The global nonlinear stability of Minkowski spacetime with self-gravitating massive Dirac fields

TL;DR

This paper proves the global nonlinear stability of Minkowski spacetime when coupled with self-gravitating massive Dirac fields, extending previous massless results and introducing gauge-invariant methods for spinor fields.

Contribution

It establishes the first gauge-invariant nonlinear stability result for Einstein-Dirac systems with massive fields, using a novel hyperboloidal-Euclidean framework and new Sobolev inequalities.

Findings

Proved global existence and stability of solutions close to Minkowski spacetime.

Developed gauge-invariant estimates for spinor fields in a hyperboloidal foliation.

Established a hierarchy of estimates for the coupled Einstein-Dirac system.

Abstract

We consider the Einstein-Dirac system for a massive field, which describes the evolution of self-gravitating massive spinor fields, and we investigate the global evolution problem, when the initial data set is sufficiently close to data describing a spacelike, asymptotically Euclidean slice of the Minkowski spacetime. We establish the gauge-invariant nonlinear stability of such fields, namely the existence of a globally hyperbolic development, which remains asymptotic to Minkowski spacetime in future timelike, null, and spacelike directions. Previous results on this problem have been limited to the Einstein-Dirac system in the massless case. Our analysis follows the asymptotically hyperboloidal-Euclidean framework introduced by LeFloch and Y. Ma for the massive Klein-Gordon-Einstein system. The structure specific to spinor fields and the Dirac equation necessitates significantly new elements in the proof. In contrast with prior approaches, our treatment of spinor fields and the Dirac equation is gauge-invariant, relying on the formalism of Lorentz Clifford algebras, principal fiber bundles, etc. Our analysis is carried out with the metric expressed in light-bending wave coordinates, as we call them. This leads us to the study of a global existence problem for a system of wave equations with constraints and a Klein-Gordon-type equations. We derive L2 estimates for the Dirac equation and its coupling with the Einstein equations, along with $pointwise estimates. New Sobolev inequalities are proven for spinor fields in a gauge-invariant manner in the hyperboloidal-Euclidean foliation. The nonlinear coupling between the massive Dirac equation and the Einstein equations is investigated, and we establish a hierarchy of estimates, which distinguish between translations, rotations, and boosts.