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 ...
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[1] 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.[2][3]
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[1] 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[4][5][6] (reviewed in [2]) 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.[7] 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.[2][3]
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.[6] 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[8] 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[9][10] and Geroch[11] (see Penrose-Hawking singularity theorems). It became clear that the search for a general solution with singularity must continue.
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