Discrete Self-Similiarity and Critical Point Behavior in Fluctuations About Extremal Black Holes

TL;DR

This paper investigates the scaling symmetry and critical phenomena in fluctuations around extremal charged black holes, revealing self-similar solutions and long-range influences unique to the extremal case.

Contribution

It demonstrates the existence of discrete self-similar solutions for charged fields near extremal black holes and analyzes the impact of scaling symmetry on correlation lengths and black hole area variations.

Findings

Existence of a one-parameter family of discretely self-similar solutions.

Long-range influence of external sources on extremal black holes.

Black hole area variations scale as the square root of the deviation from extremality.

Abstract

The issues of scaling symmetry and critical point behavior are studied for fluctuations about extremal charged black holes. We consider the scattering and capture of the spherically symmetric mode of a charged, massive test field on the background spacetime of a black hole with charge $Q$ and mass $M$. The spacetime geometry near the horizon of a $|Q|=M$ black hole has a scaling symmetry, which is absent if $|Q|<M$, a scale being introduced by the surface gravity. We show that this symmetry leads to the existence of a self-similiar solution for the charged field near the horizon, and further, that there is a one parameter family of discretely self-similiar solutions . The scaling symmetry, or lack thereof, also shows up in correlation length scales, defined in terms of the rate at which the influence of an external source coupled to the field dies off. It is shown by constructing the Greens functions, that an external source has a long range influence on the extremal background, compared to a correlation length scale which falls off exponentially fast in the $|Q|<M$ case. Finally it is shown that in the limit of $\Delta \equiv (1-{Q^2 \overM^2} )^{1\over 2} \rightarrow 0$ in the background spacetime, that infinitesimal changes in the black hole area vary like $\Delta ^{1\over 2}$.