Is it admissible to linearize the Einstein equation in the presence ofmatter?
by
Joan Girbau
Universitat Autonoma de Barcelona, Spain
Coauthors: L. Bruna

Poster

In general relativity the equation governing gravitation is the Einstein's one: G(g)=\chiT, where G is the Einstein tensor of g, G(g)=Ric(g)-(1/2)Rg, T is the stress-energy tensor and \chi is a constant. The Minkowski metric \eta in R⁴ fulfils this equation in the vacuum (when T=0). In case T is small, Einstein wrote the metric g in the form g=\eta+h with h small. Then he realized that the linear terms in h in the equation G(\eta+h)=\chiT were strongly related with the classical wave equation (gravitational waves). In the seventies a lot of papers appeared trying to answer the following question: under what mathematical conditions is it licit to linearize the Einstein equation? Most of them (Y.Choquet-Bruhat and S.Deser, Y.Choquet-Bruhat, A.Fisher and J.E.Marsden, V.Moncrief) conerned the vacuum. But little work was done in case of non-empty spaces. Recently L. Bruna and myself have studied the linearization stability of the Einstein equation in Robertson-Walker cosmological models ([1] and [2]) and proved that such a model is stable in case of null curvature and unstable in case of positive curvature. The aim of this poster is to explain and discuss these results.

Let (S, g) be a 3-Riemannian manifold with constant curvature K. Let V=S×I, where I is an \R-interval. Let [g\tilde] be a Lorentz metric in V of the form [g\tilde]=-dt²+\zeta(t)²g, where t is the coordinate of I. In V we consider a perfect fluid with u=\partial/\partialt as a velocity field, so that the stress-energy tensor of the fluid has the form [T\tilde]=(\rho+p)u^*\otimesu^*+p[g\tilde], where u^* is the 1-form associated to u by [g\tilde]. Obviously, the metric [g\tilde] and the tensor [T\tilde] are related by the Einstein equation G([g\tilde])=\chi[T\tilde], which gives well-known relations between \rho, p, \zeta and the curvatura K of g. The Lorentz manifold (V, [g\tilde]) whith the stress-energy tensor [T\tilde] is called a Robertson-Walker model. At present our main results concerning the linearization stability or the Einstein equations for Robertson-Walker models are the following:

Theorem 1.- Let (V=S×I ,  [g\tilde] ,  [T\tilde]) be a Robertson-Walker model. Suppose that S is connected and simply connected. Let K be the curvature of (S, g) (constant by definition of Robertson-Walker model).

1. a) If K=0 then the Einstein equation is stable under linearization at the initial metric [g\tilde] and the initial stress-energy tensor [T\tilde].
2. b) If K > 0 then there is no subspace F in the space of stress-energy tensors such that the Einstein equation (at the initial metric [g\tilde] and the initial stress-energy tensor [T\tilde]) is stable in the direction of F.
3. c) If K < 0 then the Einstein equation is stable under linearization (at the initial metric [g\tilde] and the initial stress-energy tensor [T\tilde]).

Of course this is a simplification of our true statement. Precise definitions of the spaces of metrics and the spaces of stress-energy tensors should be given in order to define accurately the concept of linearization stability and to state the theorem in a precise way. The proof of a) is contained in [1]. The proof of b) is contained in [2]. We have not yet published the proof of c). It is based on an unpublished result on functional analysis due to Joaquim Bruna concerning the Laplace-Beltrami operator associated to the Poincaré metric in the hiperbolic space of dimension 3.

The above theorem concerns models V=S×I with S connected and simply connected. When S is not simply connected the behaviour of the model with regard to the linearization stability of the Einstein equation may be very different and it is not due to the sign of the curvature as one might suppose, but to the existence of many or few Killing vector fields in the 3-manifold (S, g), in a similar way that the results obtained by A.Fisher, J.E.Marsden and V.Moncrief in the seventies for the vacuum spacetime. We have also proved another theorem to make this behaviour clear.

References

[1] L.Bruna and J.Girbau, Linearization stability of the Einstein equation for Robertson-Walker models (I). Journal of Math. Phys. 40, 5117-5130 (1999).

[2] L.Bruna and J.Girbau, Linearization stability of the Einstein equation for Robertson-Walker models (II). Journal of Math. Phys. 40, 5131-5137 (1999).

Date received: June 21, 2000

Copyright © 2000 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cafh-34.