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On the Dynamics of the General Bianchi IX Spacetime Near the Singularity Eur. Phys. J. C (2018) 78:691 https://doi.org/10.1140/epjc/s10052-018-6155-8 Regular Article - Theoretical Physics On the dynamics of the general Bianchi IX spacetime near the singularity Claus Kiefer1,a, Nick Kwidzinski1,b, Włodzimierz Piechocki2,c 1 Institute for Theoretical Physics, University of Cologne, Zülpicher Strasse 77, 50937 Cologne, Germany 2 Department of Fundamental Research, National Centre for Nuclear Research, Ho˙za 69, 00-681 Warsaw, Poland Received: 24 July 2018 / Accepted: 11 August 2018 © The Author(s) 2018 Abstract We show that the complex dynamics of the gen- retained in the evolution towards singularities [10] (see [11] eral Bianchi IX universe in the vicinity of the spacelike sin- for an extended physical interpretation). This motivated the gularity can be approximated by a simplified system of equa- activity of the Landau Institute in Moscow to examining the tions. Our analysis is mainly based on numerical simulations. dynamics of homogeneous spacetimes [12]. A group of rel- The properties of the solution space can be studied by using ativists inspired by Lev Landau, including Belinski, Khalat- this simplified dynamics. Our results will be useful for the nikov and Lifshitz (BKL), started to investigating the dynam- quantization of the general Bianchi IX model. ics of the Bianchi VIII and IX models near the initial space- like cosmological singularity [13]. After several years, they found that the dynamical behaviour can be generalized to 1 Introduction a generic solution of general relativity [14]. They did not present a mathematically rigorous proof, but rather a con- The problem of spacetime singularities is a central one in jecture based on deep analytical insight. It is called the BKL classical and quantum theories of gravity. Given some gen- conjecture (or the BKL scenario if specialized to the Bianchi- eral conditions, it was proven that general relativity leads to type IX model). The BKL conjecture is a locality conjecture singularities, among which special significance is attributed stating that terms with temporal derivatives dominate over to big bang and black hole singularities [1]. terms with spatial derivatives when approaching the singu- The occurrence of a singularity in a physical theory usu- larity (with the exception of possible ‘spikes’ [8,9]). Conse- ally signals the breakdown of that theory. In the case of gen- quently, points in space decouple and the dynamics then turn eral relativity, the expectation is that its singularities will out to be effectively the same as those of the (non-diagonal) disappear after quantization. Although a theory of quantum Bianchi IX universe. (In canonical gravity, this is referred to gravity is not yet available in finite form, various approaches as the strong coupling limit, see e.g. [2, p. 127].) exist within which the question of singularity avoidance can The dynamics of the Bianchi IX towards the singularity be addressed [2]. Quantum cosmological examples for such are characterized by an infinite number of oscillations, which an avoidance can be found, for example, in [3–6] and the give rise to a chaotic character of the solutions (see e.g [15]). references therein. Progress towards improving the mathematical rigour of the Independent of the quantum fate of singularities, the ques- BKL conjecture has been made by several authors (see e.g. tion of their exact nature in the classical theory, and in par- [16]), while numerical studies giving support to the conjec- ticular for cosmology, is of considerable interest and has a ture have been performed (see e.g. [17]). long history; see, for example [7,8], for recent reviews. This The dynamics of the diagonal Bianchi IX model, in the is also the topic of the present paper. Hamiltonian formulation, were studied independently from Already in the 1940s, Evgeny Lifshitz investigated the BKL by Misner [18,19]. Misner’s intention was to search for gravitational stability of non-stationary isotropic models of a possible solution to the horizon problem by a process that universes. He found that the isotropy of space cannot be he called “mixing”.1 Ryan generalized Misner’s formalism to the non-diagonal case in [20,21]. A qualitative treatment a e-mail: [email protected] b e-mail: [email protected] 1 This is how the diagonal Bianchi IX model received the name mix- c e-mail: [email protected] master universe. 0123456789().: V,-vol 123 691 Page 2 of 10 Eur. Phys. J. C (2018) 78:691 1 ¯ ¯ ¯ ¯ ¯ of the dynamics for all the Bianchi models may be found in σ =−sin(ψ)dθ + cos(ψ)sin(θ)dφ, the review article by Jantzen [22], and we make reference to σ 2 = cos(ψ)¯ dθ¯ + sin(ψ)¯ sin(θ)¯ dφ,¯ (2) it whenever we get similar results. σ 3 = (θ)¯ φ¯ + ψ.¯ Part of the BKL conjecture is that non-gravitational (‘mat- cos d d ter’) terms can be neglected when approaching the singular- The σ i , i = 1, 2, 3, are dual to the vector fields ity. An important exception is the case of a massless scalar (ψ)¯ field, which has analogies with a stiff fluid (equation of state =− (ψ)∂¯ + cos ∂ + (θ)∂¯ , X1 sin θ¯ ¯ φ¯ cos ψ¯ p = ρ) and, in Friedmann models, has the same dependence sin(θ) ρ ∝ −6 (ψ)¯ of the density on the scale factor as anisotropies ( a ). ¯ sin ¯ (3) X2 = cos(ψ)∂θ¯ + ∂φ¯ + cos(ψ)∂ψ¯ , As was rigorously shown in [23], such a scalar field will sup- sin(θ)¯ press the BKL oscillations and thus is relevant during the X3 = ∂ψ¯ , evolution towards the singularity. Arguments for the impor- tance of stiff matter in the early universe were already given which together with X0 = ∂t form an invariant basis of by Barrow [24]. the Bianchi IX spacetime. The Xi are constructed from the In our present work, we shall mainly address the general Killing vectors that generate the isometry group SO(3, R) (non-diagonal) Bianchi IX model near its singularity. Our (see [27] for more details). The basis one-forms satisfy the main motivation is to provide support for a rather simple relation asymptotic form of the dynamics that can suitably model its 1 dσ i =− Ci σ j ∧ σ k , (4) exact complex dynamics. We expect this to be of relevance in 2 jk the quantization of the general Bianchi IX model, which we with Ci = ε being the structure constants of the Lie alge- plan to investigate in later papers; see, for example [25]. Apart jk ijk so( , R) [ , ]=− k from a few particular solutions that form a set of measure bra 3 .TheXi obey the algebra Xi X j CijXk. zero in the solution space, no general analytic solutions to We parametrize the metric coefficients in this frame as fol- the classical equations of motion are known. Therefore, we lows will restrict ourselves to qualitative considerations which will k l ¯ hij = Oi O j hkl, (5) be supported by numerical simulations. The examination of the non-diagonal dynamics presented in [26], though it is where √ √ mathematically satisfactory, is based on the qualitative theory α β + β β − β − β h¯ ≡ h¯ = e2 diag e2 + 2 3 − , e2 + 2 3 − , e 4 + of differential equations, which is of little use for our purpose. ij Our paper is organized as follows. Section 2 contains ≡ diag ( 1, 2, 3) . (6) the formalism and presents our main results for a gen- The variables α, β+, and β− are known as the Misner vari- eral Bianchi IX model. We first specify the kinematics and ables. The scale factor exp(α) is related to the volume, while dynamics. We then consider a matter field in the form of the anisotropy factors β+ and β− describe the shape of this (tilted) dust. This is followed by investigating the asymptotic model universe. The variables 1, 2 and 3 were used by regime of the dynamics near the singularity. Our conclusions BKL in their original analysis [31]. are presented in Sect. 3. The numerical methods used in our j We introduced here a matrix O ≡ Oi ≡ Oθ Oφ Oψ numerical simulations are described in the Appendix. (i corresponding to rows and j corresponding to columns), which is an SO(3, R) matrix that can be parametrized by another set of Euler angles, {θ,φ,ψ} ∈[0,π]×[0, 2π]× 2 The general Bianchi IX spacetime [0,π]. Explicitly, ⎛ ⎞ 2.1 Kinematics cos(ψ) sin(ψ) 0 ⎝ ⎠ Oψ = − sin(ψ) cos(ψ) 0 , The general non-diagonal case describes a universe with 001 ⎛ ⎞ rotating principal axes. The metric in a synchronous frame 10 0 ⎝ ⎠ can be given as follows (see for the following e.g. [27,28]): Oθ = 0 cos(θ) sin(θ) , (7) 2 =− 2 2 + σ i ⊗ σ j , 0 − sin(θ) cos(θ) ds N dt hij (1) ⎛ ⎞ cos(φ) sin(φ) 0 where N is the lapse function. Spatial hypersurfaces in the ⎝ ⎠ Oφ = − sin(φ) cos(φ) 0 . spacetime are regarded topologically as S3 (describing closed 001 universes), which can be parametrized by using three angles θ,¯ φ,¯ ψ¯ ∈[0,π]×[0, 2π]×[0,π]. The basis one-forms The Euler angles θ, φ, and ψ are now dynamical quantities read and describe nutation, precession, and pure rotation of the 123 Eur. Phys. J. C (2018) 78:691 Page 3 of 10 691 principal axes, respectively. In the case of Bianchi IX space- The Lagrangian in the gauge N i = 0 then takes the form ( , R) time, the group SO 3 is the canonical choice for the 2 ˙2 ˙2 2 2 3 2 1 2 diagonalization of the metric coefficients. For a treatment of α −˙α + β+ + β− + I1 ω 3 + I2 ω 1 + I3 ω 2 L = Ne3 other Bianchi models, see [22].
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