Nonlinear stability of Schwarzschild black holes for the Einstein-massless Dirac system in spherical symmetry

Abstract

In this paper, we establish the nonlinear stability of the Schwarzschild exterior for the spherically symmetric Einstein-massless Dirac system. More precisely, we consider sufficiently small perturbations of Schwarzschild characteristic initial data and prove that the corresponding maximal development remains globally controlled in the exterior region (orbital stability), and converges, in a suitable teleological double-null gauge, to a Schwarzschild spacetime with a final mass (asymptotic stability). In particular, the maximal future development contains a complete future null infinity and a regular event horizon, with a nonempty black hole region in its interior. The main difficulties in the proof arise from the delicate coupling between the Dirac operator and the geometry, compared with the scalar wave counterpart, as well as from the failure of the Dirac stress-energy tensor to satisfy a positive energy condition. To overcome these difficulties, we rely on a bootstrap argument coupling the geometry and the Dirac field. Working in suitable teleological double-null gauges adapted to the asymptotic null boundaries, we establish weighted energy estimates combining integrated local energy decay, red-shift estimates near the event horizon, and an rp-estimates towards null infinity. A core ingredient is the use of suitable reweightings of the spinor components, which allow us to close these estimates despite the nonlinear coupling to the geometry.