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, the spacetime curvature becomes infinitely big. 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 special solutions such as the Friedmann–Lemaître–Robertson–Walker, quasi-isotropic, and Kasner solutions. The model is named after its authors Vladimir Belinski, Isaak Khalatnikov, and Evgeny Lifshitz, then working at the Landau Institute for Theoretical Physics. The picture developed by BKL has several important elements. These are:
Near the singularity the evolution of the geometry at different spatial points decouples so that the solutions of the partial differential equations can be approximated by solutions of ordinary differential equations with respect to time for appropriately defined spatial scale factors. This is called the BKL conjecture. For most types of matter the effect of the matter fields on the dynamics of the geometry becomes negligible near the singularity. Or, in the words of John Wheeler, "matter doesn't matter" near a singularity. The original BKL work posed a negligible effect for all matter but later they theorized that "stiff matter" (equation of state p = ε) equivalent to a massless scalar field can have a modifying effect on the dynamics near the singularity. The ordinary differential equations describing the asymptotics come from a class of spatially homogeneous solutions which constitute the Mixmaster dynamics: a complicated oscillatory and chaotic model that exhibits properties similar to those discussed by BKL. The study of the dynamics of the universe in the vicinity of the cosmological singularity has become a rapidly developing field of modern theoretical and mathematical physics. The generalization of the BKL model to the cosmological singularity in multidimensional (Kaluza–Klein type) cosmological models has a chaotic character in the spacetimes whose dimensionality is not higher than ten, while in the spacetimes of higher dimensionalities a universe after undergoing a finite number of oscillations enters into monotonic Kasner-type contracting regime. The development of cosmological studies based on superstring models has revealed some new aspects of the dynamics in the vicinity of the singularity. In these models, mechanisms of changing of Kasner epochs are provoked not by the gravitational interactions but by the influence of other fields present. 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). 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 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). In the open model, there is one time singularity so time is limited at one end but unlimited at the other, while in the closed model there are two singularities that limit time at both ends (the Big Bang and Big Crunch). The only physically interesting properties of spacetimes (such as singularities) are those which are stable, i.e., those properties which still occur when the initial data is perturbed slightly. It is possible for a singularity to be stable and yet be of no physical interest: stability is a necessary but not a sufficient condition for physical relevance. For example, a singularity could be stable only in a neighbourhood of initial data sets corresponding to highly anisotropic universes. Since the actual universe is now apparently almost isotropic such a singularity could not occur in our universe. A sufficient condition for a stable singularity to be of physical interest is the requirement that the singularity be generic (or general). Roughly speaking, a stable singularity is generic if it occurs near every set of initial conditions and the non-gravitational fields are restricted in some specified way to "physically realistic" fields so that the Einstein equations, various equations of state, etc., are assumed to hold on the evolved spacetimes. It might happen that a singularity is stable under small variations of the true gravitational degrees of freedom, and yet it is not generic because the singularity depends in some way on the coordinate system, or rather on the choice of the initial hypersurface from which the spacetime is evolved. For a system of non-linear differential equations, such as the Einstein equations, a 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 independent 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 of other additional general solutions that do not contain a singularity. For example, there is no reason to doubt the existence of a general solution without a 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.
Existence of physical time singularity
One of the principal problems studied by the Landau group (to which BKL belong) was whether relativistic cosmological models necessarily contain a time singularity or whether the time singularity is an artifact of the assumptions used to simplify these models. The independence of the singularity on symmetry assumptions would mean that time singularities exist not only in the special, but also in the general solutions of the Einstein equations. 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. A criterion for generality of solutions is the number of independent space coordinate functions that they contain. These include only the "physically independent" functions whose number cannot be reduced by any choice of reference frame. In the general solution, the number of such functions must be enough to fully define the initial conditions (distribution and movement of matter, distribution of gravitational field) at some moment of time chosen as initial. This number is four for an empty (vacuum) space, and eight for a matter and/or radiation-filled space. Previous work by the Landau group; reviewed in) led to the conclusion that the general solution does not contain a physical singularity. 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. 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.
At that time, the only known indication for the existence of physical singularity in the general solution was related to the form of the Einstein equations written in a synchronous frame, that is, in a frame in which the proper time x0 = t is synchronized throughout the whole space; in this frame the space distance element dl is separate from the time interval dt. 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. However, the efforts to find a general physical singularity were foregone after it became clear that the singularity mentioned above is linked with a specific geometric property of the synchronous frame: the 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, dissuaded the researchers for further investigations along these lines. 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. Other similar theorems were found later on by Hawking and Geroch (see Penrose–Hawking singularity theorems). This revived interest in the search for singular solutions.
Generalized homogeneous solution In a space that is both homogeneous and isotropic the metric is determined completely, leaving free only the sign of the curvature. Assuming only space homogeneity with no additional symmetry such as isotropy leaves considerably more freedom in choosing the metric. The following pertains to the space part of the metric at a given instant of time t assuming a synchronous frame so that t is the same synchronised time for the whole space.
The BKL conjecture In their 1970 work, BKL stated that as one approaches a singularity, terms containing time derivatives in Einstein's equations dominate over those containing spatial derivatives. This has since been known as the BKL conjecture and implies that Einstein's partial differential equations (PDE) are well approximated by ordinary differential equations (ODEs), whence the dynamics of general relativity effectively become local and oscillatory. The time evolution of fields at each spatial point is well approximated by the homogeneous cosmologies in the Bianchi classification. By separating the time and space derivatives in the Einstein equations, for example, in the way used for the classification of homogeneous spaces, and then setting the terms containing space derivatives equal to zero, one can define the so-called truncated theory of the system (truncated equations). Then, the BKL conjecture can be made more specific: Weak conjecture: As the singularity is approached the terms containing space derivatives in the Einstein equations are negligible in comparison to the terms containing time derivatives. Thus, as the singularity is approached the Einstein equations approach those found by setting derivative terms to zero. Thus, the weak conjecture says that the Einstein equations can be well approximated by the truncated equations in the vicinity of the singularity. Note that this does not imply that the solutions of the full equations of motion will approach the solutions to the truncated equations as the singularity is approached. This additional condition is captured in the strong version as follows. Strong conjecture: As the singularity is approached the Einstein equations approach those of the truncated theory and in addition the solutions to the full equations are well approximated by solutions to the truncated equations. In the beginning, the BKL conjecture seemed to be coordinate-dependent and rather implausible. Barrow and Tipler, 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. The BKL conjecture was sometimes rephrased in the literature as a statement that near the singularity only the time derivatives are important. 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. However, there exist reformulations of Einstein theory in terms of new variables involving the relevant gradients, for example in Ashtekar-like variables, for which the statement about the dominant role of the time derivatives is correct. It is true that one gets at each spatial point an effective description of the singularity in terms of a finite dimensional dynamical system described by ordinary differential equations with respect to time, but the spatial gradients do enter these equations non-trivially. 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. It is fair to say that we are still quite far from a proof of the strong conjecture. But there has been outstanding progress in simpler models. In particular, Berger, Garfinkle, Moncrief, Isenberg, Weaver, and others showed that, in a class of models, as the singularity is approached the solutions to the full Einstein field equations approach the "velocity term dominated" (truncated) ones obtained by neglecting spatial derivatives. Andersson and Rendall showed that for gravity coupled to a massless scalar field or a stiff fluid, for every solution to the truncated equations there exists a solution to the full field equations that converges to the truncated solution as the singularity is approached, even in the absence of symmetries. These results were generalized to also include p-form gauge fields. In these truncated models the dynamics are simpler, allowing a precise statement of the conjecture that could be proven. In the general case, the strongest evidence to date comes from numerical evolutions. Berger and Moncrief began a program to analyze generic cosmological singularities. While the initial work focused on symmetry reduced cases, more recently Garfinkle performed numerical evolution of space-times with no symmetries in which, again, the mixmaster behavior is apparent. Finally, additional support for the conjecture has come from a numerical study of the behavior of test fields near the singularity of a Schwarzschild black hole.
Kasner solution
The BKL approach to anisotropic (as opposed to isotropic) homogeneous spaces starts with a generalization of an exact particular solution derived by Kasner 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). Here, p1, p2, p3 are any three numbers that satisfy the following Kasner conditions
Because of these relations, only one of the three numbers is independent (two equations with three unknowns). All three numbers are never the same; two numbers are the same only in the sets of values ( − 1 3 , 2 3 , 2 3 ) {\textstyle (-{\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. This is partially proved by squaring both sides of the first condition eq. 3 and developing the square:
( p 1 + p 2 + p 3 ) 2 = ( p 1 2 + p 2 2 + p 3 2 ) + ( 2 p 1 p 2 + 2 p 2 p 3 + 2 p 1 p 3 ) = 1 {\displaystyle \left(p_{1}+p_{2}+p_{3}\right)^{2}=\left(p_{1}^{2}+p_{2}^{2}+p_{3}^{2}\right)+\left(2p_{1}p_{2}+2p_{2}p_{3}+2p_{1}p_{3}\right)=1}
The term ( p 1 2 + p 2 2 + p 3 2 ) {\displaystyle \left(p_{1}^{2}+p_{2}^{2}+p_{3}^{2}\right)} is equal to 1 by dint of the second condition eq. 3 and therefore the term with the mixed products should be zero. This is possible if at least one of the p1, p2, p3 is negative. If the numbers are arranged in increasing order, p1 < p2 < p3, they change in the intervals (Fig. 4)
The Kasner metric eq. 2 corresponds to a flat homogenous but anisotropic space in which all volumes increase with time in such a way that the linear distances along two axes y and z increase while the distance along the axis x decreases. The moment t = 0 causes a singularity in the solution; the singularity in the metric at t = 0 cannot be avoided by any reference frame transformation. At the singularity, the invariants of the four-dimensional curvature tensor go to infinity. 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. 2) into Galilean. BKL parametrize the numbers p1, p2, p3 in terms of a single independent (real) parameter u (Lifshitz-Khalatnikov parameter) as follows
The Kasner index parametrization appears mysterious until one thinks about the two constraints on the indices eq. 3. Both constraints fix the overall scale of the indices so that only their ratios can vary. It is natural to pick one of those ratios as a new parameter, which can be done in six different ways. Picking u = u32 = p3 / p2, for example, it is trivial to express all six possible ratios in terms of it. Eliminating p3 = up2 first, and then using the linear constraint to eliminate p1 = 1 − p2 − up2 = 1 − (1 + u)p2, the quadratic constraint reduces to a quadratic equation in p2
with roots p2 = 0 (obvious) and p2 = (1 + u) / (1 + u + u2), from which p1 and p3 are then obtained by back substitution. One can define six such parameters uab = pa / pb, for which pc ≤ pb ≤ pa when (c, b, a) is a cyclic permutation of (1, 2, 3). All different values of p1, p2, p3 ordered as above are obtained with u running in the range u ≥ 1. The values u < 1 are brought into this range according to
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
where
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 pl, pm, pn which, as before, are related by eq. 3, are functions of the space coordinates. The powers pl, pm, pn 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
where v = l[mn]. It is convenient to introduce the following quantities
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 Kasner metric is introduced in the Einstein equations by substituting the respective metric tensor γαβ from eq. 7 without defining a priori the dependence of a, b, c from t:
ϰ α β = 2 a ˙ a l α l β + 2 b ˙ b m α m β + 2 c ˙ c n α n β {\displaystyle \varkappa _{\alpha }^{\beta }={\frac {2{\dot {a}}}{a}}l_{\alpha }l^{\beta }+{\frac {2{\dot {b}}}{b}}m_{\alpha }m^{\beta }+{\frac {2{\dot {c}}}{c}}n_{\alpha }n^{\beta }}
where the dot above a symbol designates differentiation with respect to time. The Einstein equation eq. 11 takes the form
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 Pαβ do not include terms of order higher than two, then
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 one negative power among the 3 powers pl, pm, pn results in appearance of terms from Pαβ with an order greater than t−2. If the negative power is pl (pl = p1 < 0), then Pαβ contains the coordinate function λ and eq. 12 become
Here, the second terms are of order t−2(pm + pn − pl) whereby pm + pn − pl = 1 + 2 |pl| > 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 three Einstein equations eq. 13 contain only first order time derivatives of the metric tensor. They give three 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 four conditions. These conditions bind ten different coordinate functions: three components of each of the vectors l, m, n, and one function in the powers of t (any one of the functions pl, pm, pn, 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 three space coordinates. Therefore, the final solution contains overall 10 − 4 − 3 = 3 physically arbitrary functions which is one less than what is needed for the general solution in vacuum. The degree of generality reached at this point is not lessened by introducing matter; matter is written into the metric eq. 7 and contributes four new coordinate functions necessary to describe the initial distribution of its density and the three 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 which are the hydrodynamic equations
where ui is the 4-dimensional velocity, ε and σ are the densities of energy and entropy of matter (cf. and; also; for details see ). For the ultrarelativistic equation of state p = ε/3 the entropy σ ~ ε1/4. 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
∂ ∂ t ( − g u 0 ε 3 4 ) = 0 , 4 ε ⋅ ∂ u α ∂ t + u α ⋅ ∂ ε ∂ t = 0 , {\displaystyle {\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
where 'const' are time-independent quantities. Additionally, from the identity uiui = 1 one has (because all covariant components of uα are to the same order)
u 0 2 ≈ u n u n = u n 2 c 2 , {\displaystyle u_{0}^{2}\approx u_{n}u^{n}={\frac {u_{n}^{2}}{c^{2}}},}
where un is the velocity component along the direction of n that is connected with the highest (positive) power of t (supposing that pn = p3). From the above relations, it follows that
or
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 − 1 2 T , R α β = T α β − 1 2 δ α β T , {\displaystyle 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 R α 0 = T α 0 {\displaystyle 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 a 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. For an analysis of this case, see.
Oscillating mode towards the singularity 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. Looked at such angle, the four conditions imposed on the coordinate functions in the solution eq. 7 are of different types: three conditions that arise from the equations R α 0 {\displaystyle 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: instability caused by perturbations can break this condition. The action of such perturbation must bring the model to another, more general, mode. The 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 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. 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 for such a study. 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 {\displaystyle R_{m}^{0}} , R n 0 {\displaystyle R_{n}^{0}} , R l m {\displaystyle R_{l}^{m}} , R l n {\displaystyle R_{l}^{n}} , R m n {\displaystyle R_{m}^{n}} are identically zeros). These equations contain only functions of time; this is a condition that has to be fulfilled 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:
substituting the variable t for τ according to:
Then (subscripts denote differentiation by τ):
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:
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) (pl = 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:
(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 pl < 0); let pl = р1, pm = р2, pn = р3 so that
Then
where Λ is constant. Initial conditions for eq. 29 are redefined as
Equations eq. 29 are easily integrated; the solution that satisfies the condition eq. 32 is
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 ∼ e − Λ p 1 τ , b ∼ e Λ ( p 2 + 2 p 1 ) τ , c ∼ e Λ ( p 3