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.
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
