标题： 关于Belinski-Khalatnikov-Lifshitz场景的一些精确结果 显示英文标题

标题： Some exact results on the Belinski-Khalatnikov-Lifshitz scenario

摘要： 著名的比列斯基-哈利特尼科夫-利夫希茨（BKL）宇宙在大爆炸奇点附近的场景通过爱因斯坦方程的BKL渐近性得出的几个精确结果得到补充：（1）宇宙学奇点被证明是这些方程描述的宇宙不可避免的开始或结束。 （2）从收缩的初始条件达到奇点需要无限的时间参数$\tau$；任何类型的奇点在有限的$\tau$内都不会发生。 （3）之前找到的精确解[P.G. 和W. Piechocki，Eur. Phys. J. C 82:216 (2022)]是唯一具有定义明确的方向尺度因子比例的渐近情况，这些方向尺度因子已经适当地补偿了各向异性无限增长的影响。 在所有其他情况下，宇宙经历卡斯纳型振荡，在某些方向上将长度尺度减少到接近零，而在其他方向上则大大扩展。 加上精确解的不稳定性[同上引文]，这使得接近奇点的过程不可避免地变得混沌。 （4）提出了简化方程并显式求解以描述这些振荡在其转折点附近的行为。 在对数变量中，发现这些振荡呈现出锯齿形状。 作为副产品，动能的一个二次形式被提出，这是一个用于所有这些分析的简单几何工具。 显示英文摘要

摘要： The well-known Bielinski-Khalatnikov-Lifshitz (BKL) scenario for the universe near the cosmological singularity is supplemented with a few exact results following from the BKL asymptotic of the Einstein equations: (1) The cosmological singularity is proved to be an inevitable beginning or end of the universe as described by these equations. (2) Attaining the singularity from shrinking initial conditions requires infinite time parameter $\tau$; no singularity of any kind may occur in a finite $\tau$. (3) The previously found exact solution [P.G. and W. Piechocki, Eur. Phys. J. C 82:216 (2022)] is the only asymptotic with well-defined proportions between the directional scale factors which have been appropriately compensated against indefinite growth of anisotropy. In all other cases, the universe undergoes oscillations of Kasner type, which reduce the length scales to nearly zero in some directions, while largely extending it in the others. Together with instability of the exact solution [op. cit.], it makes the approach to the singularity inevitably chaotic. (4) Reduced equations are proposed and explicitly solved to describe these oscillations near their turning points. In logarithmic variables, the oscillations are found to have sawtooth shapes. A by-product is a quadric of kinetic energy, a simple geometric tool for all this analysis.