Einstein equations

[Note: This is an immense topic, and we do not even begin to do it justice with this very brief selection of results. Further references or expansion of this article will, of course, be very much appreciated.]

Introduction

The (vacuum) Einstein equations, for a Riemannian manifold $\,(M,g)\!$, take the form

$\,R_{\alpha \beta }=CRg_{\alpha \beta }+\Lambda g_{\alpha \beta }\!$

where $\,g\!$ is the metric for a 3+1-dimensional manifold $\,M\!$, $\,R\!$ is the Ricci curvature tensor, $\,C\!$ is an absolute constant, and $\,\Lambda \!$ the so-called cosmological constant. The Cauchy data for this problem is thus a three-dimensional Riemannian manifold together with the second fundamental form of this manifold (roughly speaking, this is like the initial position and initial velocity for the metric $g$). However, these two quantities are not completely independent; they must obey certain constraint equations. These equations are now known to be well behaved for all $s>3/2$ Max-p, Max2005 (see also earlier work in higher regularities in RenFri2000, Ren2002).

Because of the diffeomorphism invariance of the Einstein equations, these equations are not hyperbolic as stated. However, this can be remedied by choosing an appropriate choice of co-ordinate system (which is the analog of a gauge transformation in gauge theory). One popular choice is harmonic co-ordinates or wave co-ordinates, where the co-ordinate functions $x^{a}$ are assumed to obey the wave equation $\Box _{g}x^{a}=0$ with respect to the metric $g$. In this case the Einstein equations take a form which (in gross caricature) looks something like

$\Box _{g}g=\Gamma (g)Q(dg,dg)+$lower order terms

where $Q$ is some quadratic form of the first two derivatives. In other words, it becomes a quasilinear wave equation. One would then specify initial data on the initial surface $x=0$; the co-ordinate $x$ plays the role of time, locally at least.

• The critical regularity is $s_{c}=3/2$. Thus energy is super-critical, which seems to make a large data global theory extremely difficult.
• LWP is known in $H^{s}$ for $s>5/2$ by energy estimates (see HuKaMar1977, AnMc-p; for smooth data $s>4$ this is in Cq1952) - given that the initial data obeys the constraint equations, of course.
  • This result can be improved to $s>2$ by the recent quasilinear theory (see in particular KlRo-p3, KlRo-p4, KlRo-p5).
  • Some results have been recently achieved in the direction of the LWP for $s=2$ (KlRo-p6, KlRo-p7, KlRo-p8). This problem is known in the litterature as the $L^{2}$ conjecture.
  • For smooth data, one has a (possibly geodesically incomplete) maximal Cauchy development (CqGc1969).
• GWP for small smooth asymptotically flat data was shown in CdKl1993 (see also CdKl1990). In other words, Minkowski space is stable.
  • Another proof using the double null foliation is in KlNi2003, KlNi-p
  • Another proof of this fact (using the Lorenz gauge, and assuming Schwarzschild metric outside of a compact set) is in LbRo-p (see also LbRo2003 for a treatment of the asymptotic dynamics)
  • Singularities must form if there is a trapped surface (Pn1965, Haw66).
• Many special solutions (Schwarzschild space, Kerr space, etc.) The stability of these spaces is a very interesting (and difficult) question.
• The equations can simplify under additional symmetry assumptions. The $U(1)$-symmetric case reduces to a system of equations which closely resembles the two-dimensional wave maps equation (with the target manifold being hyperbolic space $H^{2}$).

Global property of solutions

It is widely known that exact solutions of vacuum Einstein equations display a singularity. Initially, this behavior was believed to be due to the particular symmetry chosen to solve them. Roger Penrose and Stephen Hawking, with some classical papers (see Pn1965 and Haw66), proved that this is a general property of solutions of Einstein equations making the role of singularities central in the understanding of these equations. We can state this result in the following way HwEl73

Theorem: Space-time is not timelike geodesically complete if:

(1) $\,R_{\alpha \beta }K^{\alpha }K^{\beta }\geq 0\!$ for every non-space-like vector $\,K\!$;

(2) there exists a compact spacelike 3-surface S (without edge);

(3) the unit normals to S are everywhere converging (or everywhere diverging) on S.

Condition (1) implies that energy is always zero or positive, condition (2) and (3) say that space-time is spatially closed and expanding or contracting.

About the analytical structure of the singularities, a conjecture was put forward by V.A.Belinski, I.M. Khalatnikov and E.M. Lifshitz in BeKhLi1970. In any case, to acquire knowledge of the analytical structure of such solutions we need to solve Einstein equations in a regime of strong field (see FraA2006).

Open problems

• Cosmic Censorship Hypothesis

• BKL Conjecture

Further reading

• For more detail, we recommend the very nice survey on existence and global dynamics of the Einstein equations by Alan Rendall.

• S. W. Hawking, G. F. R. Ellis, The large scale structure of space-time, (Cambridge University Press, Cambridge, 1973).