  In this article, we study the coupling of the Einstein field equations of
general relativity to a family of models of nonlinear electromagnetic fields.
The family comprises all covariant electromagnetic models that satisfy the
following criteria: they are derivable from a sufficiently regular Lagrangian,
they reduce to the linear Maxwell model in the weak-field limit, and their
corresponding energy-momentum tensors satisfy the dominant energy condition.
Our main result is a proof of the global nonlinear stability of the 1 +
3-dimensional Minkowski spacetime solution to the coupled system for any member
of the family, which includes the linear 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, 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.
