Homogeneous materials have identical properties at all points, while isotropic materials have identical properties in all directions.

BKL singularity. https://en.wikipedia.org/wiki/BKL_singularity

Friedmann in 1922–1924. The Universe is assumed homogeneous (space has the same metric properties (measures) in all points) and isotropic (space has A Belinski–Khalatnikov–Lifshitz (BKL) singularity is a model of the dynamic evolution of the universe near the initial gravitational singularity, described by an anisotropic, chaotic solution of the Einstein field equation of gravitation. According to this model, the universe is chaotically oscillating around a gravitational singularity in which time and space become equal to zero or, equivalently Near… The basis of modern cosmology are the special solutions of the Einstein field equations found by Alexander Friedmann in 1922–1924. The Universe is assumed homogeneous (space has the same metric properties (measures) in all points) and isotropic (space has the same measures in all directions). Friedmann's solutions allow two possible geometries for space: closed model with a ball-like, outwards-bowed space (positive curvature) and open model with a saddle-like, inwards-bowed space (negative curvature). In both models, the Universe is not standing still, it is constantly either expanding (becoming larger) or contracting (shrinking, becoming smaller). This was confirmed by Edwin Hubble who established the Hubble redshift of receding galaxies. The present consensus is that the isotropic model, in general, gives an adequate description of the present state of the Universe; however, isotropy of the present Universe by itself is not a reason to expect that it is adequate for describing the early stages of Universe evolution. At the same time, it is obvious that in the real world homogeneity is, at best, only an approximation. Even if one can speak about a homogeneous distribution of matter density at distances that are large compared to the intergalactic space, this homogeneity vanishes at smaller scales. On the other hand, the homogeneity assumption goes very far in a mathematical aspect: it makes the solution highly symmet It was proved that the cosmological models based on six main superstring models plus eleven-dimensional supergravity model exhibit the chaotic BKL dynamics towards the singularity. A connection was discovered between oscillatory BKL-like cosmological models and a special subclass of infinite-dimensional Lie algebras – the so-called hyperbolic Kac–Moody algebras. == Introduction == The basis of modern cosmology are the special solutions of the Einstein field equations found by Alexander Friedmann in 1922–1924. The Universe is assumed homogeneous (space has the same metric properties (measures) in all points) and isotropic (space has the same measures in all directions). On the other hand, the homogeneity assumption goes very far in a mathematical aspect: it makes the solution highly symmetric which can impart specific properties that disappear when considering a more general case. Another important property of the isotropic model is the inevitable existence of a time singularity: time flow is not continuous, but stops or reverses after time reaches some very large or very small value. Between singularities, time flows in one direction: away from the singularity (arrow of time). The analysis of the behavior of the model under perturbative action, performed by BKL, delineates a complex oscillatory mode on approaching the singularity. They could not give all details of this mode in the broad frame of the general case. However, BKL explained the most important properties and character of the solution on specific models that allow far-reaching analytical study. These models are based on a homogeneous space metric of a particular type. Supposing a homogeneity of space without any additional symmetry leaves a great freedom in choosing the metric. All possible homogeneous (but anisotropic) spaces are classified, according to Bianchi, in several Bianchi types (Type I to IX). (see also Generalized homogeneous solution) BKL investigate only spaces of Bianchi Types VIII and IX. If the metric has the form of eq. 7, for each type of homogeneous spaces exists some functional relation between the reference vectors l, m, n and the space coordinates. The specific form of this relation is not important. The important fact is that for Type VIII and IX spaces, the quantities λ, μ, ν eq. 10 are constants while all "mixed" products l rot m, l rot n, m rot l, etc.. are zeros. Since all 3 quantities λ, μ, ν in those Bianchi types differ from zero, the condition λ = 0 does not hold irrespective of which direction l, m, n has negative power law time dependence. The Einstein equations for the Type VIII and Type IX space models are (the remaining components R l 0 {\displaystyle R_{l}^{0}} , R m 0 Although this latter behavior is not a necessary element of metric evolution close to the singularity, it has all principal qualitative properties: metric oscillation in two spatial dimensions and monotonous change in the third dimension with a certain perturbation of this mode at the end of some time interval. However, the transitions between Kasner epochs in the general case of non-homogeneous spatial metric have not been elucidated in details. The problem connected with the possible limitations upon space geometry caused by the singularity was left aside for further study. According to the isotropic model, the observed isotropy means isotropic properties of radiation that comes to the observer from such regions of space that can not be causally connected with each other. The situation in the oscillatory evolution model near the singularity can be different. For example, in the homogeneous model for Type IX space, a signal is propagated in a direction in which for a long era, scales change by a law close to ~ t. The square of the distance element in this direction is dl2 = t2 l ¯ 2 {\displaystyle {\bar {l}}^{2}} , and the respective element of the four-dimensional interval is d s 2 = d t 2 − t 2 l ¯ 2 {\displaystyle ds^{2}=dt^{2}-t^{2}{\bar {l}}^{2}} .