5.2 Asymptotic expansions, Fuchsian methods

It is interesting to note that, in spite of the fact that the BKL picture and related proposals emphasize the importance of oscillatory behavior, the greatest successes of the BKL point of view have been obtained in the nonoscillatory setting. The main reason is that in the absence of oscillations, it is sometimes possible to characterize the asymptotic behavior in terms of asymptotic data (of course, the lack of results in the presence of oscillations is largely due to the difficulty in analyzing the asymptotic behavior in that case). In certain situations, this characterization is strong enough that the asymptotic data are in one-to-one correspondence with the solutions. If that is the case, the asymptotic data can be considered to be “initial data at the singularity”. The results in the nonoscillatory case come in different forms.

5.2.1 From solutions to asymptotics

The type of result, that is of greatest immediate interest is the one that, given a solution to the Einstein equations, provides asymptotic expansions. The means by which this is achieved vary. One method is to devise a simplified system of equations, such that the solution to the Einstein equations converges to a solution to the simplified system. In the spirit of the BKL picture, the simplified system is often obtained by omitting some (if not all) spatial derivatives. One example of a successful application is given by the analysis of Isenberg and Moncrief [50Jump To The Next Citation Point] in the polarized Gowdy case (which in some respects follows the ideas of [31]). In [50Jump To The Next Citation Point], a simplified system consisting of the Velocity Term Dominated (VTD) equations are introduced and solved, and the authors prove that solutions to the Einstein equations converge to solutions to the VTD system. Furthermore, a geometric definition for what the authors call Asymptotically Velocity-Term Dominated near the Singularity (or AVTDS) is given; see [50Jump To The Next Citation Point, pp. 88–89]. We give a brief description of the analysis of [50Jump To The Next Citation Point] in Section 7.

5.2.2 From asymptotics to solutions

Another type of result starts by specifying the asymptotics at the singularity and then proceeds to prove that there are solutions with these asymptotics. The analysis is based on reducing the particular form of the Einstein equations under consideration to an equation in Fuchsian form:

t∂tu + A(x)u = f (t,x,u,ux); (10 )
see [29Jump To The Next Citation Point, (1.5), p. 1054]; see also [53]. There is a standard theory, which deals with equations of the form of Equation (10View Equation), even with the nonregular problem of specifying initial data at t = 0, which is the case of interest in the present context. Consequently, the central problem is that of reformulating Einstein’s equations to Fuchsian form. The general procedure is as follows:

The standard Fuchsian theory is applicable in the real analytic setting. As a consequence, most of the results assume real analytic “data at the singularity” and lead to the conclusion that there are real analytic solutions with the corresponding asymptotic behavior. Clearly, the procedure is not always applicable. In particular, it is not expected to be applicable in the presence of oscillations.

5.2.3 Overview of results

Let us mention some of the results that have been obtained using Fuchsian methods. In [54Jump To The Next Citation Point], Fuchsian methods were applied to the T3-Gowdy case in the real analytic setting. The assumption of real analyticity was later relaxed to smoothness [68Jump To The Next Citation Point]. See also [86Jump To The Next Citation Point] for a similar analysis in the S2 × S1 and S3 cases (though there are some problems related to the symmetry axes in that case, and as a consequence, the results are less complete). An analysis of the polarized T2-symmetric spacetimes in the real analytic setting was carried out in [48]. In all the examples mentioned so far, the symmetry caused the suppression of oscillations. However, matter can also have the same effect. This is illustrated by [2], which consists of a study of the Einstein equations coupled to either a scalar field or a stiff fluid. The results are in the real analytic setting and associate a solution to asymptotic initial data. Finally, in [29] large classes of matter models are considered in various dimensions with similar results.

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