What is “BKL singularity” ?

BKL singularity

BKL singularity
BKL singularity
BKL singularity

A BKL 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 isn’t artificially created by the assumptions and simplifications made by the other well-known special solutions such as the Friedmann–Lemaxtre–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.

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 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 isn’t 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 isn’t continuous, but stops or reverses after time reaches some 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 isn’t 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 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 isn’t 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. Existence of a general solution with a singularity, therefore, does not preclude the existence of other additional general solutions that don’t 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 isn’t 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 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, isn’t 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.

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 didn’t have anything in common with a choice of reference frame. Other similar theorems were found later on by Hawking and Geroch. This revived interest in the search for singular solutions.

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.

Figure 1 is a plot of p1, p2, p3 with an argument 1/u. The numbers p1(u) and p3(u) are monotonously increasing while p2(u) is monotonously decreasing function of the parameter u.

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 3rd direction. The volume of the element decreases in proportion to t.

Pαβ is the 3-dimensional Ricci tensor, which is expressed by the 3-dimensional metric tensor γαβ in the same way as Rik is expressed by gik; Pαβ contains only the space derivatives of γαβ.

LK 2-5″ class=”reference”> These equations together with eq. 14 give the expressions eq. 8 with powers that satisfy eq. 3.

Here, the 2nd terms are of order t−2 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.

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Prandtl%E2%80%93Glauert singularity

Prandtl%E2%80%93Glauert singularity

Prandtl%E2%80%93Glauert singularity
Prandtl%E2%80%93Glauert singularity
Prandtl%E2%80%93Glauert singularity

The Prandtl–Glauert singularity is the prediction by the Prandtl–Glauert transformation that infinite pressures would be experienced by an aircraft as it approaches the speed of sound. Because it is invalid to apply the transformation at these speeds, the predicted singularity does not emerge. This is related to the early 20th century misconception of the impenetrability of the sound barrier.

Near the sonic speed the transformation features a singularity, although this point isn’t within the area of validity. The singularity is also called the Prandtl–Glauert singularity, and the aerodynamic forces are calculated to approach infinity. In reality, the aerodynamic and thermodynamic perturbations do get amplified strongly near the sonic speed, but they remain finite and a singularity does not occur. An explanation for this is that the Prandtl–Glauert transformation is a linearized approximation of compressible, inviscid potential flow. As the flow approaches sonic speed, the nonlinear phenomena dominate within the flow, which this transformation completely ignores for the sake of simplicity.

This obviously nonphysical result is known as the Prandtl–Glauert singularity.

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