Quiescent cosmological singularities

Bernd Schmidt, Albert Einstein Institute, Max Planck Society
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.

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 $(p=\mu)$, 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

\begin{displaymath}-dt^2+ g_{ab}(t,x^c)dx^a dx^b\ .

If we drop all spatial derivatives in the evolution equations, we obtain a system of ordinary differential equations for gab(t). In the constraints we just drop the Ricci scalar. This system and its solution, $^0g_{ab},^0k_{ab}=\partial_tg_{ab}$, is called `` velocity dominated". These equations can be integrated completely and the solutions and their singularities can be described explicitly. One has in general 0kaa=(C-t)-1. We can chose C=0 to have the singularity occur at t=0. Furthermore all mixed components 0kab are proportional to t-1. At a fixed spatial point we can simultaneously diagonalize 0kab and 0gab by a suitable choice of frame. The diagonal components of the metric in this frame are then proportional to powers of t. The equation for the matter field can also be integrated with the result $^0\phi(t,x^a)=A(x^a)\log t +B(x^a)$.

Now we can formulate the main theorem:

Theorem: Let S be a three-dimensional analytic manifold and ( $^0g_{ab}(t),^0k_{ab}(t),^0\phi(t)$) a $C^\omega$ solution of the velocity dominated Einstein-scalar field equations on $S\times (0,\infty)$ such that $t\ ^0k_a^a=-1$ and each eigenvalue $\lambda$ of $ -t\ ^0k_a^b$ is positive. Then there exists an open neighborhood U of $S\times \{0\}$ in $S\times [0,\infty)$ and a unique $C^\omega$ solution $g_{ab},k_{ab}, \phi $ of the Einstein-scalar field equations on $U\cap (S\times (0,\infty))$, such that for each compact subset $K\subset
S$ there are real positive numbers $\beta$ and $\alpha^a{}_b$ with $\beta< \alpha^a{}_b$ for which the following estimates hold uniformly on K: $
\ \ 1.\ \ \ \ ^0g^{ac}g_{cb}=\delta ^a{}_b +o(t^{\alpha^a{}_b})
$ $
\ \ 2.\ \ \ \ k^a{}_b=^0k^a{}_b+o(t^{-1+\alpha^a{}_b})
$ $
\ \ 3.\ \ \ \ \phi=^0\phi+o(t^\beta)
$ $
\ \ 4.\ \ \ \ \partial_t\phi=\partial_t\ ^0\phi+o(t^{-1+\beta})
$ together with similar estimates for spatial derivatives of gab and $\phi$.

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

\begin{displaymath}R_{ab}R^{ab}= K(x^a)\ t^{-4}+ \dots

The solution is really singular at t=0 and the BKL picture is justified in the cases considered.

The proof of the theorem relies on a result by Kichenassamy and Rendall [2]. It concerns a system of the form ( $u=(u^1\dots u^N),\ x=(x^1,\dots x^n)$)

\begin{displaymath}t\ {\partial u\over\partial t}+A(x)\ u = f(t,x,u,u_x)

Under appropriate conditions, this singular equation has a unique solution near t=0 which is continuous in t and tends to zero as $t\to 0$. To use this theorem one rewrites the field equations as equations for the ``difference between the solution and the velocity dominated solution". Hence regular equations for a singular solution are replaced by a singular equation for a regular solution.

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.


[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

Jorge Pullin