7 research outputs found

    Multidimensional integrable vacuum cosmology with two curvatures

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    The vacuum cosmological model on the manifold R×M1××MnR \times M_1 \times \ldots \times M_n describing the evolution of nn Einstein spaces of non-zero curvatures is considered. For n=2n = 2 the Einstein equations are reduced to the Abel (ordinary differential) equation and solved, when (N1=(N_1 = dim M1,N2= M_1, N_2 = dimM2)=(6,3),(5,5),(8,2) M_2) = (6,3), (5,5), (8,2). The Kasner-like behaviour of the solutions near the singularity ts+0t_s \to +0 is considered (tst_s is synchronous time). The exceptional ("Milne-type") solutions are obtained for arbitrary nn. For n=2n=2 these solutions are attractors for other ones, when ts+t_s \to + \infty. For dim M=10,11 M = 10, 11 and 3n53 \leq n \leq 5 certain two-parametric families of solutions are obtained from n=2n=2 ones using "curvature-splitting" trick. In the case n=2n=2, (N1,N2)=(6,3)(N_1, N_2)= (6,3) a family of non-singular solutions with the topology R7×M2R^7 \times M_2 is found.Comment: 21 pages, LaTex. 5 figures are available upon request (hard copy). Submitted to Classical and Quantum Gravit

    Integration of D-dimensional 2-factor spaces cosmological models by reducing to the generalized Emden-Fowler equation

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    The D-dimensional cosmological model on the manifold M=R×M1×M2M = R \times M_{1} \times M_{2} describing the evolution of 2 Einsteinian factor spaces, M1M_1 and M2M_2, in the presence of multicomponent perfect fluid source is considered. The barotropic equation of state for mass-energy densities and the pressures of the components is assumed in each space. When the number of the non Ricci-flat factor spaces and the number of the perfect fluid components are both equal to 2, the Einstein equations for the model are reduced to the generalized Emden-Fowler (second-order ordinary differential) equation, which has been recently investigated by Zaitsev and Polyanin within discrete-group analysis. Using the integrable classes of this equation one generates the integrable cosmological models. The corresponding metrics are presented. The method is demonstrated for the special model with Ricci-flat spaces M1,M2M_1,M_2 and the 2-component perfect fluid source.Comment: LaTeX file, no figure

    Erratum: How to measure the curvature of space-time

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    We propose a new method of measuring the curvature of space-time using test particles. We show that to determine the curvature of space-time in vacuum it is necessary to use at least four test particles and in general at least six

    Billiard Representation for Multidimensional Cosmology with Multicomponent Perfect Fluid near the Singularity

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    The multidimensional cosmological model describing the evolution of nn Einstein spaces in the presence of multicomponent perfect fluid is considered. When certain restrictions on the parameters of the model are imposed, the dynamics of the model near the singularity is reduced to a billiard on the (n1)(n-1)-dimensional Lobachevsky space Hn1H^{n-1}. The geometrical criterion for the finiteness of the billiard volume and its compactness is suggested. This criterion reduces the problem to the problem of illumination of (n2)(n-2)-dimensional sphere Sn2S^{n-2} by point-like sources. Some generalization of the considered scheme (including scalar field and quantum generalizations) are considered.Comment: 16 pages, LaTex. 4 figures are available upon request (hard copy

    Multidimensional Cosmological and Spherically Symmetric Solutions with Intersecting p-Branes

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