On the stability of Einstein spaces with spatial sections of negative scalar curvature

Marica Minucci (Queen Mary University of London)

In this article it is shown how the extended conformal Einstein field equations and a gauge based on the properties of conformal geodesics can be used to analyse the non-linear stability of Einstein spaces with spatial sections of negative scalar curvature. This is done by considering a de Sitter-like spacetime, which is a vacuum spacetime with a de Sitter-like value of the cosmological constant. This class of spacetimes admits a conformal extension with a spacelike conformal boundary and represent the simplest application of the conformal field equations to the analysis of global properties of spacetimes. The existence and stability theorem for this type of spacetime can be proven by means of hyperbolic reduction procedures. The method that we use relies on conformal Gaussian systems that combined with the use of conformal field equations allows us to formulate initial value problems for the perturbed de Sitter-like spacetime not only on a standard initial hypersurface at a fiduciary finite time, but also on a hypersurface corresponding to the conformal boundary of the spacetime. The appeal of considering the conformal Einstein field equations rather than the Einstein field equations for our purposes is that local results for an unphysical spacetime could in principle be translated into global results for the physical spacetimes. Furthermore, in similar manner to the case of the ADM evolution equations where the Einstein field equations are recast as a set of evolution equations for an initial value problem, we can formulate the extended conformal field equations as an initial value problem performing a 1 + 3 decomposition and obtain evolution equations along the congruence of conformal geodesics.