Stability of Minkowski space and polyhomogeneity of the metric

Abstract

We study the nonlinear stability of the $(3+1)$-dimensional Minkowski spacetime as a solution of the Einstein vacuum equation. Similarly to our previous work on the stability of cosmological black holes, we construct the solution of the nonlinear initial value problem using an iteration scheme in which we solve a linearized equation globally at each step; we use a generalized harmonic gauge and implement constraint damping to fix the geometry of null infinity. The linear analysis is largely based on energy and vector field methods originating in work by Klainerman. The weak null condition of Lindblad and Rodnianski arises naturally as a nilpotent coupling of certain metric components in a linear model operator at null infinity. Upon compactifying ${\mathbb{R}}^4$ to a manifold with corners, with boundary hypersurfaces corresponding to spacelike, null, and timelike infinity, we show, using the framework of Melrose's b-analysis, that polyhomogeneous initial data produce a polyhomogeneous spacetime metric. Finally, we relate the Bondi mass to a logarithmic term in the expansion of the metric at null infinity and prove the Bondi mass loss formula.

127 pages, 15 figures. v2 is the published version, with an improved title, a significantly expanded introduction, and many typos fixed

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• Microlocal analysis near null infinity in asymptotically flat spacetimes
• Hyperboloidal initial data without logarithmic singularities
• The Case Against Smooth Null Infinity V: Early-Time Asymptotics of Linearised Gravity Around Schwarzschild for Fixed Spherical Harmonic Modes
• Elliptic parametrices in the 0-calculus of Mazzeo and Melrose
• Physical-space estimates for axisymmetric waves on extremal Kerr spacetime
• Conformal Scattering of Maxwell Potentials
• Scattering of Maxwell potentials on curved spacetimes
• Scattering for wave equations with sources close to the lightcone and prescribed radiation fields
• The space spinor formalism and estimates for spinor fields