BKL singularity

[2] According to this model, the universe is chaotically oscillating around a gravitational singularity in which time and space become equal to zero or, equivalently, the spacetime curvature becomes infinitely big.

The model is named after its authors Vladimir Belinski, Isaak Khalatnikov, and Evgeny Lifshitz, then working at the Landau Institute for Theoretical Physics.

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.

This would also resolve another aspect of the problem: the characteristics of spacetime metric evolution in the general solution when it reaches the physical singularity, understood as a point where matter density and invariants of the Riemann curvature tensor become infinite.

This search for a broader class of solutions with a singularity has been done, essentially, by a trial-and-error method, since a systematic approach to the study of the Einstein equations was lacking.

[note 2] The Einstein equation written in synchronous frame gives a result in which the metric determinant g inevitably becomes zero in a finite time irrespective of any assumptions about matter distribution.

Several years passed before the interest in this problem waxed again when Penrose (1965) published his theorems that linked the existence of a singularity of unknown character with some very general assumptions that did not have anything in common with a choice of reference frame.

Barrow and Tipler,[21][22] for example, among the ten criticisms of BKL studies, include the inappropriate (according to them) choice of synchronous frame as a means to separate time and space derivatives.

Such a statement, taken at face value, is wrong or at best misleading since, as shown in the BKL analysis itself, space-like gradients of the metric tensor cannot be neglected for generic solutions of pure Einstein gravity in four spacetime dimensions, and in fact play a crucial role in the appearance of the oscillatory regime.

Subsequent analysis by a large number of authors has shown that the BKL conjecture can be made precise and by now there is an impressive body of numerical and analytical evidence in its support.

[33] The BKL approach to anisotropic (as opposed to isotropic) homogeneous spaces starts with a generalization of an exact particular solution derived by Kasner[34] for a field in vacuum, in which the space is homogeneous and has a Euclidean metric that depends on time according to the Kasner metric (dl is the line element; dx, dy, dz are infinitesimal displacements in the three spatial dimensions, and t is time period passed since some initial moment t0 = 0).

An exception is the case p1 = р2 = 0, р3 = 1; these values correspond to a flat spacetime: the transformation t sh z = ζ, t ch z = τ turns the Kasner metric (eq.

The key to understanding the character of metric evolution on approaching singularity is exactly this process of Kasner epoch alternation with flipping of powers pl, pm, pn by the rule eq.

The next series of Kasner epochs then flips the negative power between directions n and l or between n and m. At an arbitrary (irrational) initial value of u this process of alternation continues unlimited.

For the next era In the limitless series of numbers u, composed by these rules, there are infinitesimally small (but never zero) values x(s) and correspondingly infinitely large lengths k(s).

This difficulty is avoided if one includes in the model only the major terms of the limiting (at t → 0) metric and writes into it a matter with arbitrary initial distribution of densities and velocities.

[note 15] Replacement of α, β, and γ maxima with zeros requires that quantities ln (|p1|Λ) be small in comparison with the amplitudes of oscillations of the respective functions.

If the minimum at the beginning of an era is deep, the next minima will not become shallower; in other words, the residue |α — β| at the moment of transition between Kasner epochs remains large.

42 is corroborated by an important theorem: the probability of the appearance of anomalous cases tends asymptotically to zero as the number of iterations s → ∞ (i.e., the time t → 0) which is proved at the end of this section.

73a) it is easy to conclude that the widths of the distributions ws(x) (about other definite numbers) will then be equal to (this expression is valid only so long as it defines quantities δx(s) ≪ 1).

can be written as Reverse quantities are defined by a continued fraction with a retrograde (in the direction of diminishing indices) sequence of denominators The recurrence relation eq.

This instability is eliminated by taking the logarithm: the "doubly-logarithmic" time interval is expressed by the sum of quantities ξ(p) which have a stable statistical distribution.

is changed for xp under the summation sign and thus represent τs as The variance of this sum in the limit of large s is It is taken into account that in virtue of the statistical homogeneity of the sequence X the correlations

80 in the form and then, for the total energy change during s eras, The term with the sum by p gives the main contribution to this expression because it contains an exponent with a large power.

94 means that components γa3 are small in the sense that at any ratio of the shifts dxa and dz, terms with products dxadz can be omitted in the square of the spatial length element dl2.

It follows that both Type IX model and its generalisation contain an oscillatory mode with a single spatial scale of an arbitrary magnitude which is not selected among others by any physical conditions.

For the latter, some natural physical length must exist which determines the minimal scale at which energy exits from a system with dynamical degrees of freedom (which, for example, occurs in a liquid with a certain viscosity).

Matter distribution and fields in some moment in the evolution of Universe do not necessarily correspond to the specific conditions required for the existence of a given special solution to the Einstein equations.

It can be expected that this effect will lead to a gradual "isotropisation" of space as a result of which its characteristics come closer to the Friedman model which adequately describes the present state of the Universe.

Finally, BKL pose the problem about the feasibility of considering a "singular state" of a world with infinitely dense matter on the basis of the existing relativity theory.

A spherical body undergoing a chaotic BKL (Mixmaster) dynamics close to singularity according to rules eq. 35 . Simulation was made in Mathematica with initial . [ note 1 ]
Dynamics of Kasner metrics eq. 2 in spherical coordinates towards singularity. The Lifshitz-Khalatnikov parameter is u =2 (1/ u =0.5) and the r coordinate is 2 p α (1/ u )τ where τ is logarithmic time: τ = ln t . [ note 3 ] Shrinking along the axes is linear and anisotropic (no chaoticity).
Plot of p 1 , p 2 , p 3 with an argument 1/ u . The numbers p 1 ( u ) and p 3 ( u ) are monotonously increasing while p 2 ( u ) is a monotonously decreasing function of u .
Bianchi type VIII (open) space undergoing a chaotic BKL (Mixmaster) dynamics close to singularity according to rules eq. 35 with initial . The singularity is at the central pinch of the hyperboloid surface.
Variation of α, β, and γ as functions of the logarithmic time Ω during one era. The vertical dash lines denote alterations of the Kasner epochs, corresponding to the linear segments of the curves. On the top are indicated the values of the parameter u that determine the Kasner exponents. The last epoch has a longer duration if x is small. In the first epoch of the next era, γ begins to increase and α becomes a monotonically decreasing function.
The probability distribution function P (δ). Red line: exact function eq. 79h . Blue line: approximate solution of the integral equation in. [ 48 ] Both curves appear to be strikingly similar and the means of both distributions are 0.50. [ note 19 ]