BKL Method in the Bianchi IX Universe Model Revisited: Advanced Study

Given the extreme difficulty in finding analytical solutions to Einstein’s equations for universe models,
such as the Bianchi type IX, different physical-mathematical techniques have been designed to
attempt to find, at least approximate solutions that could have relevance in modern cosmology.
Belinskii, Khalatnikov and Lifshitz (BKL) have developed a method to study a particular time evolution
dynamic of the Bianchi type IX universe model. Herein, it is shown that this method contains pitfalls.
These pitfalls, among other things, prevent the conclusion that this model provides chaotic behaviour.
We show that the initial presumption made by BKL turns out to be inconsistent with the assumed
initial conditions, since it cannot be kept along with the time-extended BKL process. That is, the
Kasner stipulations corresponding to    , taken as initial conditions, are at odds with the initial
assumption imposed on Einstein’s field equations.