–23], for the parametrization ofthe SU (2) algebra in [4] Ay´on-Beato, Canfora and Zanelli found that for the four-dimensional spacetime, ds = − F ( r ) ( dt + cos θdϕ ) + N ( r ) dr + ρ ( r ) (cid:0) dθ + sin θdϕ (cid:1) , (5)the conservation equation, T µν ; ν = 0 , is satisfied always and the energy-momentum tensor, (4), is expressed only interms of the fields F , ρ and N . Specifically, the matter field equations (cid:0) U − U ; µ (cid:1) ; ν g µν = 0, are identically satisfied sothat one has only to deal with the Einstein’s field equations while the solution provides a nontrivial topological sector[4].Spacetime, (5), is a locally rotational spacetime and for arbitrary functions, F and ρ , admits a four-dimensionalKilling Algebra which comprises the autonomous symmetry, A = { ∂ t } , and SO (3), i.e., the Killing vectors form the A ⊕ SO (3) Lie Algebra [24].In the minisuperspace approach the gravitational field equations (3) can arise from the Euler-Lagrange equationsof the singular Lagrangian, L ( F, F ′ , ρ, ρ ′ , N ) = 4 N (cid:18) √ F F ′ + ρ √ F F ′ ρ ′ (cid:19) + Λ ρ N √ F − K N F − (cid:0) ρ − F (cid:1) − ρ − N √ F (cid:0) F + 4 ρ (cid:1) , (6)where the equation ∂L∂N = 0 (7)gives the constraint equation or the G rr − T rr = 0 (8)component of (3). In (6) we can see the dynamical terms which corresponds to the R (3) curvature term of (5) of thecosmological constant and of the σ − model.Because the field equations are singular there could exist a nonlocal conservation law which is generated by theconformal Killing vectors of the minisuperspace, for details see [26, 27]. In our consideration, as the minisuperspace of(6) has dimension two, there exists an infinite number of conformal Killing vectors and an infinite number of nonlocalconservation laws. Hence with the use of a nonlocal conservation law the two second-order differential equations andthe first-order differential equation, which describe the gravitational field equations, can be reduced to the second-ordernonautonomous equation [25],0 = ry (cid:0) Kr − r (cid:0) K − r Λ (cid:1) y + 2 y (cid:1) y ′′ − (cid:18) Kr (cid:0) y (cid:1) ′ + Kr ( y ′ ) − (cid:0) y (cid:1) ′ − r (cid:0) y (cid:1) ′ (cid:0) r Λ + y ′ (cid:1)(cid:19) , (9)where y = y ( r ) = ρF, N ( r ) is N ( r ) = (cid:0) y − Kr (cid:1) y (4 r + y − r ) + Ky ( r − r y ) h ′ ( r ) V eff , (10)in which (cid:16) r (cid:0) y (cid:1) ′ (cid:17) ( h ( r )) = (cid:0) K r −
16 Λ r (cid:1) (11)+ 4 yr ((4 − K ) + 4 y ) , (12)and the new gauge has been selected to be so that ρ = r . The term V eff in (10) includes all the potential terms of(6) so that the Lagrangian (6) is that of geodesic equations in a two-dimensional manifold.The nonautonomous equation (9) can always be written in the form of an autonomous third-order differentialequation by introducing the new variables r → Y ( x ) and y → Y ,x . The third-order equation is0 = Y Y x (cid:16)
8Λ ( Y ,xx ) − Y ,x (3 KY ,xx + 8Λ Y ,xxx ) (cid:17) + Y (cid:16) KY ,x Y ,xxx − K ( Y ,xx ) (cid:17) + 2( K − Y Y ,x (cid:16) ( Y ,xx ) − Y ,x Y ,xxx (cid:17) + 2 Y ( Y ,x ) Y ,xxx + (cid:0) Y + 6 ( Y ,x ) (cid:1) ( Y ,x ) Y ,xx , (13)for which reduction with the autonomous symmetry, ∂ x , leads to the original equation, (9).We found that (13), except the autonomous one, does not admit any other point symmetry vector for any value ofthe cosmological constant. That is an interesting result because it indicates that there exists a unique relation amongequations (9) and (13). On the other hand, it has been found in [25] that equation (9) admits a rescaling symmetrywhen the cosmological constant is zero. The application of this symmetry vector reduced equation (9) to a first-orderAbel equation. However, for nonvanishing cosmological constant only special solutions of the form y ( r ) = σ r havebeen derived, where σ = − , K . Those special solutions correspond to specific initial conditions which describe theasymptotic behaviour of the general evolution of the system.Below we assume the case of nonvanishing cosmological constant and we perform the singularity analysis. Notethat, if (13) is integrable, then (9) is also integrable which means that the gravitational field equations for the Einstein-nonlinear SU (2) σ − model are also integrable, that is, the dynamical system which follows from the action integral,(1), is integrable.We define the new variable Φ ( x ) = Y ( x ) and we search for power-law solutions Φ ( x ) = αχ p in (13) from wherewe have the following possible sets ( χ = x − x ), where x is the location of the putative movable singularity) p = − α = − K , α = 12 (14)and p = −
12 with α = ± i r
32Λ . (15)We consider the values of (14) from which we can see that these are the power-law solutions of (9) which wedescribed above. The next step, in order to test if (13) passes the singularity test, is to determine the resonances. Let α = − K , which is the case in which the solution leads to a Lorentzian signature spacetime, for details see [25]. Thenby substituting Φ ( χ ) = αχ − + γχ − s (16)into (13) and taking the terms linear in γ , we have s (cid:0) s − s − (cid:1) = 0, which gives the triple solution s = − , s = 0 , s = 32 . (17)Here we remark that the resonances are the same and for α = , from where we can say that (13) passes the singularitytest and it is integrable.As far as the solution is concerned, we can write it in a series form in which from s , we have that the powers of χ in the series increase by . Therefore the solution isΦ ( χ ) = m χ − + m χ − + m + m χ + + ∞ X I =+4 m I χ − mI , (18)where m and m are arbitrary constants and m , m , m I have to be determined. In particular they are functionsof m , m , K and Λ . Recall that there exists a resonance with value zero.
Hence we substitute (18) into (13) and find for the coefficient constants that m = 0, m = 38 m Λ ( m (4 + K (2 m − −
2) (19)and m = − m − m Λ (cid:0)
12 + 8 ( K − m + K m (1 + m ) (cid:1) (20)from which we can see that for m = − K it follows that m = 0and m = 0. For values of χ such that χ − >> χ the solution takes the following form Φ ( χ ) ≃ − K χ − + m χ . (21)Now the spacetime, (5), has the following form ds = − Φ ( χ ) ,χ
2Φ ( χ ) ( dt + cos θdϕ ) + N (Φ , Φ ,χ ) Φ ( χ ) ,χ p Φ ( χ ) dχ + Φ ( χ ) (cid:0) dθ + sin θdϕ (cid:1) . (22)Furthermore from the second dominant behavior p = − we find the resonances s = − , s = − , s = 32 (23)which provides us with the second solutionΦ ( χ ) = −∞ X I = − n J χ − J + n − χ − + n − x − + n x − + n + n + n x + + ∞ X I =+4 n I χ − I (24)which is a right and left Laurent expansion. The free parameters of solution (24) are the n − , and n .One important issue of the general solution of (22) is the number of constants which are two from the solution(18); the { m , m } , { n − , n } and the constant K which corresponds to the matter source. One would expect threeintegration constants for the system (3). The latter is hidden in χ , as for the singularity analysis we move to thecomplex plane in which χ = x − x and x is the position of the singularity. However, that is not an essential constantbecause it can always be absorbed with the transformation of the coordinate, x, which means that at the end thereare only two constants.On the other hand, as we discussed above, the asymptotic behaviour of the general solution (18) is Φ ( χ ) ≃ χ − ,which is the dominant term around the singularity of (13). If we start far from that solution, that is, from differentinitial conditions, then in the asymptotic behaviour of the solution only the terms of lower powers of χ contribute in thesolution. Hence the solution can be described well from the first terms of the Laurent series. We demonstrate that in fig.1 where the numerical solution of (13) (using Mathematica’s NDSolve routine) is given and compared with analyticalsolutions from the Laurent expansion (18) where we considered the first four terms of (18), ( m − m ) (AnalyticSol.1), the six first terms ( m − m ) (Analytic Sol..2) and the seven first terms ( m − m ) (Analytic Sol..3). Figure2 presents the absolute error between the analytical solutions and the numerical solution. We observe that thosespecific initial conditions, where the singularity x is at x ≃ . , are very close in the region of the singularity theapproximation works well. Of course eventually the error becomes big and other terms of the Laurent expansion haveto be considered.We remark that in the case of vanishing cosmological constant the singularity analysis provides that the resonancesdepend on the parameter K . Last but not least an important observation which we can extract from the singularityanalysis about the stability of the leading order term is that the solution is unstable. The reason for that is theexistence of the terms given by the right Laurent expansion which dominate as far as we are moving from thesingularity, see also [28] and references therein. Acknowledgments
This work has been funded by the FONDECYT grants no. 3160121 (AP) and 3140123 (TT). The Centro de EstudiosCientificos (CECs) is funded by the Chilean Government through the Centers of Excellence Base Financing Program
FIG. 1: Numerical simulation and analytical approximations close to the singulartity for the master equation.FIG. 2: Relative error of the analytical approximations with numerical simulation for the master equation close to the singu-lartity. of Conicyt. PGL Leach thanks the Instituto de Ciencias F´ısicas y Matem´aticas of the UACh for the hospitalityprovided while this work carried out and acknowledges the National Research Foundation of South Africa and theUniversity of KwaZulu-Natal for financial support. [1] N. Nagaosa and Y. Tokura, Nature Nanotechnology
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