Stability of Minkowski space and polyhomogeneity of the metric
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Stability of Minkowski space and polyhomogeneity of the metric Peter Hintz1,2
· András Vasy3
Received: 27 November 2017 / Accepted: 23 January 2020 / Published online: 8 February 2020 © Springer Nature Switzerland AG 2020
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 R4 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. Mathematics Subject Classification Primary 35B35 · Secondary 35C20 · 83C05 · 83C35
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Peter Hintz [email protected] András Vasy [email protected]
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Department of Mathematics, University of California, Berkeley, CA 94720-3840, USA
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Present Address: Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139-4307, USA
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Department of Mathematics, Stanford University, Stanford, CA 94305-2125, USA
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P. Hintz, A. Vasy
1 Introduction We prove the nonlinear stability of (3 + 1)-dimensional Minkowski space as a vacuum solution of Einstein’s field equation and obtain a precise full expansion of the solution, in a mildly generalized harmonic gauge, in all asymptotic regions, i.e. near spacelike, null, and timelike infinity. On a conceptual level, we show how some of the methods we developed for our proofs of black hole stability in cosmological spacetimes [53,60] apply in this more familiar setting, studied by Christodoulou–Klainerman [27], Lindblad–Rodnianski [79,80], and many others: this includes the use of an iteration scheme for the construction of the metric in which we solve a linear equation globally at each step, keeping track of the precise asymptotic behavior of the iterates by working on a suitable compactification M of the spacetime, and the implementation of constraint damping. The estimates we prove for the linear equations—which arise as linearizations of the gauge-fixed Einstein equation around metrics which lie in the precise function space in which we seek the solution—are largely based on energy estimates and a version of the vector field method [64]. The estimates are rather refined in te
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