Quiescent cosmological singularities

Bernd Schmidt, Albert Einstein Institute, Max Planck Society

schmidt@aei-potsdam.mpg.de

In 1964 Penrose and Hawking showed that singularities are a general
feature of classes of solutions of
Einstein's field equations. Their result said nothing about the structure
of those singularities. In the
following years much effort was directed to define and analyze
``singularities" of spacetimes.
However, it turned out that without really using the field equations there
were far too many and absurdly
complicated possibilities such that it seemed hopeless to attempt a useful
classification. In 1970
Belinskii, Khalatnikov and Lifshitz (BKL) gave a heuristic description of
a class of singularities based
on formal expansions of the metric near a singularity. It remained,
however, unclear whether the use of
the field equation together with the formal expansions could be justified.
schmidt@aei-potsdam.mpg.de

A recent theorem by L. Andersson and A. Rendall [1] shows rigorously that in a particular case, when the matter is described by a scalar field or a stiff perfect fluid , the BKL picture is correct. In this particular case there is no oscillatory behavior near the singularity, i.e. a quiescent singularity.

I will describe the theorem and make some remarks about the way it is
proved. The theorem uses the notion
of ``velocity dominated solution" which I will define first. Suppose we
have a metric

If we drop all spatial derivatives in the evolution equations, we obtain a system of ordinary differential equations for

Now we can formulate the main theorem:

**Theorem:** *Let **S** be a three-dimensional analytic
manifold and
(*
*) a ** solution of the
velocity dominated Einstein-scalar
field equations on *
* such that *
* and
each eigenvalue ** of
*
* is positive. Then there exists an open neighborhood
**U** of *
* in *
* and a unique **
solution *
* of the Einstein-scalar field equations on
*
*, such that for each compact subset *
*
there are real positive numbers ** and *
*
with *
* for which the following estimates hold
uniformly
on **K**:*
*together with similar estimates for spatial derivatives of **g*_{ab}*
and
**.*

Each velocity dominated solution is approached by a unique solution of the
full equations. Hence, the
singularity structure of the full solution is the same as that of the
velocity dominated solution. There are
indications that conversely, each solution approaches a velocity dominated
solution. The behavior of the
curvature tensor near the singularity is

The solution is really singular at

The proof of the theorem relies on a result by Kichenassamy and Rendall
[2]. It concerns a system of the
form (
)

Under appropriate conditions, this singular equation has a unique solution near

New mathematical tools allow fresh investigations of the properties of singularities of solutions of Einstein's field equations. Hopefully, this technique can also be used to treat the more complicated cases of an oscillatory behavior of the metric near the singularity.

*References:*

[1] Andersson, L. and Rendall, A. D. Quiescent cosmological
singularities.
`gr-qc/0001047`.

[2] Kichenassamy, S. and Rendall, A. D. (1998) Analytic description of
singularities in Gowdy spacetimes. Class.
Quantum Grav. 15, 1339-1355