# Curvature invariants for the Bianchi IX spacetime filled with tilted dust

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## Abstract

We present an analysis of the Kretschmann and Weyl squared scalars for the general Bianchi IX model filled with tilted dust. Particular attention is given to the asymptotic regime close to the singularity for which we provide heuristic considerations supported by numerical simulations. The present paper is an extension of our earlier publication (Kiefer et al., in Eur Phys J C 78:691, 2018).

## 1 Introduction

Einstein’s theory of general relativity, GR, suffers from singularities (see, e.g. [2, 3, 4, 5]). These are pathologies of spacetime, which determine the limits of the validity of GR. They also signal the occurrence of interesting phenomena like black holes and the cosmological big bang, which seem to be supported by observational data.

The present paper is an extension of our recently published paper [1] on the evolution towards the singularity of the *general* Bianchi IX spacetime. The dynamics of this model underlies the dynamics of the Belinski–Khalatnikov–Lifshitz (BKL) scenario (for an overview see e.g. [6]), which is conjectured to describe the approach towards a *generic* spacelike singularity of GR. As far as we know, the time evolution of particular curvature invariants of the general BIX has not been considered yet.

The commonly known singularity theorems (see e.g. [2, 3, 4]) predict the existence of incomplete geodesics, but say little about other features of singularities. Studying particular curvature invariants gives more characteristics of these pathologies. The blowing up of the Kretschmann scalar was proved rigorously by Ringstrom [7] in the vacuum case of the BIX model (mixmaster universe). Barrow and Hervik [8] have studied the Weyl tensor and provided asymptotic expression for homogeneous cosmological models filled with non-tilted perfect fluids. The behaviour of the Weyl squared scalar is of particular interest since it is conjectured that this invariant acts as a measure of gravitational entropy [8, 9].

In our paper we examine the time evolution of the Kretschmann scalar for the non-diagonal (general) BIX spacetime which requires the coupling, for instance, to a *tilted* matter field. The latter is chosen to be dust (for simplicity). The difference in the dynamics between the diagonal and non-diagonal cases has been considered in [10].

Our paper is organized as follows: We first review in Sect. 2 the Hamiltonian formulation of the dust filled Bianchi IX model which was employed for the numerical simulation of the dynamics in [1]. In Sect. 3 we compute an expression for the Kretschmann scalar which allows for a numerical evaluation via the framework based on the Hamiltonian formulation.

Section 4 is devoted to the asymptotic regime close to the singularity. We provide heuristic considerations predicting the behaviour of the Kretschmann scalar and support our result with a numerical analysis. We conclude in Sect. 5.

## 2 Hamiltonian formulation of the dust filled Bianchi IX model

^{1}satisfy the relation

^{2}The scale factor \(\exp (\alpha )\) is related to the volume, while the anisotropy factors \(\beta _+\) and \(\beta _-\) describe the shape of the universe. The variables \(\Gamma _1\), \(\Gamma _2\) and \(\Gamma _3 \) were used by BKL in their original analysis [19]. In addition, we have introduced the \(SO(3,{\mathbb {R}})\) matrix \(O\equiv \left\{ O_i{}^{j}\right\} \equiv O_{\theta }O_{\phi }O_{\psi }\), which will be parameterized by a set of Euler angles, \(\left\{ \theta ,\phi ,\psi \right\} \in [0,\pi ]\times [0,2\pi ]\times [0,\pi ]\). Explicitly,

*T*, that is, the proper time measured by observes co-moving with the dust particles. The three-curvature scalar on spatial hypersurfaces of constant coordinate time

*t*is given by

*t*over some finite time interval \([t_1, t_2]\). We restrict our attention to the so called tumbling case, that is, all \(v_i\) are non-zero initially. This case might be considered as the most generic one in the context of the model under consideration. For convenience, we restrict ourselves in this work to the numerical solution which was already considered in [1]. The part of the solution plotted in Fig. 2b extends over one Kasner era. We regard this solution to occur at the transition into the asymptotic regime.

The numerical accuracy of our solution has been confirmed by checking the (approximate) preservation of the Hamiltonian constraint \({\mathcal {H}}=0\), and the constant of motion \(C^2-1=0\). Both \({\mathcal {H}}\) and \(C^2-1\) stay at order \(10^{-15}\). We believe that the effect of chaos is negligible in a single Kasner era. For further details see [1].

## 3 Calculation of the Kretschmann scalar

*R*the Ricci scalar. For our purposes it is convenient to make use of the constraints and the equations of motion to simplify the expressions such that they are suited for a numerical evaluation. We will do so throughout the calculation in this section and bring our expression into a form that is ready for a numerical evaluation. This means that all expressions should only involve the variables (11) as well as the constants \(p_T'\equiv 12p_T\) and

*C*. Furthermore we shall use the quasi-Gaussian gauge \(N^i=0\) while keeping the lapse

*N*unspecified. We now proceed by calculating the terms on the right-hand side of equation (12).

It is well known that the Weyl squared scalar vanishes for the Friedmann models. The dust filled closed Friedmann universe is included in the model under consideration as the particular case for which \(\Gamma _1=\Gamma _2=\Gamma _3\) and \(C=0\). As a consistency check of our calculation we convinced ourselves that the Weyl squared scalar vanishes for these restrictions. We find that \(B_{ij}B^{ij}\) and \(E_{ij}E^{ij}\) vanish separately and hence \(C_{\mu \nu \lambda \sigma }C^{\mu \nu \lambda \sigma }=0\) as expected.

^{3}is shown in Fig. 1. We note that the bare Kretschmann scalar appears to roughly blow up exponentially in

*t*and it rapidly exceeds the range of numbers that are accessible in Matlab. This is why from now on we turn to numerically evaluating the so-called Hubble normalized Kretschmann scalar \(R_{\mu \nu \lambda \sigma }R^{\mu \nu \lambda \sigma }/|K^i{}_i|\). This quantity has the virtue of being dimensionless and numerically well behaved. The expansion scalar is given by

## 4 The asymptotic regime close to the singularity

We remark again that we are considering the tumbling case. We expect, however, that a similar discussion also holds for the non-tumbling and non-rotating cases.

*effectively*diagonal. For our variables this means that the dust velocities \(v_i\) assume constant values. The main purpose of the article [1] was to provide a numerical verification for the two assumptions.

According to the phrase “matter does not matter” we expect the matter terms in the Kretschmann scalar to be negligible in the asymptotic regime, that is, the Weyl part should dominate over the Ricci part.

*u*following the considerations in [19]. Doing so and using the assumption (29) we obtain that the Hubble-normalized Kretschmann scalar can be approximated by

*u*in Fig. 2a. It is important that the function has a maximum in \(u=1\).

^{4}

*transformations of the first kind*while they call bounces from centrifugal walls

*transformations of the second kind*. Transformations of the first kind change the Lifshitz-Khalatnikov parameter according to \( u\overset{1}{\rightarrow } u-1 \). Transformations of the second kind interchange the values of the velocities according to \((\log \Gamma _1)^\cdot \overset{2}{\rightarrow } (\log \Gamma _2)^\cdot \), \((\log \Gamma _2)^\cdot \overset{2}{\rightarrow } (\log \Gamma _1)^\cdot \) and leave the value of

*u*unchanged, i.e. \(u \overset{2}{\rightarrow }u\). It follows that \(\overset{1}{\rightarrow }\) changes the value of the Hubble normalized Kretschmann scalar (31) while \(\overset{2}{\rightarrow }\) does not. According to the analysis in [19] a typical Kasner era can be expressed as a sequence of

*n*Kasner epochs which starts with an epoch that has a maximum

*u*-value larger than 1 when evolving towards the singularity. The value of

*u*decreases with each transformation of the first kind and ends with the epoch for which

*u*becomes smaller than 1 for the first time, e.g.

*u*-map was found to be asymptotically exact for particular cases (for a collection of rigorous results concerning the

*u*-map see [5]). A solution of the discrete mixmaster map and a detailed study of its chaotic nature for the vacuum Bianchi IX case can be found in [22].

It was helpful, in this paper, to support the computations by using the tensor algebra package *xAct* [23]. Numerical calculations were performed using MATLAB R2016b.

## 5 Summary

The main purpose of this paper is to provide a description of the temporal behaviour of the Kretschmann scalar in the asymptotic regime. In this regime the volume density, being proportional to the product of the three directional scale factors, evolves towards zero [19], but this is not a satisfactory indication of the singularity as it depends on the choice of coordinates. The blowing up of curvature invariants, on the other hand, shows that we are dealing with a genuine curvature singularity.

During Kasner epochs, \(R_{\mu \nu \lambda \sigma }R^{\mu \nu \lambda \sigma }\) increases like the expansion to the power four. Over the course of a single Kasner era the value of the Hubble normalized Kretschmann scalar increases until it drops down to a finite value when it ends. This process will repeat itself with the beginning of the next Kasner era until the system approaches the singularity.

The present paper is supposed to be an extension of our previous paper [1], which considers, for simplicity, only the tilted dust field as a source. The discussion of other tilted fluids goes beyond the scope of our present programme. The effect of tilted radiation, which has been studied analytically and numerically in the recent paper [24], is particularly interesting.

The asymptotic regimes of the Bianchi IX and BVIII models are quite similar [25]. Both models have been used to derive the BKL scenario [26].

The asymptotic regime approximates well the dynamics near the singularity. This is why it was recently used in the struggle for removing the singularity of the BKL scenario by quantization [27].

## Footnotes

- 1.
Throughout this work we choose the units such that \(\frac{3}{4\pi G}\int \sigma ^1\wedge \sigma ^2\wedge \sigma ^3=1\), which correspond the setting \(\kappa ={8\pi G}/{c^4}=6\).

- 2.
Note that Misner originally used the variable \(\Omega =-\alpha \).

- 3.
The existence of the recollapse was proven by Lin and Wald [21].

- 4.
This maximum implies an upper bound for the Hubble normalized Kretschmann scalar during Kasner eras given by 64 / 27.

## Notes

### Acknowledgements

This paper profited from correspondence with Vladimir Belinski and Claes Uggla. Moreover we would like to thank Claus Kiefer for helpful discussions. We are also grateful to the anonymous referee for constructive criticism. This work was supported by the German-Polish bilateral project DAAD and MNiSW, No 57391638.

## References

- 1.C. Kiefer, N. Kwidzinski, W. Piechocki, On the dynamics of the general Bianchi IX spacetime near the singularity. Eur. Phys. J. C
**78**, 691 (2018)ADSCrossRefGoogle Scholar - 2.S.W. Hawking, G.F.R. Ellis,
*The Large Scale Structure of Space-Time*(Cambridge University Press, Cambridge, 1973)CrossRefGoogle Scholar - 3.S. Hawking, R. Penrose,
*The Nature of Space and Time*(Princeton University Press, Princeton, 1996)zbMATHGoogle Scholar - 4.J.M.M. Senovilla, Singularity theorems and their consequences. Gen. Relativ. Gravit.
**30**, 701 (1998)ADSMathSciNetCrossRefGoogle Scholar - 5.C. Uggla, Spacetime singularities: recent developments. Int. J. Mod. Phys. D
**22**, 1330002 (2013)ADSMathSciNetCrossRefGoogle Scholar - 6.V. Belinski, M. Henneaux,
*The Cosmological Singularity*(Cambridge University Press, Cambridge, 2017)CrossRefGoogle Scholar - 7.H. Ringström, Curvature blow up in Bianchi VIII and IX vacuum spacetimes. Class. Quantum Gravity
**17**, 713 (2000)ADSMathSciNetCrossRefGoogle Scholar - 8.J.D. Barrow, S. Hervik, The Weyl tensor in spatially homogeneous cosmological models. Class. Quantum Gravity
**19**, 155 (2002)ADSMathSciNetCrossRefGoogle Scholar - 9.R. Penrose, Singularities and time-asymmetry, in
*General Relativity: An Einstein Centenary Survey*, ed. by S.W. Hawking, W. Israel (Cambridge University Press, Cambridge, 1979), pp. 581–638Google Scholar - 10.E. Czuchry, N. Kwidzinski, W. Piechocki, Comparing the dynamics of diagonal and general Bianchi IX spacetime. Eur. Phys. J. C
**(accepted for publication)**Google Scholar - 11.A.R. King, G.F.R. Ellis, Tilted homogeneous cosmological models. Commun. Math. Phys.
**31**, 209 (1973)ADSMathSciNetCrossRefGoogle Scholar - 12.R.A. Matzner, L.C. Shepley, J.B. Warren, Dynamics of SO(3, R)-homogeneous cosmologies. Ann. Phys. (N.Y.)
**57**, 401 (1970)ADSCrossRefGoogle Scholar - 13.L.P. Grishchuk, A.G. Doroshkevich, V.N. Lukash, The model of mixmaster universe with arbitrarily moving matter. J. Exp. Theor. Phys.
**34**, 1 (1972)ADSGoogle Scholar - 14.M.P. Ryan, Qualitative cosmology: diagrammatic solutions for Bianchi type IX universes with expansion, rotation, and shear. I. The symmetric case. Ann. Phys. (N.Y.)
**65**, 506 (1971)ADSCrossRefGoogle Scholar - 15.M.P. Ryan, Qualitative cosmology: diagrammatic solutions for Bianchi type IX universes with expansion, rotation, and shear. II. The general case. Ann. Phys. (N.Y.)
**68**, 541 (1971)ADSCrossRefGoogle Scholar - 16.R.T. Jantzen, Spatially homogeneous dynamics: a unified picture. arXiv:gr-qc/0102035. Originally published in the Proceedings of the International School Enrico Fermi, Course LXXXVI (1982) on Gamov Cosmology, edited by R. Ruffini and F. Melchiorri (North Holland, Amsterdam, 1987), pp. 61–147
- 17.M.P. Ryan,
*Hamiltonian Cosmology*(Springer, Berlin, 1972)Google Scholar - 18.M.P. Ryan, L.C. Shepley,
*Homogeneous Relativistic Cosmologies*(Princeton University Press, Princeton, 1975)Google Scholar - 19.V.A. Belinskii, I.M. Khalatnikov, M.P. Ryan, The oscillatory regime near the singularity in Bianchi-type IX universes. Preprint
**469**(1971). Landau Institute for Theoretical Physics, Moscow (unpublished); published as sections 1 and 2 in M.P. Ryan, Ann. Phys. (N.Y.)**70**, 301 (1971)Google Scholar - 20.M. Alcubierre,
*Introduction to 3+1 Numerical Relativity*(Oxford University Press, Oxford, 2008)CrossRefGoogle Scholar - 21.X. Lin, R.M. Wald, Proof of the closed-universe recollapse conjecture for general bianchi type-IX cosmologies. Phys. Rev. D
**44**, 2444 (1990)ADSCrossRefGoogle Scholar - 22.L.P. Chernoff, V.N. Barrow, The model of mixmaster universe with arbitrarily moving matter. J. Exp. Theor. Phys.
**34**, 1 (1972)Google Scholar - 23.J.M. Martín-García, xAct: efficient tensor computer algebra for the Wolfram language. http://www.xact.es
- 24.C. Ganguly, J.D. Barrow, Evolution of cyclic mixmaster universes with noncomoving radiation. Phys. Rev. D
**96**, 123534 (2017)ADSMathSciNetCrossRefGoogle Scholar - 25.V.A. Belinski, private communicationGoogle Scholar
- 26.V.A. Belinskii, I.M. Khalatnikov, E.M. Lifshitz, Oscillatory approach to a singular point in the relativistic cosmology. Adv. Phys.
**19**, 525 (1970)ADSCrossRefGoogle Scholar - 27.A. Góźdź, W. Piechocki, G. Plewa, Quantum Belinski–Khalatnikov–Lifshitz scenario. Eur. Phys. J. C
**79**, 45 (2019)ADSCrossRefGoogle Scholar

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