Comparing the dynamics of diagonal and general Bianchi IX spacetime

We make comparison of the dynamics of the diagonal and nondiagonal Bianchi IX models in the evolution towards the cosmological singularity. Apart from the original variables, we use the Hubble normalized ones commonly applied in the examination of the dynamics of homogeneous models. Applying the dynamical systems method leads to the result that in both cases the continuous space of critical points is higher dimensional and they are of the nonhyperbolic type. This is a generic feature of the dynamics of both cases and seems to be independent on the choice of phase space variables. The topologies of the corresponding critical spaces are quite different. We conjecture that the nondiagonal case may carry a new type of chaos different from the one specific to the usually examined diagonal one.


I. INTRODUCTION
The Belinskii-Khalatnikov-Lifshitz (BKL) scenario is thought to be a generic solution to the Einstein equations near spacelike singularity [1,2]. It has been shown [1,2] that symmetries of spacetime such as isotropy and homogeneity are dynamically unstable in the evolution towards the singularity. The existence of the general solution to the Einstein equations with a cosmological singularity means that General Relativity is incomplete. One believes that an imposition of quantum rules onto this scenario may heal the singularity. Finding the nonsingular quantum BKL scenario would mean solving, to some extent, the generic cosmological singularity problem. Such a quantum theory could be used as a realistic model of the very early Universe.
Quantization of the BKL scenario should be preceded by the quantization of the Bianchi IX model. This seems to be a reasonable strategy because the BKL scenario has been obtained via analysis of the dynamics of the Bianchi IX model. The best prototype for the BKL scenario is the nondiagonal Bianchi IX model [2][3][4].
The quantization of the Bianchi IX model requires full understanding of its classical dynamics in terms of variables convenient for quantization procedure. Our recent paper [5] has initiated such analysis. As far as we know, the existing comprehensive analyses of the dynamics of the Bianchi IX model, with the exception of the paper [6], concern the diagonal case only (see, e.g. [7] and references therein). The examination of the dynamics presented in [6], concerning the nondiagonal case, is mathematically satisfactory but seems to be too difficult to use in the canonical quantization scheme which we plan to apply in our quantization programme.
The three metric of the Bianchi IX model (in the synchronous reference system) can be chosen to be diagonal or nondiagonal one (during the entire evolution of the system) depending on the matter field. It turns out that the evolutions corresponding to both cases are essentially different near the cosmological singularity [3]. The aim of this paper is examination of this difference in more details.
In this paper we use two quite different sets of variables parameterizing the dynamics: the original BKL [4] and the Hubble normalized [8]. They cannot be connected by canonical transformation. In both cases we obtain the spaces of the nonhyperbolic type of critical points.
The paper is organized as follows: Section II concerns the nondiagonal case. We introduce quasi Hubble normalized variables, examine the dynamics in these (and BKL) variables, and identify the spaces of critical points of the corresponding vector fields. The diagonal case is considered in Sec. III, where we follow the steps of Sec. II. We conclude in Sec. IV. There are serious differences in dynamics between diagonal and nondiagonal cases, and the sets of critical points have quite different topologies. Appendix A concerns the issue of an effective form of the metric near the singularity. The choice of quotient coordinates, presented in App. B, enables making an extension of the interpretation of our results. We present the relationship between the BKL and our new variables in App. C.

II. THE NONDIAGONAL CASE
The general form of a line element of the nondiagonal Bianchi IX model, in the synchronous reference system, reads where Latin indices a, b, . . . run from 1 to 3 and label the frame vectors e a α , and Greek indices α, β, . . . take values 1, 2, 3 and concern space coordinates, and where γ ab is a spatial metric.
It was shown in [1,2] that near the cosmological singularity the general form of the metric γ ab should be considered. Consequently, one cannot globally diagonalize the metric, i.e. for all values of time. After making use of the Bianchi identities, freedom in the rotation of the metric γ ab and frame vectors e a α , one arrives at the well-defined, but complicated system of equations specifying the dynamics of the nondiagonal Bianchi IX model [4]. The assumption that the anisotropy of space may grow without bound, when approaching the singularity, enables considerable simplification of the dynamics. Finally, the asymptotic form (near the cosmological singularity) of the dynamical equations of the nondiagonal Bianchi IX model reads [4,5] where a, b, c are functions of time τ , satisfying the constraint d ln a dτ and where τ is connected with the cosmological time variable t as follows (γ denotes the determinant of γ ab ). Turning the above dynamics into Hamilton's dynamics, one can examine qualitatively the mathematical structure of the corresponding physical phase space by using the dynamical system methods (DSM). It has been found that the critical points of the system have the following properties: (i) define a three-dimensional continuous subspace ofR 6 defined by the relation a ≫ b ≫ c > 0, with a → 0 (see, Eq. (38) of [5] for more details), and (ii) are of the nonhyperbolic type.
The property (i) was already found long time ago [4] without using the DSM. The characteristic (ii) has been identified recently [5]. The latter property means that getting insight into the structure of the space of orbits near such critical set requires further examination of the exact nonlinear dynamics. The information obtained from linearization of the dynamics cannot be conclusive (see, e.g., [10]).
The above identification suggests that we have a sort of an effective diagonal metric g αβ near the cosmological singularity, i.e., in the asymptotic region of spacetime. It is known that in this limit the nondiagonal components of the exact metric freeze [4] so that the essential dynamics is carried by the three independent scale factors a, b and c, which gives some support to our assumption (7). More arguing in favour of this interpretation is given in App. A. The effective 3-metric (7) is used below to introduce quasi-HN (qHN) variables.
The second fundamental form k αβ associated with (7) is defined to be where due to (4) we have and where g := det[g αβ ], so v is the spatial volume density. If we take k α β := g αγ k γβ , the trace of the matrix k αβ reads We define the shear as follows and one has σ 1 + σ 2 + σ 3 = 0. Defining the expansion θ by we get θ := −tr(k). The volume changes according to dv/dt = θ v. Following the considerations in [8,11], we define the Hubble variable H := θ/3. Now, we are ready to define the qHN variables (Σ α , N α ) for our nondiagonal Bianchi IX model by using the HN formalism [8]: (thus, Σ 1 + Σ 2 + Σ 3 = 0), and which coincide with Egs. (5) and (6).
B. Dynamics

Critical points of the dynamics
Direct inspection of the system (28) -(32) leads to the following identification of the set of the critical points: in such a way that N 3 << N 2 << N 1 and N 2 /N 1 << N 2 1 → 0, and N 3 /N 2 << N 2 /N 1 → 0, which imply that One can avoid taking the uncommon form of the limits (34) by introducing new variables, which we consider in App. B. However, this does not change the character of critical points. The stay to be the nonhyperbolic ones.

The linearization of the vector field
One may verify, with some effort, that some elements of the Jacobian J of the system (28) -(32), evaluated at any point of S qHN , are diverging. This behavior comes from differentiating square roots. However, when calculating characteristic polynomial of the Jacobian J at any point those divergencies cancel out giving so the eigenvalues are (0, 0, 0, 0, 0). Since the real parts of all eigenvalues of the Jacobian are equal to zero, we are dealing with the nonhyperbolic critical points.
Our system evolves asymptotically, as time goes to zero (when the system approaches the cosmological singularity), to the nonhyperbolic critical subspace with the coordinates (Σ 1 , Σ 2 , N 1 , N 2 , N 3 ) given by Further analysis should be based on making use of the exact form of our vector field.

III. THE DIAGONAL CASE
In what follows we demonstrate that the asymptotic forms of the dynamics of the non-diagonal and diagonal Bianchi IX model are quite different.
The diagonal case can be obtained from the nondiagonal one by the assumption that near the singularity the rotation stops. It means that the Euler angles approach three constants (see, [3] and Eq. (2.23) in [4] for more details), i.e.
The difference results from the fact that Eqs. (2) -(3) has been obtained by imposition onto the original set of equations defining the nondiagonal dynamics (see, Eqs. (2.14)-(2.20) in [4]) the condition which implies Eq. (37), but not vice versa.
which turns the vector field (53) -(58) intȯ (69) The above system has the same critical subspaces as the one without the constraint built into it. The Jacobian associated with the system (65) -(69) is found to be The characteristic polynomial evaluated at the critical subspaces reads: Hence, we can conclude that the character of the critical hypersurfaces (60) -(63) is the nonhyperbolic one.

D. Critical points
The critical points of the system (79) -(84), satisfying (78), define the set of critical hypersurfaces: The Jacobian associated with the vector field (79) -84), satisfying (78), evaluated at any point of {S 0 , S 1 , S 2 , S 3 } has diverging components arising from differentiating terms of the type √ M 1 M 2 and in the limit M 1 → 0 (or other M's going to zero). However, calculating characteristic polynomial and taking the value of its coefficient at the critical subspaces leads to the following result: Hence, we can conclude that the character of the critical hypersurfaces (85) -(88) is the nonhyperbolic one.

IV. CONCLUSIONS
According to [1,2], near the cosmological singularity, an evolution of the diagonal model is an infinite sequence of the so called eras each of which consists of the Kasner type epochs. Each epoch can be described, e.g., by the relationΓ 1 ∼Γ 2 >Γ 3 (where ∼ means coupled) called an oscillation 1 .
The dynamics of the nondiagonal model has sophisticated structure: the oscillation of the diagonal type, e.g., Γ 1 ∼ Γ 2 > Γ 3 enters sooner or later the relation Γ 1 > Γ 2 > Γ 3 , which turns into the strong relation Γ 1 >> Γ 2 >> Γ 3 . One may speculate that the latter implies approaching the singularity in a finite proper time. Such situation does not occur in the diagonal case, where the oscillations may last for ever.
The difference between the dynamics of the diagonal and nondiagonal cases leads to different topological structures of the corresponding sets of critical points. In the former case, this set consists of three hypersurfaces inR 6 having the same topology, Eqs. (86)-(88), and one set, Eq. (85), with the simple topology ofR 3 . In the latter case, the set of critical points has sophisticated topology, Eq. (33), quite different from the diagonal case. Similar relationship occurs between the critical sets expressed in term of the BKL variables. However, in both cases the critical sets consist of the nonhyperbolic type of critical points.
The nonhyberbolicity of the diagonal case is expected to be linked with the well known chaoticity of dynamics of the Bianchi IX model (see, e.g. [12,13]). We expect that there may exist some difference between the BKL scenario derived from the nondiagonal case [1,2] and the scenario that might be obtained within the diagonal one. On the other hand, the attractor of the dynamics of the diagonal case (see, e.g. [8]) may not occur in the nondiagonal case, or it may have completely different form. Further studies are required to get insight into this fascinating issue. 1 The meaning of Γ ′ s is given in the appendix A. There can also occur small oscillationsΓ 1 ∼ Γ 2 >>Γ 3 , but they last for a finite interval of time and can be ignored.