Multidimensional integrable vacuum cosmology with two curvatures

V. R. Gavrilov; V. D. Ivashchuk; V. N. Melnikov

doi:10.1088/0264-9381/13/11/018

paper

Multidimensional integrable vacuum cosmology with two curvatures

arXiv:gr-qc/9602063 · doi:10.1088/0264-9381/13/11/018

Abstract

The vacuum cosmological model on the manifold $R \times M_{1} \times \dots \times M_{n}$ describing the evolution of $n$ Einstein spaces of non-zero curvatures is considered. For $n = 2$ the Einstein equations are reduced to the Abel (ordinary differential) equation and solved, when $(N_{1} =$ dim $M_{1}, N_{2} =$ dim $M_{2}) = (6, 3), (5, 5), (8, 2)$ . The Kasner-like behaviour of the solutions near the singularity $t_{s} \to + 0$ is considered ( $t_{s}$ is synchronous time). The exceptional ("Milne-type") solutions are obtained for arbitrary $n$ . For $n = 2$ these solutions are attractors for other ones, when $t_{s} \to + \infty$ . For dim $M = 10, 11$ and $3 \leq n \leq 5$ certain two-parametric families of solutions are obtained from $n = 2$ ones using "curvature-splitting" trick. In the case $n = 2$ , $(N_{1}, N_{2}) = (6, 3)$ a family of non-singular solutions with the topology $R^{7} \times M_{2}$ is found.

21 pages, LaTex. 5 figures are available upon request (hard copy). Submitted to Classical and Quantum Gravity

Multidimensional integrable vacuum cosmology with two curvatures

Abstract

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