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A BKL (Belinsky–Khalatnikov–Lifshitz) singularity is a model of the dynamic evolution of the Universe near the initial singularity, described by an anisotropic, homogeneous, chaotic 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 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 is 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 brilliantly 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. 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 adequacy of the isotropic model in describing the present state of the 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 give the solution specific properties that disappear when considering a more general case. 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. 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. 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. 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 group〔 〕〔; Physical Review Letters, 6, 311 (1961)〕 (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 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 the Einstein equations written in a synchronous frame, that is, in a frame in which the proper time ''x''0 = ''t'' is synchronized throughout the whole space; in this frame the space distance element ''dl'' is separate from the time interval ''dt''.〔The convention used by BKL is the same as in the 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 = −''g''αβ 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 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.〔〔 This indication, however, was dropped after it became clear that it 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, removed the incentive among the researchers for further investigations along these lines. However, the interest in this problem waxed again in the 1960s 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). This revived interest in the search for singular solutions. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「BKL singularity」の詳細全文を読む スポンサード リンク
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