Stability of Schwarzschild-AdS for the spherically symmetric Einstein-Klein-Gordon system
Gustav Holzegel, Jacques Smulevici
TL;DR
The paper analyzes the global dynamics of the spherically symmetric Einstein-Klein-Gordon system with negative cosmological constant and establishes that Schwarzschild-AdS is both orbitally and asymptotically stable under small perturbations with Dirichlet boundary conditions. The authors deploy a bootstrap on the exterior region, leverage the redshift effect near the horizon, and utilize Hardy-type inequalities to overcome the non-monotonic Hawking mass, obtaining integrated decay for the scalar field and exponential decay of energies in the AdS setting. A vector-field multiplier framework centered on the Kodama structure yields degenerate and nondegenerate decay estimates, which are then extended to higher-order bounds via commutation with the T-field $T[\phi]$. The results confirm a rigorous nonlinear stability paradigm for Schwarzschild-AdS in spherical symmetry and offer a detailed methodology that may guide future studies of AdS stability and hair in related models.
Abstract
In this paper, we study the global behavior of solutions to the spherically symmetric coupled Einstein-Klein-Gordon (EKG) system in the presence of a negative cosmological constant. We prove that the Schwarzschild-AdS spacetimes (the trivial black hole solutions of the EKG system for which $φ=0$ identically) are asymptotically stable: Small perturbations of Schwarzschild-AdS initial data again lead to regular black holes, with the metric on the black hole exterior approaching a Schwarzschild-AdS spacetime. The main difficulties in the proof arise from the lack of monotonicity for the Hawking mass and the asymptotically AdS boundary conditions, which render even (part of) the orbital stability intricate. These issues are resolved in a bootstrap argument on the black hole exterior, with the redshift effect and weighted Hardy inequalities playing the fundamental role in the analysis. Both integrated decay and pointwise decay estimates are obtained.