XLI Workshop on Geometric Methods in Physics Białystok, 1-6.07.2024 XIII School on Geometry and Physics Białystok, 8-12.07.2024

Piotr P. Goldstein

Variants of asymptotic behaviour in the BKL scenario as described with a quadric of kinetic-energy

The Belinski-Khalatnikov-Lifshitz (BKL) scenario [1], which describes asymptotic behaviour of an anisotropic universe near the cosmological singularity, is found to manifest several interesting variants of behaviour. These variants are classified and analysed in terms of the model's diagonal Hamitonian variables. The geometric picture of the dynamics may be seen as advancing of the system in the 3-dimensional momentum space. This motion takes place within a quadric of the "kinetic energy". As the latter is an indefinite quadratic form, the quadric is a cone. Since the governing equations are time-reversible, the description may be used for a collapse towards or expansion from the singularity.

We find that the possible "subscenarios" of the collapse are:
  1. unstable squeeze of the universe in all directions to a point;
  2. stable approaching a limit in which the universe collapses in one or two directions, while infinitely stretching in the remaining two or one, like in Kasner's models [2];
  3. an oscillatory approach to such a limit; finally
  4. an approach to a limit in two directions with unstable oscillations of increasing amplitude in the third.


  1. V. A. Belinskii, I. M. Khalatnikov, and E. M. Lifshitz, Oscillatory approach to a singular point in the relativistic cosmology, Adv. Phys., 1970, 19 (80), 525–573.
  2. C. W. Misner, K. S. Thorne and J. A. Wheeler, Gravitation, W.H. Freeman & Co., San Francisco 1973, p. 801ff.
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