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Covariant Mixmaster Dynamics Giovanni Imponente and Giovanni Montani ICRA-International Center for Relativistic Astrophysics and Physics Department, G9, University of Rome “La Sapienza”, 2 piazza A.Moro 5 00185 Rome, Italy 0 0 e-mail: [email protected] 2 n Abstract a J We provide a Hamiltonian analysis of the Mixmaster Universe dy- 1 namics on the base of a standard Arnowitt-Deser-Misner Hamiltonian 3 approach, showing the covariant nature of its chaotic behaviour with respecttothechoiceofanytimevariable,fromthepointofvieweither 1 ofthedynamicalsystemstheory,eitherofthestatisticalmechanicsone. v 3 0 1 1 Introduction 1 0 SinceBelinski-Kalatnikov-Lifshitz(BKL)derivedtheoscillatoryregimewhich 2 0 characterizes the behaviour of the Bianchi type VIII and IX cosmological / models [1] (the so-called Mixmaster universe [2]) near a physical singularity, c q a wide literature faced over the years this subject in order to provide the - r best possible understanding of the resulting chaotic dynamics. g The research activity developed overall in two different, but related, di- : v rections: on one hand the dynamical analysis was devoted to remove the i X limits of the BKL approach due to its discrete nature, on the other one to r get a better characterization of the Mixmaster chaos (especially in view of a its properties of covariance). The first line of investigation provided satisfactory representations of the Mixmaster dynamics in terms of continuous variables (leading to the construction of an invariant measure for the system [3], [5]). The efforts in the second direction found non-trivial difficulties due to the impossibility, to apply the standard chaos indicators to relativistic systems. The existence of difficulties related to the covariance of various ap- proachesprevented,uptonow,tosayadefinitivewordabouttheMixmaster chaoticity (apart from the indication provided by [4], see also [7]). The aim of this work is to show how in the points of view of the theory of dynamical systems and statistical mechanics the representation of the Mixmaster chaoticity is independent on the choice of a time gauge. 1 The description of the system evolution as a “stochastic scattering” isomorphic to a billiard on the Lobachevsky space can be constructed independently of the choice of a time variable, simply providing very general Misner-Chitré-like coordinates. On the other hand, we show how the derivation of an invariant measure for the Mixmaster model (performed in [5, 6] within the framework of the statistical mechanics) can be extended to a generic time gauge. In fact, asymptotically close to the cosmological singularity, the Mixmaster dynamics can be modeled by a two-dimensional point-Universe randomizing in a closed domain with fixed “energy” (just theADM kinetic energy); since it is natural to represent such a system by a microcanonical ensemble, then the corresponding invariant measure is induced by the Liouville one. Up to the limit of the adopted approximation on the form of the potential term, our analysis shows, without any ambiguity, that the Mixmaster stochasticity can not be removed by any redefinition of the time variable. 2 The Hamiltonian Formulation ThegeometricalstructureoftheBianchitypeVIIIandIXspacetimes, i.e. of theso-called Mixmaster Universe models, is summarizedby thelineelement [1] ds2 = N(η)2dη2 +e2α e2β σiσj (1) − ij (cid:16) (cid:17) where N(η) denotes the lapse function, σi are the dual 1-forms associated with the anholonomic basis 1 and β is a traceless 3 3 symmetric matrix ij × diag(β ,β ,β ); α, N, β are functions of η only. Parameterizing the 11 22 33 ij matrix β by the usual Misner variables [2] ij β11 = β+ +√3β−, β22 = β+ √3β−, β33 = 2β+ (2) − − 1 The dual 1-forms of theconsidered models are given by: σ1 =−sinhψsinhθdφ + coshψdθ (Bianchi VIII): σ2 =−coshψsinhθdφ + sinhψdθ ( σ3 = coshθdφ + dψ σ1 = sinψsinθdφ + cosψdθ (Bianchi IX): σ2 =−cosψsinθdφ + sinψdθ ( σ3 = cosθdφ + dψ 2 thedynamicsoftheMixmaster modelisdescribedbyacanonical variational principle δI = δ Ldη = 0, with Lagrangian L R 6D N ′2 ′2 ′2 L = α +β+ +β− V (α,β+,β−). (3) N − − D h i Here ()′ = ddη, D ≡deteα+βij = e3α and the potential V (α,β+,β−) reads 1 V = D4H1 +D4H2 +D4H3 D2H1+2H2 D2H2+2H3 D2H3+2H1, (4) 2 − ± ± (cid:16) (cid:17) where (+) and ( ) refers respectively to Bianchi type VIII and IX, and the − anisotropy parameters H (i = 1,2,3) denote the functions [5] i 1 β+ +√3β− 1 β+ √3β− 1 2β+ H = + , H = + − , H = . (5) 1 2 3 3 3α 3 3α 3 − 3α In the limit D 0 the second three terms of the above potential turn → out to be negligible with respect to the first one. Let’s introduce the new (Misner-Chitré-like) variables α = ef(τ)ξ, β+ = ef(τ) ξ2 1cosθ, β− = ef(τ) ξ2 1sinθ, (6) − − − q q with f denoting a generic functional form of τ, 1 ξ < and 0 θ < 2π. ≤ ∞ ≤ Then the Lagrangian (3) reads L = 6D efξ′ 2 + efθ′ 2 ξ2 1 ef ′2 NV (f(τ),ξ,θ). (7) N (cid:16)ξ2 (cid:17)1 − − − D − (cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17)     In terms of f (τ), ξ and θ we have D =exp 3ξ ef(τ) (8) − · n o and since D 0 toward the singularity, independently of its particular → form, in this limit f must approach infinity. The Lagrangian (3) leads to the conjugate momenta 12D df 2 12D ξ′ 12D p = ef τ′, p = e2f , p = e2fθ′ ξ2 1 τ − N · dτ ξ N ξ2 1 θ N − (cid:18) (cid:19) − (cid:16) (9(cid:17)) 3 which by a Legendre transformation make the initial variational principle assume the Hamiltonian form Ne−2f ′ ′ ′ δ p ξ +p θ +p τ dη = 0, (10) ξ θ τ − 24D H! Z being p 2 p 2 = τ +p 2 ξ2 1 + θ +24Ve2f. (11) H − df 2 ξ − ξ2 1 dτ (cid:16) (cid:17) − (cid:16) (cid:17) By variating (10) with respect to N we get the constraint = 0, which H solved provides df df p = ε2+24Ve2f (12) τ ADM − ≡ dτ ·H dτ · p where p 2 ε2 = ξ2 1 p 2+ θ . (13) − ξ ξ2 1 (cid:16) (cid:17) − In terms of (12) the variational principle (10) reduces to ′ ′ ′ δ p ξ +p θ f dη =0. (14) ξ θ ADM − H Z (cid:0) (cid:1) Since the equation for the temporal gauge actually reads 12D df N (η) = e2f τ′, (15) dτ ADM H our analysis remains fully independent of the choice of the time variable ′ until the form of f and τ is not fixed. ′ The variational principle (14) provides the Hamiltonian equations for ξ and θ′ 2 ′ ′ f f p ξ′ = ξ2 1 p , θ′ = θ . (16) − ξ (ξ2 1) ADM ADM H (cid:16) (cid:17) H − Furthermore can be straightforward derived the important relation ′ ′ ′ ′ d( f ) ∂( f ) d( f ) ∂( f ) ADM ADM ADM ADM H = H = H = H , (17) dη ∂η ⇒ df ∂f i.e. explicitly ∂ e2f ∂V ADM H = 24 2V + . (18) ∂f 2 · ∂f HADM (cid:18) (cid:19) 2In this study thecorresponding equations for p′ and p′ are not relevant. ξ θ 4 In this reduced Hamiltonian formulation, the functional f(η) plays simply therole of a parametric function of time and actually the anisotropy parameters H (i = 1,2,3) are functions of the variables ξ,θ only i 1 ξ2 1 H = − cosθ+√3sinθ 1 3 − 3ξ p (cid:16) (cid:17) 1 ξ2 1 H = − cosθ √3sinθ (19) 2 3 − 3ξ − p (cid:16) (cid:17) 1 ξ2 1 H = +2 − cosθ. 3 3 3ξ p Finally, toward the singularity (D 0 i.e. f ) by the expressions → → ∞ (4, 8, 19), we see that 3 ∂V = O efV . (20) ∂f (cid:16) (cid:17) Since in the domain Γ all the H are simultaneously greater than 0, the H i potential term U e2fV can be modeled by the potential walls ≡ U∞ = Θ∞(Hl(ξ,θ))+Θ∞(Hm(ξ,θ))+Θ∞(Hn(ξ,θ)) (21) + if x < 0 Θ∞(x)= ∞ 0 if x > 0 (cid:26) thereforeinΓ theADMHamiltonian becomes(asymptotically) anintegral H of motion = √ε2+24 U =ε = E = const. ∀{ξ,θ}∈ ΓH ( H∂HA∂ADfDMM = ∂∂Ef = 0. · ∼ (22) The key point for the use of the Misner-Chitré-like variables relies on the independence of the time variable for the anisotropy parameters H . i 3 The Jacobi Metric and the Billiard Representa- tion Sinceabove we have shown that asymptotically to thesingularity (f , → ∞ i.e. α ) d /df = 0 i.e. = ǫ = E = const., the variational ADM ADM → −∞ H H 3By O() we mean terms of thesame order of theinclosed ones. 5 principle (14) reduces to δ (p dξ+p dθ Edf)= δ (p dξ +p dθ)= 0 , (23) ξ θ ξ θ − Z Z where we dropped the third term in the left hand side since it behaves as an exact differential. ByfollowingthestandardJacobiprocedure[10]toreduceourvariational principle to a geodesic one, we set xa′ gabp , and by the Hamiltonian b ≡ equation (16) we obtain the metric ′ ′ f f 1 gξξ = ξ2 1 , gθθ = . (24) E − E ξ2 1 (cid:16) (cid:17) − By these and by the fundamental constraint relation p 2 ξ2 1 p 2+ θ = E2, (25) − ξ ξ2 1 (cid:16) (cid:17) − we get g xa′xb′ = f′ ξ2 1 p 2+ pθ2 = f′E. (26) ab E ( − ξ ξ2 1) (cid:16) (cid:17) − By the definition xa′ = dxa ds uads, the (26) rewrites ds dη ≡ dη ds 2 g uaub = f′E, (27) ab dη (cid:18) (cid:19) which leads to the key relation g uaub ab dη = ds. (28) s f′E ′ ′ ′ Indeed the expression (28) together with p ξ +p θ = Ef allows us to put ξ θ the variational principle (23) in the geodesic form: δ f′E dη = δ g uaubf′E ds = δ G uaub ds = 0 (29) ab ab Z Z q Z q wherethemetricG f′Eg satisfiesthenormalizationconditionG uaub = 1 ab ab ab ≡ and therefore 4 ds ds ′ = Ef = E. (30) dη ⇒ df 4Wetakethepositive root since we requirethat thecurvilinear coordinate s increases monotonically with increasing value of f, i.e. approaching the initial cosmological singularity. 6 Summarizing, in the region Γ the considered dynamical problem reduces H to a geodesic flow on a two dimensional Riemanniann manifold described by the line element dξ2 ds2 = E2 + ξ2 1 dθ2 . (31) "ξ2 1 − # − (cid:16) (cid:17) Theabovemetric(31)hasnegativecurvature,sincetheassociated curvature scalar reads R = 2 ; therefore the point-universe moves over a negatively −E2 curved bidimensional space on which the potential wall (4) cuts the region Γ . By a way completely independent of the temporal gauge we provided H a satisfactory representation of the system as isomorphic to a billiard on a Lobachevsky plane [10]. The invariant Lyapunov exponent defined as [11] reads 2 ln Z2+ dZ ds 1 λv = supsl→im∞ (cid:18) 2s(cid:16) (cid:17) (cid:19) = E > 0, (32) where Z is the geodesic deviation field (Jacobi) orthogonal to the geodesic one. When the point-universe bounces against the potential walls, it is reflected from a geodesic to another one thus making each of them unstable. By itself, the positive Lyapunov number (32) is not enough to ensure the system chaoticity, since its derivation remains valid for any Bianchi type model; the crucial point is that for the Mixmaster (type VIII and IX) the potential walls reduce the configuration space to a compact region (Γ ), H ensuring that the positivity of λ implies a real chaotic behaviour, provided v the factorized coordinate transformation in the configuration space α = ef(τ)a(θ,ξ) , β+ = ef(τ)b+(θ,ξ) , β− = ef(τ)b−(θ,ξ) , (33) − where f,a,b± denote generic functional forms of the variables τ,θ,ξ. 4 Statistical Mechanics Approach In order to reformulate the description of the Mixmaster stochasticity in terms of the Statistical Mechanics, we adopt in (14) the restricted time ′ gauge τ = 1, leading to the variational principle dξ dθ δ p +p df = 0. (34) ξ θ ADM df df −H Z (cid:18) (cid:19) 7 In spite of this restriction, for any assigned time variable τ (i.e. η) there exists a corresponding function f(τ) (i.e. a set of Misner-Chitré-like variables abletoprovidetheschemepresentedinSection2)definedbythe(invertible) relation df = HADMN (τ)e−2f. (35) dτ 12D As a consequence of the variational principle (34) we have again the expression (18). In agreement with this scheme, in the region Γ where the potential H vanishes, we have by (35) d /df = 0, i.e. ε = E = const. (by other ADM H words the ADM Hamiltonian approaches an integral of motion). Hence the analysis to derive theinvariant measurefor the system follows the same lines presented in [5, 6]. IndeedwegotagainarepresentationoftheMixmasterdynamicsinterms ofatwo-dimensionalpoint-universemovingwithinclosedpotentialwallsand over a negative curved surface(the Lobachevsky plane [5]), describedby the line element (31). Due to the bounces against the potential walls and to the instability of the geodesic flow on such a plane, the dynamics acquires a stochastic feature. This system, admitting an “energy-like” constant of motion (ε= E), is well-described by a microcanonical ensemble, whose Li- ouville invariant measure reads d̺ = Aδ(E ε)dξdθdp dp , A= const. (36) ξ θ − where δ(x) denotes the Dirac function. After the natural positions ε p = cosφ, p = ε ξ2 1sinφ, (37) ξ θ ξ2 1 − − q p being 0 φ 2 π, and the integration over all possible values of ε ≤ ≤ (depending on the initial conditions, they do not contain any information about the system chaoticity), we arrive to the uniform invariant measure 1 dµ = dξdθdφ . (38) 8π2 The validity of our potential approximation is legitimated by imple- menting the Landau-Raichoudhury theorem 5 near the initial singularity (placed by convention in T = 0, where T denotes the synchronous time, 5Such a theorem, within the mathematical assumptions founding Einsteinian dynamics, states that in a synchronous reference it always exists a given instant of time in correspondence to which themetric determinant vanishesmonotonically. 8 i.e. dT = N (τ)dτ), we easily get that D vanishes monotonically (i.e. for − T 0 we have dlnD/dT > 0). In terms of the adopted time variable τ → (D 0 f(τ) ), we have → ⇒ → ∞ dlnD dlnDdT dlnD = = N (τ) (39) dτ dT dτ − dT and therefore D vanishes monotonically for increasing τ as soon as, by (35), dΓ/dτ > 0. Now the key point of our analysis is that any stationary solution of the Liouville theorem, like (18), remains valid forany choice of thetime variable τ; indeed in [6] the construction of the Liouville theorem with respect to the variables(ξ,θ,φ)showstheexistenceofsuchpropertiesevenfortheinvariant measure (38). Weconcludebyremarkinghow,whenapproachingthesingularityf → ∞ (i.e. E), the time gauge relation (35) simplifies to ADM H → df Ee−2f+3ξef = N (τ)e−2f; (40) dτ 12 in agreement with the analysis presented in [6], during a free geodesic motion the asymptotic functions ξ(f),θ(f),φ(f), are provided by the simple system dξ dθ sinφ dφ ξsinφ = ξ2 1cosφ, = , = . (41) df − df ξ2 1 df − ξ2 1 q − − p p Once getting ξ(f) as the parametric solution ρ ξ(φ) = sinφ2 ρcosφ arctanh 1sinφ2+a2(1+cosφ2) 1 2 aρcosφ f(φ) = a (cid:18) (cid:19)+b − −2 aρcosφ       ρ a2+sin2φ a,b = const. (42) ≡ ∈ℜ q it reduces, for a free geodesic motion, equation (40) to a simple differential one for the function f (τ). However, the global behaviour of ξ along the whole geodesic flow, is described by the invariant measure (38) and therefore relation (40) takes a stochastic character. If we assign one of the two functions f(τ) or N (f) 9 analytically, the other one acquires a stochastic behaviour. We see how the one-to-one correspondence between any lapse function N (η) and the associated set of Misner-Chitè-like variables, ensures the covariant nature with respect to the time-gauge of the Mixmaster universe stochastic behaviour. We are very grateful to Remo Ruffini for his valuable comments on this subject. References [1] Belinski V A, Khalatnikov I M and Lifshitz E M (1970) Adv. Phys.,19, 525. [2] Misner C W (1969) Phys. Rev. Lett. ,22, 1071. [3] Chernoff D F and Barrow J D (1983) Phys. Rev. Lett, 50, 134. [4] CornishNJandLevinJJ(1997)Phys. Rev. Lett.,78,998; (1997)Phys. Rev. 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