تفرد بي كي إل
ملحوظات

1 2 3  The convention used by BKL is the same as in the Landau & Lifshitz (1988) book. The Latin indices run through the values 0, 1, 2, 3; Greek indices run through the space values 1, 2, 3. The metric gik has the signature (+ − − −); γαβ = −gαβ is the 3-dimensional space metric tensor. BKL use a system of units, in which the speed of light and the Einstein gravitational constant are equal to 1.
↑  The expression for r is derived by logarithming the power coefficients in the metric: ln [t2pα(1/u)] = 2pα(1/u) ln t.
↑  When (p1, p2, p3) = (0, 0, 1) the spacetime metric قالب:EquationNote with dl2 from قالب:EquationNote transforms to Galilean metric with the substitution t sh z = ζ, t ch z = τ, that is, the singularity is fictional and the spacetime is flat.
1 2  Here and below all symbols for vector operations (vector products, the operations rot, grad, etc.) should be understood in a very formal way as operations over the covariant components of the vectors l, m, n such that are performed in نظام إحداثي ديكارتي x1, x2, x3.
↑  Excepting the case (p1, p2, p3) = (0, 0, 1), in which the metric singularity is fictitious.
↑  The constants λ, μ, ν are the so-called structural constants of the space movement group.
↑  In their exact form, the Einstein equations for homogeneous space contain, in general, 6 different functions of time γab(t) in the metric. The fact that in the present case a consistent system of exact equations is obtained for the metric which contains only 3 functions of time (γ11 = а2, γ22 = b2, γ33 = c2) is related to a symmetry that leads to the disappearance of 6 Ricci tensor components.
↑  The asymptotic values of ατ, βτ, γτ at τ → −∞ can be found without fully solving قالب:EquationNote. It suffices to note that the first of these equations has a form of a "particle" moving in one dimension in the field of an exponential potential wall with α playing the role of a constant. In this analogy, the Kasner mode refers to a free movement with constant velocity ατ = Λp1. After reflection from the wall, the particle moves freely with velocity ατ = −Λp1. Also noting that from قالب:EquationNote ατ + βτ = const, and ατ + γτ = const, one can see that βτ and γτ take the values βτ = Λ(p2 − 2p1), γτ = Λ(p3 − 2p1).
↑  Introduction of non-diagonal components of γab(t) imparts some new features to the BKL model: rotations of axes corresponding to the Kasner epoch powers; this problem is studied in Belinsky, Khalatnikov & Lifshitz (1971)
↑  The constant in the sine argument, of course, is not necessarily the same as ξ0 in قالب:EquationNote and قالب:EquationNote; however, making them the same does not change in any way the solution character.
↑  In a more precise calculation, a slowly changing logarithmic term appears in the sine argument, and a multiplier appears in front the exponent in the expression for с(ξ), see Belinsky, Khalatnikov & Lifshitz 1970.
↑  If in قالب:EquationNote, one substitutes sh 2χ with 2χ and solves it for all values of ξ, one obtains χ = c1J0(ξ) + c2N0(ξ) where J0, N0 are Bessel functions of the I and II kind. This solution interpolates between the two limiting cases and allows to relate by an order of magnitude the constant parameters in قالب:EquationNote and قالب:EquationNote.
↑  Since a, b, c have the dimension of length, their logarithms are defined only up to an additive constant which depends on the choice of the length units; in this sense قالب:EquationNote has a conditional meaning corresponding to a certain choice of the zero value of α, β, γ.
↑  According to قالب:EquationNote, transitions are large with small |p1| (i. e. large u) and are ≈1/|p1| ~ u. But even in this case Δn ~ un |αn|  \gg  un
↑  Fixing the limits of the era according to eq. قالب:EquationNote is meaningful because in such case the era contains all epochs in which the third function, γ(t) decreases monotonously. If the era is defined by the sequence of u values from k + x to 1 + x, then the monotonous decrease of γ(t) will continue during the first epoch of the next era.
↑  Epoch lengths are great in comparison to transitions between epochs. According to قالب:EquationNote transition lengths are great at small |p1| (i.e. large u) and are ∝ 1/|p1| ∝ u. But even in this case Δn ∝ un|αn|  \gg  un.
↑  قالب:EquationNote was known already to Gauss, and an equation of type قالب:EquationNote was considered in this connection by Rodion Kuzmin (see Gauss–Kuzmin distribution). Further information on the chaotic behaviour and entropy of continued fractions in Linas Vepstas. 2008. Entropy of Continued Fractions (Gauss-Kuzmin Entropy) نسخة محفوظة 2020-10-29 على موقع واي باك مشين.
↑  The plot of the function P(δ) in Fig. 2 in Lifshitz, Lifshitz & Khalatnikov 1970 is incorrect for several reasons. Apparently some errors were admitted in preparing the program for numerical solution of the integral equation. Also a "forced" reduction of the values P(0) and P(1) was performed in view of the incorrect footnote in Lifshitz, Lifshitz & Khalatnikov 1970. The finite probability of the value δ = 0 does not mean the possibility of the initial amplitude of oscillation becoming zero (which would be in contradiction to the regular course of evolution shown in Fig. 4). From قالب:EquationNote δs+1 tends to zero with xs → 0 proportional to xs; but the amplitude is given by the product δs+1Ωs+1, which tends to a finite limit since the expression قالب:EquationNote contains a term with 1/xs.
↑  Note that this metric allows arbitrary transformations of type ξ′ + z″ = f1 (ξ + z), ξ′ − z′ = f2 (ξ − z), x′a = fa (x1, x2).
↑  The equation  R_{0}^{0}+R_{3}^{3}=0 is a direct result of قالب:EquationNote–قالب:EquationNote if  {\dot {G}}\neq 0 or  G^{\prime }\neq 0. The case  {\dot {G}}=G^{\prime }=0 does not require a special treatment: it can be shown that the spacetime metric in this case converges (in first approximation) to Galilean.
↑  It is possible to look for a solution in the form of Fourier integrals; this issue has not been studied in detail. Therefore, BKL do not require Fourier series decomposition as a mandatory condition for the coordinate dependence of functions σ and φ
↑  Squared H terms in قالب:EquationNote result only in small (≈1/ξ) corrections in σ and φ. Calculation with cubic terms leads to appearance of a weak dependence of A, B from ξ that can be presented as an appearance of logarithmic phases in the oscillating factors in قالب:EquationNote. These calculations for the case ρ = 0 are given in Belinsky & Khalatnikov (1970) (cf. the analogous situation for homogeneous models, Belinsky, Khalatnikov & Lifshitz (1970)).
↑  The regular decomposition of the general solution of Einstein equations contains (in addition to the four three-dimensional functions) three independent functions of two coordinates (cf. Petrov 1969; Lifshitz & Khalatnikov (1963))