An Extension of the Stability Theorem of the Minkowski Space in General Relativity

TL;DR

The paper extends the global nonlinear stability of Minkowski space for the Einstein vacuum equations to more general, weaker-asymptotics initial data by employing a robust bootstrap framework built on invariant formulations and Bel-Robinson energy methods. It replaces reliance on full spacetime symmetries with a controlled quasi-conformal action generated by $T$, $S$, and $ar K$, and develops a detailed system of weighted norms for curvature, geometry, and the optical function to track decay. The main result proves global existence and geodesic completeness for asymptotically flat data under a global smallness condition, with precise decay bounds for the Weyl curvature components and geometric quantities; the work also exposes borderline decay cases, indicating sharpness of the assumptions. This generalizes Christodoulou–Klainerman’s result by permitting weaker fall-off and fewer derivatives, while preserving a precise description of asymptotic behavior and gravitational radiation features in the framework of the EV equations.

Abstract

In this paper, we sketch the proof of the extension of the stability theorem of the Minkowski space in General Relativity done explicitly in previous work by the present author. We discuss solutions of the Einstein vacuum (EV) equations. We solve the Cauchy problem for more general, asymptotically flat initial data than in the pioneering work `The Global Nonlinear Stability of the Minkowski Space' of D. Christodoulou and S. Klainerman or than in any other work. Moreover, we describe precisely the asymptotic behaviour. Our relaxed assumptions on the initial data yield a spacetime curvature which is not bounded in $L^{\infty}(M)$. As a major result, we encounter in our work borderline cases, which we discuss in this paper as well. The fact that certain of our estimates are borderline in view of decay indicates that the conditions in our main theorem are sharp in so far as the assumptions on the decay at infinity on the initial data are concerned. Thus, the borderline cases are a consequence of our relaxed assumptions on the data. They are not present in the other works, as all of them place stronger assumptions on their data. We work with an invariant formulation of the EV equations. Our main proof is based on a bootstrap argument. To close the argument, we have to show that the spacetime curvature and the corresponding geometrical quantities have the required decay. In order to do so, the Einstein equations are decomposed with respect to specific foliations of the spacetime. This result generalizes the work `The Global Nonlinear Stability of the Minkowski Space' of D. Christodoulou and S. Klainerman.