# Comparing the dynamics of diagonal and general Bianchi IX spacetime

## Abstract

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.

## 1 Introduction

According to the singularity theorems of General Relativity (GR), the evolution of an expanding universe is geodesically past-incomplete. The Belinskii, Khalatnikov and Lifshitz (BKL) [1, 2] scenario predicts that on approach to a space-like cosmological singularity the dynamics of gravitaional field simplifies as time derivatives in Einstein equations dominate over spatial derivatives (see [3] for numerical support for BKL). In this regime the evolution of the Universe becomes strongly non-linear and chaotic, comprising expanding and contracting oscillatory phases around the singular point. 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 spacetime. The three metric on space of the Bianchi IX model (in the synchronous reference system) is in general nondiagonal for general matter models. However in the case of vacuum or simple fluids it can be diagonalized during the entire evolution of the system. We refer to these two cases as nondiagonal and diagonal Bianchi IX models, respectively. The best prototype for the BKL scenario is the nondiagonal Bianchi IX model [2, 4, 5] corresponding to general matter fields.

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 [6] has initiated such analysis. As far as we [7, 8] and references therein). The examination of the dynamics presented in [9] of the nondiagonal case is mathematically satisfactory, but seems to be too complicated to be used in any quantization scheme.

Recent analysis indicate that the dynamics of the nondiagonal case has asymptotic regime near the singularity [10]. The dynamics of this regime looks similarly to the dynamics of the diagonal case (devoid of asymptotic regime). However, the symmetry aspects of both set of equations defining the corresponding dynamics are quite different, which leads to the different topologies of the corresponding spaces of solutions. The aim of this paper is the examination of these differences in more details.

In this paper we use two quite different sets of variables parameterizing the dynamics: original BKL type [4, 5] and quasi Hubble normalized [11]. Making use the scale invariance of Einstein equations one can introduce variables which divided by the Hubble parameter become scale invariant [12]. The Hubble parameter, which in general spacetime is a geometrical average of expansion rates in three space directions, becomes infinite approaching the singularity. Although gravitational field variables like orthonormal frame variables also diverge approaching singularity, normalized by Hubble parameter remain finite and more useful for analytical analysis [8, 12], and they enabled successful numerical verification [3].

However, original BKL variables and Hubble normalized ones cannot be connected by canonical transformation. In both cases, applying dynamical systems method enables identification of the spaces of non-isolated critical (equilibrium) points, which are of nonhyperbolic type. Topologies of these spaces are quite different, and making them explicit constitutes one of the main results of this paper. Additional result is expressing the asymptotic nondiagonal Bianchi IX model in terms of non-divergent variables similar to the Hubble normalized variables, thus enabling future more detailed investigations.

Our paper is organized as follows: Sect. 2 concerns the nondiagonal case. We introduce quasi Hubble normalized variables, examine the asymptotic dynamics in these (and BKL) variables, and identify the spaces of critical points of the corresponding vector fields. The diagonal case is considered in Sect. 3, where we follow the steps of Sect. 2. The numerical simulations of the dynamics is presented in Sect. 4. We conclude in Sect. 5. Appendix A concerns the issue of an effective form of the metric near the singularity. The choice of quotient coordinates, presented in “Appendix B”, enables making an extension of the interpretation of our results. We present the relationship between the BKL and our new variables in “Appendix C”. Finally, we apply the Poincaré sphere to deal with the space of critical points in finite region of phase space in “Appendix D”.

## 2 The nondiagonal case

*a*,

*b*,

*c*are functions of time \(\tau \), satisfying the constraint

*t*as follows

Turning the above dynamics into Hamiltonian dynamics, one can examine qualitatively the mathematical structure of the corresponding physical phase space by using the dynamical systems method (DSM). It has been found that the *critical* points of the system have the following properties: (i) define a three-dimensional continuous subspace of \({\bar{{\mathbb {R}}}}^6\) defined by the relation \(a \gg b \gg c > 0\), with \(a \rightarrow 0\) (see, Eq. (38) of [6] for more details), and (ii) are of the nonhyperbolic type.

The property (i) was already found long time ago [5] without using the DSM. The characteristic (ii) has been identified recently [6]. 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. So the results obtained from inearization of the dynamics cannot be conclusive (see, e.g., [14]).

### 2.1 Quasi Hubble normalized variables

*H*is proportional to the expansion \( H= \theta /3\) and is related to changes of the spatial volume density via \(d \sqrt{g}/dt= 3 H \sqrt{g}\), where \(g=\text {det} g_{\alpha \beta }\). One can also define variables \(n_\alpha \)

*H*:

*H*factors out the overall expansion. Analysing dynamics of the Bianchi IX spacetimes near its singularity in terms of HN variables brought a lot of important and interesting results (see, e.g., [8, 11] and references therein).

In the general case, the Euler angles \((\theta , \varphi , \psi )\) are time dependent and describe the rotation with respect to the frame vectors \(e^a\), which are fixed. In the asymptotic regime the Euler angles become time independent, but \(\Gamma _\alpha \) stay being functions of time.

*C*is a constant of motion. The metric (11) describes only the oscillatory modes devoid of the rotation. Since

*a*,

*b*and

*c*satisfy Eqs. (2)–(3), derived from the

*exact*system of equations with nondiagonal form of 3-metric, they have encoded nondiagonal aspects of the metric, and the line element:

*effective*3-metric. This identification suggests that we have a sort of an effective diagonal metric \(g_{\alpha \beta }\) near the cosmological singularity, i.e., in the asymptotic region of spacetime.

*a*,

*b*,

*c*) satisfy Eqs. (2) and (3). Thus, \(\Sigma _1 + \Sigma _2 +\Sigma _3 = 0\) identically, and \(N_1> 0, N_2> 0, N_3 > 0\) as \(a b c \rightarrow 0\) near the singularity.

In what follows we will present similarities between the set of defined above variables and original HN ones.

*v*is the spatial volume density.

### 2.2 Dynamics

#### 2.2.1 Finding the vector field

#### 2.2.2 Critical points of the dynamics

#### 2.2.3 The linearization of the vector field

*J*of the system (36)–(40), 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 due to relations (42) giving

*J*and characteristic polynomial, we exhibit only the result after embedding conditions (42). Since the real parts of all eigenvalues of the Jacobian are equal to zero, we are dealing with the

*nonhyperbolic*critical points.

*exact*form of our vector field.

## 3 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.

### 3.1 Dynamical system analysis

*strong*relations among \(x_1,~x_2 \) and \(x_3\) in each of the above sets, contrary to the nondiagonal case (see the statement following Eq. (B9)).

*nonhyperbolic*one.

### 3.2 Introducing the qHN variables

### 3.3 The vector field

### 3.4 Critical points

*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:

*nonhyperbolic*one.

## 4 Numerical simulations of the dynamics

In this section we present the numerical simulations of both evolutions, defined by Eqs. (2)–(3) and (50)–(53), to give support to some assumptions of the preceding sections. The numerical method we employed here is the same as described in [10]. Our simulations concern the dynamics with the initial data satisfying the strong inequality defined by Eq. (56). Since the product of the three scale factors is proportional to the volume density of the space, decreasing volume means evolution towards the singularity.

Figure 1a presents the plots of the directional scale factors corresponding to the dynamics of the nondiagonal case. Taking the initial data satisfying (56) leads to the evolution towards the singularity that *maintains* this strong inequality. This result gives support to the claim that this dynamics has the special *asymptotic* regime. Further support can be found in [10], where the simulations have been performed by using the exact dynamics of the general Bianchi IX model filled with a tilted pressureless fluid.

Figure 1b presents the evolution of the directional scale factors of the diagonal case with almost the same initial data as in the nondiagonal case.^{1} No special regime occurs in this case. One can see the permutation symmetry of the relation (54) during the evolution of the system, *contrary* to the nondiagonal case. The permutation of the initial data leads to the same solutions (recoloring the plots), which is consistent with the permutation symmetry of the dynamics (50)–(53).

In fact, the permutation symmetry (54) was used to check the correctness of the numerical simulations.

We were able to keep the numerical error in solving the Hamiltonian constraints, (3) or (53), as low as the order of \(10^{-16}\). This is illustrated in Fig. 2. Further increase of the precision of calculations keeps the plots unchanged.

## 5 Conclusions

Near the cosmological singularity, an evolution of the Bianchi IX model is an infinite sequence of the so called eras each of which consists of the Kasner type epochs [1]. In the *diagonal* case, each epoch can be described, e.g., by the relation \({\tilde{\Gamma }}_1 \sim {\tilde{\Gamma }}_2 > {\tilde{\Gamma }}_3\) (where \(\sim \) means coupled) called an oscillation.^{2} The dynamics of the *nondiagonal* model has essentially different structure [4, 5]: the oscillation of the diagonal type, e.g., \(\Gamma _1 \sim \Gamma _2 > \Gamma _3\) enters sooner or later the relation \(\Gamma _1> \Gamma _2 > \Gamma _3\), which turns into the strong relation \(\Gamma _1 \gg \Gamma _2 \gg \Gamma _3\). Finally, the system approaches the singularity in a finite proper time.

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 in \({{\bar{{\mathbb {R}}}}}^6\) having the same topology, Eqs. (91)–(93), and one set, Eq. (90), with the simple topology of \({{\bar{{\mathbb {R}}}}}^3\). In the latter case, the set of critical points has sophisticated topology, defined by Eq. (41), 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 is expected to be directly linked with the chaoticity of the dynamics of the Bianchi IX model. We conjecture that due to the different topologies of the critical spaces the chaoticity aspects of both cases can be different. Further studies are required to get insight into this intriguing issue.

Our main concern is the nonhyperbolicity of equilibrium points in both diagonal and general cases. They do not define a set of *isolated* points, but a three-dimensional *continuous* space. Thus, our choice of phase space variables seems to be unsatisfactory. We have already tried [6] to use the so-called blowing up technic initiated by McGehee [19] to avoid this obstacle, but with no success. More sophisticated approach based on \(\sigma \)-process of algebraic geometry proposed in [7] may bring some progress, but it leads to a noncanonical variables that we try to avoid. Another framework proposed for the spacially inhomogeneous models [12], within Hubble-normalized approach, can be probably specialized to the homogeneous models. However, this formulation is again a noncanonical one which we do not favour.

The way out seems to be giving up the insistence on dealing entirely with canonical formulations and planning making use of coherent states quantization methods (based on phase space structure of the underlying system) that we have recently applied to the diagonal Bianchi IX model [20, 21]. In such a case making use of the results of [12] to elucidate mathematical structure of the physical phase space specific to the dynamics of the Bianchi IX model (in both considered cases) would make sense. This is supposed to be the next step of our investigation and the results of the present paper could be used as a starting point. Another approach would be based on modification of the definition of the Hubble-normalized variables that we use in the present paper.

The fact that some critical points occur at infinity is not an obstacle. The mapping of the set of critical points onto the Poincaré sphere (considered, e.g., for the nondiagonal case, in “Appendix C”) des not change the type of the criticality. It stays to be of nonhyperbolic type. Thus, compactification of phase space does not help.

It seems that the nonhyperbolicity of the equilibrium points distributed in a continuous way in higher dimensional space is a generic feature of the dynamics of the Bianchi IX model and cannot be avoided. These properties may correspond to mathematical structure [13, 22] underlying chaotic behaviour of considered dynamics (see, e.g., [23, 24]), and needs to be further examined.

## Footnotes

## Notes

### Acknowledgements

We are grateful to Claes Uggla for the suggestion to use the Hubble normalized variables and to Juliette Hell for valuable discussions concerning the dynamics of the Bianchi IX model. Finally, we appreciate inspiring discussions with Vladimir Belinski. This work was partially supported by the German-Polish bilateral project DAAD and MNiSW, No 57391638, “Model of stellar collapse towards a singularity and its quantization”.

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