Stability of Minkowski Space

The Minkowski space, which is the simplest solution of the Einstein field equations in vacuum, that is, in the absence of matter, plays a fundamental role in modern physics as it provides the natural mathematical background of the special theory of relativity. It is most reasonable to ask whether it is stable under small perturbations. In other words, can arbitrary small perturbations of flat initial conditions lead to developments which are radically different, in the large, from the flat Minkowski space? It turns out to be a highly nontrivial problem as the Einstein equations are of a quasilinear hyperbolic character. Typical systems of this type, in three space dimensions, do form singularities in finite time even for small disturbances of their trivial initial data. To avoid finite-time singularities, we must require that sufficiently small perturbations of Minkowski space are geodesically complete. This, however, is not enough; one should also insist that the corresponding spacetimes become flat along all possible directions, that is, globally asymptotically flat. This is measured by the decay of the curvature tensor to zero. The precise rate of decay is also of interest. One expects that various null-frame components of the curvature tensor decay at different rates along outgoing null hypersurfaces; this goes under the name of "peeling estimates." It turns out in fact that we cannot prove geodesic completeness without establishing at the same time sufficiently fast rates of decay to flatness corresponding to at least some peeling.