Analysis & PDE
- Anal. PDE
- Volume 7, Number 4 (2014), 771-901.
The global stability of the Minkowski spacetime solution to the Einstein-nonlinear system in wave coordinates
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 -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.
Anal. PDE, Volume 7, Number 4 (2014), 771-901.
Received: 29 November 2010
Revised: 18 September 2012
Accepted: 21 May 2013
First available in Project Euclid: 20 December 2017
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
Primary: 35A01: Existence problems: global existence, local existence, non-existence 35Q76: Einstein equations
Secondary: 35L99: None of the above, but in this section 35Q60: PDEs in connection with optics and electromagnetic theory 35Q76: Einstein equations 78A25: Electromagnetic theory, general 83C22: Einstein-Maxwell equations 83C50: Electromagnetic fields
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
Speck, Jared. The global stability of the Minkowski spacetime solution to the Einstein-nonlinear system in wave coordinates. Anal. PDE 7 (2014), no. 4, 771--901. doi:10.2140/apde.2014.7.771. https://projecteuclid.org/euclid.apde/1513731541