On the Horizon Instability of Extremal Black Holes and Black Branes

As an important and challenging problem, the study of black hole stability has attracted considerable interest over the last 60 years. With many stability results appearing over the years, it came as a surprise when Aretakis discovered a peculiar horizon instability of extremal black holes. He proved that while perturbations on an extremal black hole background themselves decay, sufficiently high-order transverse derivatives of the fields grow at late times precisely on the horizon. In this thesis, we have advanced our understanding of this instability. General proofs of the instability rely on the extremal horizon having compact topology. We find the first example of a system having a planar (non-compact) horizon topology that exhibits this instability---the planar Reissner-Nordstrӧm black hole in five spacetime dimensions that is asymptotically Anti-de-Sitter (AdS). Asymptotically AdS black holes are particularly interesting in light of the AdS/CFT correspondence (holography) where these systems can be mapped to strongly coupled field theories that ‘live’ on the boundary of these spacetimes. We find that the horizon instability is crucial in preserving temporal conformal symmetry of correlators in the corresponding strongly coupled field theory. To obtain a glimpse of a unifying picture of the horizon instability of extremal black holes, we investigate the instability in the simplest possible setting---the BTZ black hole. We show that the instability is associated with a class of null geodesics that orbit near the event horizon arbitrarily many times before falling in. This provides a physical interpretation of perturbations being trapped near the event horizon for arbitrarily long periods, causing the horizon to be unstable. Prior work has associated the appearance of the horizon instability to an AdS 2-factored near horizon geometry, a feature common to all extremal black holes. However, extremal black branes that arise in string theory have different near-horizon geometries. We study the extremal D3 brane, which has an AdS 5-factored near horizon geometry. We find evidence that such an instability does not arise in this geometry, which bolsters the idea that an AdS 2-factored near horizon geometry is crucial for the horizon instability to occur