Multidimensional mixmaster models

Szydłowski, Marek; Biesiada, Marek; Szczęsny, Jerzy

doi:10.1016/0370-2693(87)90880-x

Cited by 12 publications

(8 citation statements)

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“…For a cosmological application of the eleven space-times models see [383]. A wide analysis of the dynamics concerning homogeneous and inhomogeneous multidimensional cosmologies see [336,42,337,312,478,176,491,30,412,123,337,334,333,335,332,331,330].…”

Section: Multidimensional Oscillatory Regimementioning

confidence: 99%

Classical and Quantum Features of the Mixmaster Singularity

Montani

Battisti

Benini

et al. 2008

Int. J. Mod. Phys. A

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This review article is devoted to analyze the main properties characterizing the cosmological singularity associated to the homogeneous and inhomogeneous Mixmaster model. After the introduction of the main tools required to treat the cosmological issue, we review in details the main results got along the last forty years on the Mixmaster topic. We firstly assess the classical picture of the homogeneous chaotic cosmologies and, after a presentation of the canonical method for the quantization, we develop the quantum Mixmaster behavior. Finally, we extend both the classical and quantum features to the fully inhomogeneous case. Our survey analyzes the fundamental framework of the Mixmaster picture and completes it by accounting for recent and peculiar outstanding results.are obtained requiring the action for the matter S m introduced in Eq. (2.1.11) to be invariant under diffeomorphisms. For the Einstein equations, the Bianchi identities ∇ i G ij = 0 may be viewed as a consequence of the invariance of the Hilbert action under diffeomorphisms. Let us now list the principal aspects of the three fields under consideration.• The energy-momentum tensor of a perfect fluid is given bywhere u i is a unit time-like vector field representing the four-velocity of the fluid. The scalar functions p and ρ are the energy density and the pressure, respectively, as2.8b) where j k is the current density four-vector of electric charge and [ ] denote the antisymmetric sum, or in the forms language equations (2.2.8) are written as d ⋆ F = 4π ⋆ j (2.2.9a) dF = 0 (2.2.9b)where d is the exterior derivative and ⋆ is the Hodge star operator [505,370].

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Section: Multidimensional Oscillatory Regimementioning

confidence: 99%

Classical and Quantum Features of the Mixmaster Singularity

Montani

Battisti

Benini

et al. 2008

Int. J. Mod. Phys. A

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“…The iteration of this map represents the so called Mixmaxter attractor, the subsequent transitions between Kasner epochs as we approach the big-bang singularity in more general (Bianchi IX) models (see [12] for a recent discussion). It is reasonable to think that a similar map may exist for Lovelock gravity models as well, despite the fact that the previous chaotic behaviour disappears in higher dimensions for vacuum Einstein gravity [13][14][15]. The characterization of the space of solutions performed in this section can thus be taken as the starting point for the more general analysis of big-bang singularities for this class of theories.…”

Section: B Structure Of Vacuum Solution Space: the Kasner Shamrockmentioning

confidence: 70%

Pure Lovelock Kasner metrics

Camanho

Dadhich

Molina

2015

Class. Quantum Grav.

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We study pure Lovelock vacuum and perfect fluid equations for Kasner-type metrics. These equations correspond to a single N th order Lovelock term in the action in d = 2N + 1, 2N + 2 dimensions, and they capture the relevant gravitational dynamics when aproaching the big-bang singularity within the Lovelock family of theories. Pure Lovelock gravity also bears out the general feature that vacuum in the critical odd dimension, d = 2N + 1, is kinematic; i.e. we may define an analogue Lovelock-Riemann tensor that vanishes in vacuum for d = 2N + 1, yet the Riemann curvature is non-zero. We completely classify isotropic and vacuum Kasner metrics for this class of theories in several isotropy types. The different families can be characterized by means of certain higher order 4th rank tensors. We also analyze in detail the space of vacuum solutions for five and six dimensional pure Gauss-Bonnet theory. It possesses an interesting and illuminating geometric structure and symmetries that carry over to the general case. We also comment on a closely related family of exponential solutions and on the possibility of solutions with complex Kasner exponents. We show that the latter imply the existence of closed timelike curves in the geometry.PACS numbers: I. INTRODUCTIONLovelock gravity is the most natural extension of general relativity (GR) in dimension higher than four, as it retains the basic character of the theory -the equation of motion remains second order despite the action being higher order in Riemann curvature. No other purely gravitational theory preserves this crucial feature. Yet another interesting feature of GR is the fact that it is kinematic in three dimensions and it turns dynamical in the next even dimension, i.e. d = 4. In three dimensions the Riemann curvature tensor can be written in terms of the Ricci so that there exist no non-trivial vacuum solution. If we want this property to remain true as we go to higher odd dimensions there is a unique choice that corresponds to pure Lovelock gravity [1,2]. The action reduces in this case to a single N th order Lovelock term, with or without cosmological constant [19] in dimensions d = 2N + 1, 2N + 2. This is the maximal order term in the Lovelock series, higher order terms being either topological or zero. There is no sum over lower order terms and in particular there is no Einstein term in the action.It is possible to define an analogue of the Riemann tensor for N th order Lovelock gravity, its characterizing property being that the trace of its Bianchi derivative vanishes, yielding the corresponding divergence free analogue of the Einstein tensor [3]. This is exactly the same as the one obtained from the variation of the N th order Lovelock action term. For the appropriate definition of Lovelock-Riemann tensor and zero cosmological constant, any pure Lovelock vacuum in odd d = 2N + 1 dimensions is Lovelock flat, i.e. any vacuum solution of the theory has vanishing Lovelock-Riemann tensor [1,4]. Likewise, even for non-zero cosmological constant, the Weyl curvat...

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“…where (a, by c) E {(4,5,6), (7,8,9)}, and eigenvalues n, are given in table 1. Table 1 implies that the microspace is a product of class A Bianchi models ( a = 0), whereas the macrospace can either be a class A or a class B model.…”

Section: Classification Of Homogeneous Multidimensional Cosmological ...mentioning

confidence: 99%

“…Tensor components of Einstein equations can be written in a suitable Lagrange form if we use the least-action principle based on the well known identity: m , n = 0 , 1 , ..., n. a(RdTiT) = (Rmn -4gmnRW'kl Sgmn +GI g""SRnm (9) The second term in ( 9) is a full divergence; however, it should be taken into account since, from the definition of homogeneity, variations of the metric need not be finite.…”

Section: The Hamiltonian Formalismmentioning

confidence: 99%

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The dynamics of homogeneous multidimensional cosmological models

Szydłowski

Pajdosz

1989

Class. Quantum Grav.

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The authors present a systematic method of investigating the dynamics of homogeneous multidimensional cosmological models. The method is based on the full classification of homogeneous multidimensional Kaluza-Klein models. The space sections are homogeneous diffeomorphic to the coset space G/H, where H is a discrete subgroup (dim(H)=0). Here they set H=I (the identity group) and consider the group manifold MD+3=G3+D as a model space. Additionally, they assume that MD+3=M3*BD and that the spatial metric is decomposable, i.e. equivalent to independent metrics on the macrospace factor manifold M3 and microspace BD which are orthogonal to each other. The spatial metric matrix g has then a block diagonal form g=(g3,gD). They further assume that metric gD is also in block diagonal form with matrix blocks in each microspace sector. They reduce the investigation of dynamics to the investigation of multidimensional dynamical systems and then draw some general conclusions about chaotic behaviour and genericity of homogeneous multidimensional cosmological models.

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Multidimensional mixmaster models

Cited by 12 publications

References 12 publications

Classical and Quantum Features of the Mixmaster Singularity

Classical and Quantum Features of the Mixmaster Singularity

Pure Lovelock Kasner metrics

The dynamics of homogeneous multidimensional cosmological models

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