What happens at the horizon(s) of an extreme black hole?

Abstract

A massless scalar field exhibits an instability at the event horizon of an extreme black hole. We study numerically the nonlinear evolution of this instability for spherically symmetric perturbations of an extreme Reissner-Nordstrom (RN) black hole. We find that generically the endpoint of the instability is a non-extreme RN solution. However, there exist fine-tuned initial perturbations for which the instability never decays. In this case, the perturbed spacetime describes a time-dependent extreme black hole. Such solutions settle down to extreme RN outside, but not on, the event horizon. The event horizon remains smooth but certain observers who cross it at late time experience large gradients there. Our results indicate that these dynamical extreme black holes admit a C 1 extension across an inner (Cauchy) horizon.

Lux in obscuro: photon orbits of extremal black holes revisited

“…However, it turns out that an exactly extremal black hole behaves quite differently from a near-extremal black hole. For example, an (exactly) extremal black hole suffers from the -classical -Aretakis instability but near-extremal ones do not [23][24][25][26][27][28][29]. Furthermore, the procedure of taking the extremal limit is quite subtle [30][31][32].…”

“…In fact, the instability of the photon orbit for Kerr black holes outside their horizon was a crucial ingredient used in the proof that slowly rotating Kerr black holes are stable when considering linear wave equation on the background [80]. The fact that static extremal black holes have stable photon orbits, on the contrary, implies that these black holes are unstable (even without considering the Aretakis instability [23][24][25][26][27][28][29]). The question is then: what exactly happens to the geometry due to this backreaction?…”

Linear Waves in the Interior of Extremal Black Holes I

“…They found that for fine-tuned initial data the corresponding future developments are black hole spacetimes containing no (marginally) trapped surfaces of symmetry that approach extremal Reissner-Nordström along the event horizon. 5 Moreover, the numerics in [42] suggest that the interior region of these spacetimes has a non-empty Cauchy horizon across which the metric is extendible in C 0 with Christoffel symbols in L 2 loc at the Cauchy horizon. Additionally, both the scalar field φ and all its first-order derivatives remain bounded at the Cauchy horizon.…”

Boundedness of Massless Scalar Waves on Reissner-Nordström Interior Backgrounds

“…(Furthermore, note that the quantity M − e 2 r is always positive in R.) As defined in (35) the r coordinate in the noshift region N ranges between r red defined by (36) and r blue , defined below, strictly bigger than r − . In N we exploit the fact that J −∂ r and K −∂ r are invariant under translations along ∂ t .…”