A BKL (Belinsky-Khalatnikov-Lifshitz) singularity is a model of the dynamic evolution of the Universe near the initial singularity, described by a non-symmetric, chaotic, vacuum solution to Einstein's field equations of gravitation. According to this model, the Universe is oscillating (expanding and contracting) around a singular point (singularity) in which time and space become equal to zero. This singularity is physically real in the sense that it is a necessary property of the solution, and will appear also in the exact solution of those equations. The singularity is not artificially created by the assumptions and simplifications made by the other well-known special solutions such as the Friedmann-Lemaître-Robertson-Walker, quasi-isotropic, and Kasner solutions.

The Mixmaster universe is a solution to general relativity that exhibits properties similar to those discussed by BKL.

Existence of time singularity

The basis of modern cosmology are the special solutions of Einstein's field equations found by Alexander Friedmann in 1922 and 1924 that describe a completely homogeneous and isotropic Universe ("closed" or "open" model, depending on closeness or infiniteness of space). The principal property of these solutions is their non-static nature. The concept of an inflating Universe that arises from Friedmann's solutions is fully supported by astronomical data and the present consensus is that the isotropic model, in general, gives an adequate description of the present state of the Universe.

Another important property of the isotropic model is the existence of a time singularity in the spacetime metric. In other words, the existence of such time singularity means finiteness of time. However, the adequacy of the isotropic model in describing the present state of the Universe by itself is not a reason to expect that it is so adequate in describing the early stages of Universe evolution. The problem initially addressed by the BKL paper is whether the existence of such time singularity is a necessary property of relativistic cosmological models. There is the possibilty that the singularity is generated by the simplifying assumptions, made when constructing these models. Independence of singularity on assumptions would mean that time singularity exists not only in the particular but also in the general solutions of the Einstein equations. A criterion for generality of solutions is the number of arbitrary space coordinate functions that they contain. These include only the "physically arbitrary" functions whose number cannot be reduced by any choice of reference frame. In the general solution, the number of such functions must be sufficient for arbitrary definition of initial conditions (distribution and movement of matter, distribution of gravitational field) in some moment of time chosen as initial. This number is four for vacuum and eight for a matter and/or radiation filled space.

For a system of non-linear differential equations, such as the Einstein equations, general solution is not unambiguously defined. In principle, there may be multiple general integrals, and each of those may contain only a finite subset of all possible initial conditions. Each of those integrals may contain all required arbitrary functions which, however, may be subject to some conditions (e.g., some inequalities). Existence of a general solution with a singularity, therefore, does not preclude the existence also of other general solutions that do not contain a singularity. For example, there is no reason to doubt the existence of a general solution without singularity that describes an isolated body with a relatively small mass.

It is impossible to find a general integral for all space and for all time. However, this is not necessary for resolving the problem: it is sufficient to study the solution near the singularity. 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. The BKL paper concerns only the cosmological aspect. This means, that the subject is a time singularity in the whole spacetime and not in some limited region as in a gravitational collapse of a finite body.

Previous work by the Landau-Lifshitz group (reviewed in ) led to a conclusion that the general solution does not contain a physical singularity. This search for a broader class of solutions with singularity has been done, essentially, by a trial-and-error method, since a systemic approach to the study of the Einstein equations is lacking. A negative result, obtained in this way, is not convincing by itself; a solution with the necessary degree of generality would invalidate it, and at the same time would confirm any positive results related to the specific solution.

It is reasonable to suggest that if a singularity is present in the general solution, there must be some indications that are based only on the most general properties of the Einstein equations, although those indications by themselves might be insufficient for characterizing the singularity. At that time, the only known indication was related to the form of Einstein equations written in a synchronous reference frame, that is, in a frame in which the interval element is

$ds^2\; =\; dt^2\; -\; dl^2\; ,\; dl^2\; =\; gamma\_\{alpha\; beta\}\; dx^\{alpha\}\; dx^\{beta\},$ (eq. 1)

where the space distance element dl is separate from the time interval dt, and x⁰ = t is the proper time synchronized throughout the whole space. The Einstein equation $scriptstyle\{R\_0^\{0\}=T\_0^\{0\}-frac\{1\}\{2\}T\}$ 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.

This indication, however, was dropped after it became clear that it is linked with a specific geometric property of the synchronous frame: crossing of time line coordinates. This crossing takes place on some encircling hypersurfaces which are four-dimensional analogs of the caustic surfaces in geometrical optics; g becomes zero exactly at this crossing. Therefore, although this singularity is general, it is fictitious, and not a physical one; it disappears when the reference frame is changed. This, apparently, stopped the incentive for further investigations.

However, the interest in this problem waxed again after Penrose 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. Other similar theorems were found later on by Hawking and Geroch (see Penrose-Hawking singularity theorems). It became clear that the search for a general solution with singularity must continue.

Generalized Kasner solution

Further generalization of solutions depended on some solution classes found previously. The Friedmann solution, for example, is a special case of a solution class that contains three physically arbitrary coordinate functions. In this class the space is anisotropic; however, its compression when approaching the singularity has "quasi-isotropic" character: the linear distances in all directions diminish as the same power of time. Like the fully homogeneous and isotropic case, this class of solutions exist only for a matter-filled space.

Much more general solutions are obtained by a generalization of an exact particular solution derived by Kasner for a field in vacuum, in which the space is homogeneous and has Euclidean metric that depends on time according to the Kasner metric

$dl^2=t^\{2p\_1\}dx^2+t^\{2p\_2\}dy^2+t^\{2p\_3\}dz^2$ (eq. 2)

(see ). Here, p₁, p₂, p₃ are any 3 numbers that are related by

$p\_1+p\_2+p\_3=p\_1^2+p\_2^2+p\_3^2=1.$ (eq. 3)

Because of these relationships, only 1 of the 3 numbers is independent. All 3 numbers are never the same; 2 numbers are the same only in the sets of values $scriptstyle\{(-frac\; \{1\}\{3\},frac\{2\}\{3\},frac\; \{2\}\{3\})\}$ and (0, 0, 1). In all other cases the numbers are different, one number is negative and the other two are positive. If the numbers are arranged in increasing order, p₁ < p₂ < p₃, they change in the ranges

$-frac\; \{1\}\{3\}\; le\; p\_1\; le\; 0,\; 0\; le\; p\_2\; le\; frac\{2\}\{3\},\; frac\{2\}\{3\}\; le\; p\_3\; le\; 1.$ (eq. 4)

The numbers p₁, p₂, p₃ can be written parametrically as

$p\_1(u)=frac\; \{-u\}\{1+u+u^2\},\; p\_2(u)=frac\; \{1+u\}\{1+u+u^2\},\; p\_3(u)=frac\; \{u(1+u)\}\{1+u+u^2\}$ (eq. 5)

All different values of p₁, p₂, p₃ ordered as above are obtained by changing the value of the parameter u in the range u ≥ 1. The values u < 1 are brought into this range according to

$p\_1\; left(frac\; \{1\}\{u\}\; right\; )=p\_1(u),\; p\_2\; left(frac\; \{1\}\{u\}\; right\; )=p\_3(u),\; p\_3\; left(frac\; \{1\}\{u\}\; right\; )=p\_2(u)$ (eq. 6)

Figure 1 is a plot of p₁, p₂, p₃ with an argument $scriptstyle\{frac\; \{1\}\{u\}\}$. The numbers p₁(u) and p₃(u) are monotonously increasing while p₂(u) is monotonously decreasing function of the parameter u.

In the generalized solution, the form corresponding to (eq. 2) applies only to the asymptotic metric (the metric close to the singularity t = 0), respectively, to the major terms of its series expansion by powers of t. In the synchronous reference frame it is written in the form of (eq. 1) with a space distance element

$dl^2=(a^2l\_\{alpha\}l\_\{beta\}+b^2m\_\{alpha\}m\_\{beta\}+c^2n\_\{alpha\}n\_\{beta\})dx^\{alpha\}dx^\{beta\},$ (eq. 7)

where $a=t^\{p\_l\},\; b=t^\{p\_m\},\; c=t^\{p\_n\}$ (eq. 8)

The three-dimensional vectors l, m, n define the directions at which space distance changes with time by the power laws (eq. 8). These vectors, as well as the numbers p_l, p_m, p_n which, as before, are related by (eq. 3), are functions of the space coordinates. The powers p_l, p_m, p_n are not arranged in increasing order, reserving the symbols p₁, p₂, p₃ for the numbers in (eq. 5) that remain arranged in increasing order. The determinant of the metric of (eq. 7) is

$-g=a^2b^2c^2v^2=t^2v^2\; ,$ (eq. 9)

where v = l[mn]. It is convenient to introduce the following quantitities

$lambda=frac\{mathbf\{l\}\; mathrm\{rot\}\; mathbf\{l\}\}\{v\},\; mu=frac\{mathbf\{m\}\; mathrm\{rot\}\; mathbf\{m\}\}\{v\},\; nu=frac\{mathbf\{n\}\; mathrm\{rot\}\; mathbf\{n\}\}\{v\}.$ (eq. 10)

The space metric in (eq. 7) is anisotropic because the powers of t in (eq. 8) cannot have the same values. On approaching the singularity at t = 0, the linear distances in each space element decrease in two directions and increase in the third direction. The volume of the element decreases in proportion to t.

The Einstein equations in vacuum in synchronous reference frame are

$R\_0^0=-frac\{1\}\{2\}frac\{partial\; varkappa\_\{alpha\}^\{alpha\}\}\{partial\; t\}-frac\{1\}\{4\}\; varkappa\_\{alpha\}^\{beta\}\; varkappa\_\{beta\}^\{alpha\}=0,$ (eq. 11)

$R\_\{alpha\}^\{beta\}=-left\; (frac\{1\}\{2\}sqrt\{-g\}\; right\; )\; frac\{partial\}\{partial\; t\}\; left\; (sqrt\{-g\}\; varkappa\_\{alpha\}^\{beta\}\; right\; )-P\_\{alpha\}^\{beta\}=0,$ (eq. 12)

$R\_\{alpha\}^\{0\}=frac\{1\}\{2\}\; left\; (varkappa\_\{alpha;beta\}^\{beta\}-\; varkappa\_\{beta;alpha\}^\{beta\}right\; )=0,$ (eq. 13)

where $scriptstyle\{varkappa\_\{alpha\}^\{beta\}\}$ is the 3-dimensional tensor $scriptstyle\{varkappa\_\{alpha\}^\{beta\}=frac\{partial\; gamma\_\{alpha\}^\{beta\}\}\{partial\; t\}\}$, and P_αβ is the 3-dimensional Ricci tensor, which is expressed by the 3-dimensional metric tensor γ_αβ in the same way as R_ik is expressed by g_ik; P_αβ contains only the space (but not the time) derivatives of γ_αβ.

The Kasner metric is introduced in the Einstein equations by substituting the respective metric tensor γ_αβ from (eq. 7) without defining a priori the dependence of a, b, c from t:

$varkappa\_\{alpha\}^\{beta\}=left\; (frac\{2\; dot\; a\}\{a\}\; right\; )l\_\{alpha\}l^\{beta\}+left\; (frac\{2\; dot\; b\}\{b\}\; right\; )m\_\{alpha\}m^\{beta\}+left\; (frac\{2\; dot\; c\}\{c\}\; right\; )n\_\{alpha\}n^\{beta\}$

where the dot above a symbol designates differentiation with respect to time. The Einstein equation (eq. 11) takes the form

$-R\_0^0=frac\{ddot\; a\}\{a\}+frac\{ddot\; b\}\{b\}+frac\{ddot\; c\}\{c\}=0.$ (eq. 14)

All its terms are to a second order for the large (at t → 0) quantity 1/t. In the Einstein equations (eq. 12), terms of such order appear only from terms that are time-differentiated. If the components of P_αβ do not include terms of order higher than 2, then

$-R\_l^l=frac\{(dot\; a\; b\; c)dot\{\; \}\}\{abc\}=0,\; -R\_m^m=frac\{(a\; dot\; b\; c)dot\{\; \}\}\{abc\}=0,\; -R\_n^n=frac\{(a\; b\; dot\; c)dot\{\; \}\}\{abc\}=0$ (eq. 15)

where indices l, m, n designate tensor components in the directions l, m, n. These equations together with (eq. 14) give the expressions (eq. 8) with powers that satisfy (eq. 3).

However, the presence of 1 negative power among the 3 powers p_l, p_m, p_n results in appearance of terms from P_αβ with an order greater than t⁻². If the negative power is p_l (p_l = p₁ < 0), then P_αβ contains the coordinate function λ and (eq. 12) become

$begin\{align\}$

-R_l^l & =frac{(dot a b c)dot{ }}{abc}+frac{lambda^2 a^2}{2b^2 c^2}=0, -R_m^m & =frac{(a dot b c)dot{ }}{abc}-frac{lambda^2 a^2}{2b^2 c^2}=0, -R_n^n & =frac{(a b dot c)dot{ }}{abc}-frac{lambda^2 a^2}{2b^2 c^2}=0. end{align} (eq. 16)

Here, the second terms are of order t^{−2(p_m + p_n − p_l)} whereby p_m + p_n − p_l = 1 + 2 |p_l| > 1. To remove these terms and restore the metric (eq. 7), it is necessary to impose on the coordinate functions the condition λ = 0.

The remaining 3 Einstein equations (eq. 13) contain only first order time derivatives of the metric tensor. They give 3 time-independent relations that must be imposed as necessary conditions on the coordinate functions in (eq. 7). This, together with the condition λ = 0, makes 4 conditions. These conditions bind 10 different coordinate functions: 3 components of each of the vectors l, m, n, and one function in the powers of t (any one of the functions p_l, p_m, p_n, which are bound by the conditions (eq. 3)). When calculating the number of physically arbitrary functions, it must be taken into account that the synchronous system used here allows time-independent arbitrary transformations of the 3 space coordinates. Therefore, the final solution contains overall 10 − 4 − 3 = 3 physically arbitrary functions which is 1 less than what is needed for the general solution in vacuum.

The degree of generality reached until now is not lessened by introducing matter; matter is written into the metric (eq. 7) and contributes 4 new coordinate functions necessary to describe the initial distribution of its density and the 3 components of its velocity. This makes possible to determine matter evolution merely from the laws of its movement in an a priori given gravitational field. These movement laws are the hydrodynamic equations

$frac\{1\}\{sqrt\{-g\}\}frac\{partial\}\{partial\; x^i\}\; left\; (sqrt\{-g\}sigma\; u^i\; right\; )\; =\; 0,$ (eq. 17)

$(p+varepsilon)\; u^k\; left\; \{\; frac\{partial\; u\_i\}\{partial\; x^k\}-frac\{1\}\{2\}\; u^l\; frac\{partial\; g\_\{kl\}\}\{partial\; x^i\}\; right\; rbrace\; =-frac\{partial\; p\}\{partial\; x^i\}-u\_i\; u^k\; frac\{partial\; p\}\{partial\; x^k\},$ (eq. 18)

where u ⁱ is the 4-dimensional velocity, ε and σ are the densities of energy and entropy of matter. For the ultrarelativistic equation of state p = ε/3 the entropy σ ~ ε^1/4. The major terms in (eq. 17) and (eq. 18) are those that contain time derivatives. From (eq. 17) and the space components of (eq. 18) one has

$frac\{partial\}\{partial\; t\}\; left\; (sqrt\{-g\}\; u\_0\; varepsilon^\{frac\{3\}\{4\}\}\; right\; )\; =\; 0,\; 4\; varepsilon\; cdot\; frac\{partial\; u\_\{alpha\}\}\{partial\; t\}+u\_\{alpha\}\; cdot\; frac\{partial\; varepsilon\}\{partial\; t\}\; =\; 0,$

resulting in

$abc\; u\_0\; varepsilon^\{frac\{3\}\{4\}\}=\; mathrm\{const\},\; u\_\{alpha\}\; varepsilon^\{frac\{1\}\{4\}\}=\; mathrm\{const\},$ (eq. 19)

where 'const' are time-independent quantities. Additionally, from the identity u_iuⁱ = 1 one has (because all covariant components of u_α are to the same order)

$u\_0^2\; approx\; u\_n\; u^n\; =\; frac\{u\_n^2\}\{c^2\},$

where u_n is the velocity component along the direction of n that is connected with the highest (positive) power of t (supposing that p_n = p₃). From the above relations, it follows that

$varepsilon\; sim\; frac\{1\}\{a^2\; b^2\},\; u\_\{alpha\}\; sim\; sqrt\{ab\}$ (eq. 20)

or

$varepsilon\; sim\; t^\{-2(p\_1+p\_2)\}=t^\{-2(1-p\_3)\},\; u\_\{alpha\}\; sim\; t^\{frac\{(1-p\_3)\}\{2\}\}.$ (eq. 21)

The above equations can be used to confirm that the components of the matter stress-energy-momentum tensor standing in the right hand side of the equations

$R\_0^0\; =\; T\_0^0\; -\; frac\{1\}\{2\}T,\; R\_\{alpha\}^\{beta\}\; =\; T\_\{alpha\}^\{beta\}-\; frac\{1\}\{2\}delta\_\{alpha\}^\{beta\}T,$

are, indeed, to a lower order by 1/t than the major terms in their left hand sides. In the equations $scriptstyle\{R\_\{alpha\}^0\; =\; T\_\{alpha\}^0\}$ the presence of matter results only in the change of relations imposed on their constituent coordinate functions.

The fact that ε becomes infinite by the law (eq. 21) confirms that in the solution to (eq. 7) one deals with a physical singularity at any values of the powers p₁, p₂, p₃ excepting only (0, 0, 1). For these last values, the singularity is non-physical and can be removed by a change of reference frame.

The fictional singularity corresponding to the powers (0, 0, 1) arises as a result of time line coordinates crossing over some 2-dimensional "focal surface". As pointed out in , a synchronous reference frame can always be chosen in such way that this inevitable time line crossing occurs exactly on such surface (instead of a 3-dimensional caustic surface). Therefore, a solution with such simultaneous for the whole space fictional singularity must exist with a full set of arbitrary functions needed for the general solution. Close to the point t = 0 it allows a regular expansion by whole powers of t.

Oscillating mode towards the singularity

The four conditions that had to be imposed on the coordinate functions in the solution (eq. 7) are of different types: three conditions that arise from the equations $scriptstyle\{R\_\{alpha\}^0\}$ = 0 are "natural"; they are a consequence of the structure of Einstein equations. However, the additional condition λ = 0 that causes the loss of one derivative function, is of entirely different type.

The general solution by definition is completely stable; otherwise the Universe would not exist. Any perturbation is equivalent to a change in the initial conditions in some moment of time; since the general solution allows arbitrary initial conditions, the perturbation is not able to change its character. In other words, the existence of the limiting condition λ = 0 for the solution of (eq. 7) means instability caused by perturbations that break this condition. The action of such perturbation must bring the model to another mode which thereby will be most general. Such perturbation cannot be considered as small: a transition to a new mode exceeds the range of very small perturbations.

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 9 classes. 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. For Type IX spaces, the quantities λ, μ, ν have the same sign and one can write λ = μ = ν = 1 (the simultaneous sign change of the 3 constants does not change anything). For Type VIII spaces, 2 constants have a sign that is opposite to the sign of the third constant; one can write, for example, λ = − 1, μ = ν = 1.

The study of the effect of the perturbation on the "Kasner mode" is thus confined to a study on the effect of the λ-containing terms in the Einstein equations. Type VIII and IX spaces are the most suitable models exactly in this connection. Since all 3 quantities λ, μ, ν 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

$begin\{align\}$

-R_l^l & =frac{left(dot a b cright)dot{ }}{abc}+frac{1}{2}left (a^2b^2c^2right )left [lambda^2 a^4-left (mu b^2-nu c^2right )^2right ]=0, -R_m^m & =frac{(a dot{b} c)dot{ }}{abc}+frac{1}{2}left(a^2b^2c^2right )left [mu^2 b^4-left(lambda a^2-nu c^2right)^2right]=0, -R_n^n & =frac{left(a b dot cright)dot{ }}{abc}+frac{1}{2}left(a^2b^2c^2right)left[nu^2 c^4-left(lambda a^2-mu b^2right)^2right]=0, end{align} (eq. 22)

$-R\_0^0=frac\{ddot\; a\}\{a\}+frac\{ddot\; b\}\{b\}+frac\{ddot\; c\}\{c\}=0$ (eq. 23)

(the remaining components $scriptstyle\{R\_l^0\}$, $scriptstyle\{R\_m^0\}$, $scriptstyle\{R\_n^0\}$, $scriptstyle\{R\_l^m\}$, $scriptstyle\{R\_l^n\}$, $scriptstyle\{R\_m^n\}$ are identically zeros). These equations contain only functions of time; this is a condition that has to be fulfiled in all homogeneous spaces. Here, the (eq. 22) and (eq. 23) are exact and their validity does not depend on how near one is to the singularity at t = 0.

The time derivatives in (eq. 22) and (eq. 23) take a simpler form if а, b, с are substituted by their logarithms α, β, γ:

$a=e^alpha,\; b=e^beta,\; c=e^gamma,$ (eq. 24)

substituting the variable t for τ according to:

$dt=abc\; dtau,$ (eq. 25).

Then:

$begin\{align\}$

2alpha_{tautau} & =left (mu b^2-nu c^2right )^2-lambda^2 a^4=0, 2beta_{tautau} & =left (lambda a^2-nu c^2right )^2-mu^2 b^4=0, 2gamma_{tautau} & =left (lambda a^2-mu b^2right )^2-nu^2 c^4=0, end{align} (eq. 26)

$frac\{1\}\{2\}left(alpha+beta+gamma\; right)\_\{tautau\}=alpha\_tau\; beta\_tau\; +alpha\_tau\; gamma\_tau+beta\_tau\; gamma\_tau.$ (eq. 27)

Adding together equations (eq. 26) and substituting in the left hand side the sum (α + β + γ)_{τ τ} according to (eq. 27), one obtains an equation containing only first derivatives which is the first integral of the system (eq. 26):

$alpha\_tau\; beta\_tau\; +alpha\_tau\; gamma\_tau+beta\_tau\; gamma\_tau\; =\; frac\{1\}\{4\}left(lambda^2a^4+mu^2b^4+nu^2c^4-2lambda\; mu\; a^2b^2-2lambda\; nu\; a^2c^2-2mu\; nu\; b^2c^2\; right).$ (eq. 28)

This equation plays the role of a binding condition imposed on the initial state of (eq. 26). The Kasner mode (eq. 8) is a solution of (eq. 26) when ignoring all terms in the right hand sides. But such situation cannot go on (at t → 0) indefinitely because among those terms there are always some that grow. Thus, if the negative power is in the function a(t) (p_l = p₁) then the perturbation of the Kasner mode will arise by the terms λ²a⁴; the rest of the terms will decrease with decreasing t. If only the growing terms are left in the right hand sides of (eq. 26), one obtains the system:

$alpha\_\{tautau\}=-frac\{1\}\{2\}lambda^2e^\{4alpha\},\; beta\_\{tautau\}=gamma\_\{tautau\}=frac\{1\}\{2\}lambda^2e^\{4alpha\}$ (eq. 29)

(compare (eq. 16); below it is substituted λ² = 1). The solution of these equations must describe the metric evolution from the initial state, in which it is described by (eq. 8) with a given set of powers (with p_l < 0); let p_l = р₁, p_m = р₂, p_n = р₃ so that

$a\; sim\; t^\{p\_1\},\; b\; sim\; t^\{p\_2\},\; c\; sim\; t^\{p\_3\}.$ (eq. 30)

Then

$abc=Lambda\; t,\; tau=Lambda^\{-1\}ln\; t+mathrm\{const\}$ (eq. 31)

where Λ is constant. Initial conditions for (eq. 29) are redefined as

$alpha\_tau=Lambda\; p\_1,\; beta\_tau=Lambda\; p\_2,\; gamma\_tau=Lambda\; p\_3\; mathrm\{at\}\; tau\; to\; infty$ (eq. 32)

Equations (eq. 29) are easily integrated; the solution that satisfies the condition (eq. 32) is

$begin\{cases\}a^2=frac\{2|p\_1|Lambda\}\{operatorname\{ch\}(2|p\_1|Lambdatau)\},\; b^2=b\_0^2e^\{2Lambda(p\_2-|p\_1|)tau\}operatorname\{ch\}(2|p\_1|Lambdatau),$

c^2=c_0^2e^{2Lambda(p_2-|p_1|)tau}operatorname{ch}(2|p_1|Lambdatau),end{cases} (eq. 33)

where b₀ and c₀ are two more constants.

It can easily be seen that the asymptotic of functions (eq. 33) at t → 0 is (eq. 30). The asymptotic expressions of these functions and the function t(τ) at τ → −∞ is

$a\; sim\; e^\{-Lambda\; p\_1tau\},\; b\; sim\; e^\{Lambda(p\_2+2p\_1)tau\},\; c\; sim\; e^\{Lambda(p\_3+2p\_1)tau\},\; t\; sim\; e^\{Lambda(1+2p\_1)tau\}.$

Expressing a, b, c as functions of t, one has

$a\; sim\; t^\{p\text{'}\_l\},\; b\; sim\; t^\{p\text{'}\_m\},\; c\; sim\; t^\{p\text{'}\_n\}$ (eq. 34)

where

$p\text{'}\_l=frac${1-2|p_1, p'_m=-frac{2|p_1|-p_2}{1-2|p_1, p'_n=frac{p_3-2|p_1{1-2|p_1.> (eq. 35)

Then

$abc=Lambda\text{'}\; t,\; Lambda\text{'}=(1-2|p\_1|)Lambda.$ (eq. 36)

The above shows that perturbation acts in such way that it changes one Kasner mode with another Kasner mode, and in this process the negative power of t flips from direction l to direction m: if before it was p_l < 0, now it is p'_m < 0. During this change the function a(t) passes through a maximum and b(t) passes through a minimum; b, which before was decreasing, now increases: a from increasing becomes decreasing; and the decreasing c(t) decreases further. The perturbation itself (λ²a^4α in (eq. 29)), which before was increasing, now begins to decrease and die away. Further evolution similarly causes an increase in the perturbation from the terms with μ² (instead of λ²) in (eq. 26), next change of the Kasner mode, and so on.

It is convenient to write the power substitution rule (eq. 35) with the help of the parametrization (eq. 5):

$begin\{matrix\}$

mathbf{if} & p_l=p_{1}(u) & p_m=p_{2}(u) & p_n=p_{3}(u) mathbf{then} & p'_l=p_{2}(u-1) & p'_m=p_{1}(u-1) & p'_n=p_{3}(u-1) end{matrix} (eq. 37)

The greater of the two positive powers remains positive.

BKL call this flip of negative power between directions a Kasner epoch. The key to understanding the character of metric evolution on approaching singularity is exactly this process of Kasner epoch alternation with flipping of powers p_l, p_m, p_n by the rule (eq. 37).

The successive alternations (eq. 37) with flipping of the negative power p₁ between directions l and m (Kasner epochs) continues by depletion of the whole part of the initial u until the moment at which u < 1. The value u < 1 transforms into u > 1 according to (eq. 6); in this moment the negative power is p_l or p_m while p_n becomes the lesser of two positive numbers (p_n = p₂). 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.

In the exact solution of the Einstein equations, the powers p_l, p_m, p_n lose their original, precise, sense. This circumstance introduces some "fuzziness" in the determination of these numbers (and together with them, to the parameter u) which, although small, makes meaningless the analysis of any definite (for example, rational) values of u. Therefore, only these laws that concern arbitrary irrational values of u have any particular meaning.

The larger periods in which the scales of space distances along two axes oscillate while distances along the third axis decrease monotonously, are called eras; volumes decrease by a law close to ~ t. On transition from one era to the next, the direction in which distances decrease monotonously, flips from one axis to another. The order of these transitions acquires the asymptotic character of a random process. The same random order is also characteristic for the alternation of the lengths of successive eras (by era length, BKL understand the number of Kasner epoch that an era contains, and not a time interval).

The era series become denser on approaching t = 0. However, the natural variable for describing the time course of this evolution is not the world time t but its logarithm, ln t, by which the whole process of reaching the singularity is extended to −∞.

According to (eq. 33), one of the functions a, b, c, that passes through a maximum during a transition between Kasner epochs, at the peak of its maximum is

$a\_max=sqrt\{2Lambda|p\_1(u)$ (eq. 38)

where it is supposed that a_max is large compared to b₀ and c₀; in (eq. 38) u is the value of the parameter in the Kasner epoch before transition. It can be seen from here that the peaks of consecutive maxima during each era are gradually lowered. Indeed, in the next Kasner epoch this parameter has the value u' = u - 1, and Λ is substituted according to (eq. 36) with Λ' = Λ(1 − 2|p₁(u)|). Therefore, the ratio of 2 consecutive maxima is

$frac\{a\text{'}\_max\}\{a\_max\}=left[frac\{p\_1(u-1)\}\{p\_1(u)\}left(1-2|p\_1(u)|right)right]^\{frac\{1\}\{2\}\};$

and finally

$frac\{a\text{'}\_max\}\{a\_max\}=sqrt\{frac\{u-1\}\{u\}\}equiv\; sqrt\{frac\{u\text{'}\}\{u\}\}.$ (eq. 39)

The above are solutions to Einstein equations in vacuum. As for the pure Kasner mode, matter does not change the qualitative properties of this solution and can be written into it disregarding its reaction on the field.

However, if one does this for the model under discussion, understood as an exact solution of the Einstein equations, the resulting picture of matter evolution would not have a general character and would be specific for the high symmetry imminent to the present model. Mathematically, this specificity is related to the fact that for the homogeneous space geometry discussed here, the Ricci tensor components $scriptstyle\{R\_alpha^0\}$ are identically zeros and therefore the Einstein equations would not allow movement of matter (which gives non-zero stress energy-momentum tensor components $scriptstyle\{T\_alpha^0\}$).

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. Then the course of evolution of matter is determined by its general laws of movement (eq. 17) and (eq. 18) that result in (eq. 21). During each Kasner epoch, density increases by the law

$varepsilon=t^\{-2(1-p\_3)\},$ (eq. 40)

where p₃ is, as above, the greatest of the numbers p₁, p₂, p₃. Matter density increases monotonously during all evolution towards the singularity.

To each era (s-th era) correspond a series of values of the parameter u starting from the greatest, $scriptstyle\{u\_\{max\}^\{(s)\}\}$, and through the values $scriptstyle\{u\_\{max\}^\{(s)\}\}$ − 1, $scriptstyle\{u\_\{max\}^\{(s)\}\}$ − 2, …, reaching to the smallest, $scriptstyle\{u\_\{min\}^\{(s)\}\}$ < 1. Then

$u\_\{min\}^\{(s)\}=x^\{(s)\},\; u\_\{max\}^\{(s)\}=k^\{(s)\}+x^\{(s)\},$ (eq. 41)

that is, k^(s) = [$scriptstyle\{u\_\{max\}^\{(s)\}\}$] where the brackets mean the whole part of the value. The number k^(s) is the era length, measured by the number of Kasner epochs that the era contains. For the next era

$u\_\{max\}^\{(s+1)\}=frac\{1\}\{x^\{(s)\}\},\; k^\{(s+1)\}=left[frac\{1\}\{x^\{(s)\}\}right].$ (eq. 42)

In the limiteless 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).

Metric evolution

Very large u values correspond to Kasner powers

$p\_1\; approx\; -frac\{1\}\{u\},\; p\_2\; approx\; frac\{1\}\{u\},\; p\_2\; approx\; 1-frac\{1\}\{u^2\},$ (eq. 43)

which are close to the values (0, 0, 1). Two values that are close to zero, are also close to each other, and therefore the changes in two out of the three types of "perturbations" (the terms with λ, μ and ν in the right hand sides of (eq. 26)) are also very similar. If in the beginning of such long era these terms are very close in absolute values in the moment of transition between two Kasner epochs (or made artificially such by assigning initial conditions) then they will remain close during the greatest part of the length of the whole era. In this case (BKL call this the case of small oscillations), analysis based on the action of one type of perturbations becomes incorrect; one must take into account the simultaneous effect of two perturbation types.

Two perturbations

Consider a long era, during which 2 out of the 3 functions a, b, c (let them be a and b) undergo small oscillations while the third function (c) decreases monotonously. The latter function quickly becomes small; consider the solution just in the region where one can ignore c in comparison to a and b. The calculations are first done for the Type IX space model by substituting accordingly λ = μ = ν = 1.

After ignoring function c, the first 2 equations (eq. 26) give

$alpha\_\{tautau\}+beta\_\{tautau\}=0,,$ (eq. 44)

$alpha\_\{tautau\}-beta\_\{tautau\}=e^\{4beta\}-e^\{4alpha\},,$ (eq. 45)

and as a third equation, (eq. 28) can be used, which takes the form

$gamma\_\{tautau\}left(alpha\_\{tautau\}+beta\_\{tautau\}right)=-alpha\_taubeta\_tau+frac\{1\}\{4\}left(e^\{2alpha\}-e^\{2beta\}right)^2.$ (eq. 46)

The solution of (eq. 44) is written in the form

$alpha+beta=left(frac\{2a\_0^2\}\{xi\_0\}right)left(tau-tau\_0right)+2ln\; a\_0,$

where α₀, ξ₀ are positive constants, and τ₀ is the upper limit of the era for the variable τ. It is convenient to introduce further a new variable (instead of τ)

$xi=xi\_0exp\; left[frac\{2a\_0^2\}\{xi\_0\}left(tau-tau\_0\; right)right].$ (eq. 47)

Then

$alpha+beta=ln\; left(frac\{xi\}\{xi\_0\}right)+2ln\; a\_0.$ (eq. 48)

Equations (eq. 45) and (eq. 46) are transformed by introducing the variable χ = α − β:

$chi\_\{xixi\}=frac\{chi\_xi\}\{xi\}+frac\{1\}\{2\}operatorname\{sh\}2chi=0,$ (eq. 49)

$gamma\_xi=-frac\{1\}\{4\}xi+frac\{1\}\{8\}xileft(2chi\_xi^2+operatorname\{ch\}2chi-1right).$ (eq. 50)

Decrease of τ from τ₀ to −∞ corresponds to a decrease of ξ from ξ₀ to 0. The long era with close a and b (that is, with small χ), considered here, is obtained if ξ₀ is a very large quantity. Indeed, at large ξ the solution of (eq. 49) in the first approximation by 1/ξ is

$chi=alpha-beta=left(frac\{2A\}\{sqrt\{xi\}\}right)sin\; left(xi-xi\_0right),$ (eq. 51)

where A is constant; the multiplier $tfrac\{1\}\{sqrt\{xi\}\}$ makes χ a small quantity so it can be substituted in (eq. 49) by sh 2χ ≈ 2χ.

From (eq. 50) one obtains

$gamma\_xi=frac\{1\}\{4\}xileft(2chi\_xi^2+chi^2right)=A^2,\; gamma=A^2left(xi-xi\_0right)+mathrm\{const\}.$

After determining α and β from (eq. 48) and (eq. 51) and expanding e^α and e^β in series according to the above approximation, one obtains finally:

$begin\{cases\}$

a

b

end{cases}=a_0sqrt{frac{xi}{xi_0}}left[1pm frac{A}{sqrt{xi}}sin left(xi-xi_0right)right], (eq. 52)

$c=c\_0\; e^\{-A^2left(xi\_0-xiright)\}.$ (eq. 53)

The relation between the variable ξ and time t is obtained by integration of the definition dt = abc dτ which gives

$frac\{t\}\{t\_0\}=e^\{-A^2left(xi\_0-xiright)\}.$ (eq. 54)

The constant c₀ (the value of с at ξ = ξ₀) should be now c₀ $scriptstyle\{ll\}$ α₀·

Let us now consider the domain ξ $scriptstyle\{ll\}$ 1. Here the major terms in the solution of (eq. 49) are:

$chi=alpha-beta=kln\; xi+mathrm\{const\},,$

where k is a constant in the range − 1 < k < 1; this condition ensures that the last term in (eq. 49) is small (sh 2χ contains ξ^2k and ξ^−2k). Then, after determining α, β, and t, one obtains

$a\; sim\; xi^\{frac\{1+k\}\{2\}\},\; b\; sim\; xi^\{frac\{1-k\}\{2\}\},\; c\; sim\; xi^\{-frac\{1-k^2\}\{4\}\},\; t\; sim\; xi^\{frac\{3+k^2\}\{4\}\}.$ (eq. 55)

This is again a Kasner mode with the negative t power coming into the function c(t).

These results picture an evolution that is qualitatively similar to that, described above. During a long period of time that corresponds to a large decreasing ξ value, the two functions a and b oscillate, remaining close in magnitude $tfrac\{a-b\}\{a\}\; sim\; tfrac\{1\}\{sqrt\{xi\}\}$; in the same time, both functions a and b slowly ($scriptstyle\{sim\; sqrt\{xi\}\}$) decrease. The period of oscillations is constant by the variable ξ : Δξ = 2π (or, which is the same, with a constant period by logarithmic time: Δ ln t = 2πΑ²). The third function, c, decreases monotonously by a law close to c = c₀t/t₀.

This evolution continues until ξ ~ 1 and formulas (eq. 52) and (eq. 53) are no longer applicable. Its time duration corresponds to change of t from t₀ to the value t₁, related to ξ₀ according to

$A^2xi\_0=ln\; frac\{t\_0\}\{t\_1\}.$ (eq. 56)

The relationship between ξ and t during this time can be presented in the form

$frac\{xi\}\{xi\_0\}=frac\{ln\; tfrac\{t\}\{t\_1\}\}\{ln\; tfrac\{t\_0\}\{t\_1\}\}.$ (eq. 57)

After that, as seen from (eq. 55), the decreasing function c starts to increase while functions a and b start to decrease. This Kasner epoch continues until terms c²/a²b² in (eq. 22) become ~ t² and a next series of oscillations begins.

The law for density change during the long era under discussion is obtained by substitution of (eq. 52) in (eq. 20):

$varepsilon\; sim\; left(frac\{xi\_0\}\{xi\}right)^2.$ (eq. 58)

When ξ changes from ξ₀ to ξ ~ 1, the density increases $scriptstyle\{xi^2\_0\}$ times.

It must be stressed that although the function c(t) changes by a law, close to c ~ t

The small-time domain

Statistics near the singularity

General long era solution with small oscillations

Notes

References