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 [50] in the polarized Gowdy case (which in some
respects follows the ideas of [31]). In [50
], 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 [50
, pp. 88–89]. We give a brief description of the analysis of [50
] in
Section 7.
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:
see [29The 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.
Let us mention some of the results that have been obtained using Fuchsian methods. In [54], Fuchsian
methods were applied to the T3-Gowdy case in the real analytic setting. The assumption of real analyticity
was later relaxed to smoothness [68
]. See also [86
] 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.
http://www.livingreviews.org/lrr-2010-2 |
Living Rev. Relativity 13, (2010), 2
![]() This work is licensed under a Creative Commons License. E-mail us: |