On the Global Uniqueness for the Einstein–Maxwell-Scalar Field System with a Cosmological Constant

Abstract

This paper is the second part of a trilogy dedicated to the following problem: given spherically symmetric characteristic initial data for the Einstein–Maxwell-scalar field system with a cosmological constant ${\Lambda}$, with the data on the outgoing initial null hypersurface given by a subextremal Reissner–Nordström black hole event horizon, study the future extendibility of the corresponding maximal globally hyperbolic development as a “suitably regular” Lorentzian manifold. In the first paper of this sequence (Costa et al., Class Quantum Gravity 32:015017, 2015), we established well posedness of the characteristic problem with general initial data. In this second paper, we generalize the results of Dafermos (Ann Math 158:875–928, 2003) on the stability of the radius function at the Cauchy horizon by including a cosmological constant. This requires a considerable deviation from the strategy followed in Dafermos (Ann Math 158:875–928, 2003), focusing on the level sets of the radius function instead of the red-shift and blue-shift regions. We also present new results on the global structure of the solution when the free data is not identically zero in a neighborhood of the origin. In the third and final paper (Costa et al., On the global uniqueness for the Einstein–Maxwell-scalar field system with a cosmological constant. Part 3. Mass inflation and extendibility of the solutions. arXiv:1406.7261, 2015), we will consider the issue of mass inflation and extendibility of solutions beyond the Cauchy horizon.