Vol. 7, No. 4, 2014

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The global stability of the Minkowski spacetime solution to the Einstein-nonlinear system in wave coordinates

Jared Speck

Vol. 7 (2014), No. 4, 771–901
Abstract

We study the coupling of the Einstein field equations of general relativity to a family of nonlinear electromagnetic field equations. The family comprises all covariant electromagnetic models that satisfy the following criteria: (i) they are derivable from a sufficiently regular Lagrangian; (ii) they reduce to the standard Maxwell model in the weak-field limit; (iii) their corresponding energy-momentum tensors satisfy the dominant energy condition. Our main result is a proof of the global nonlinear stability of the $\left(1+3\right)$-dimensional Minkowski spacetime solution to the coupled system for any member of the family, which includes the standard Maxwell model. This stability result is a consequence of a small-data global existence result for a reduced system of equations that is equivalent to the original system in our wave-coordinate gauge. Our analysis of the spacetime metric components is based on a framework recently developed by Lindblad and Rodnianski, which allows us to derive suitable estimates for tensorial systems of quasilinear wave equations with nonlinearities that satisfy the weak null condition. Our analysis of the electromagnetic fields, which satisfy quasilinear first-order equations that have a special null structure, is based on an extension of a geometric energy-method framework developed by Christodoulou together with a collection of pointwise decay estimates for the Faraday tensor developed in the article. We work directly with the electromagnetic fields and thus avoid the use of electromagnetic potentials.

Keywords
Born–Infeld, canonical stress, energy currents, global existence, Hardy inequality, Klainerman–Sobolev inequality, Lagrangian field theory, nonlinear electromagnetism, null condition, null decomposition, quasilinear wave equation, regularly hyperbolic, vector field method, weak null condition
Mathematical Subject Classification 2010
Primary: 35A01, 35Q76
Secondary: 35L99, 35Q60, 35Q76, 78A25, 83C22, 83C50
Milestones
Received: 29 November 2010
Revised: 18 September 2012
Accepted: 21 May 2013
Published: 27 August 2014
Authors
 Jared Speck Department of Mathematics Massachusetts Institute of Technology 77 Massachusetts Avenue, Room 2-163 Cambridge, MA 02139-4307 United States http://math.mit.edu/~jspeck/