Analytic Evidence for Continuous Self Similarity of the Critical Merger Solution

TL;DR

This paper provides analytic evidence that the double cone solution exhibits continuous self-similarity and acts as a critical attractor in black-hole/black-string phase transitions, supported by detailed spectral and non-linear dynamical analyses.

Contribution

It offers the first detailed spectral analysis of the double cone's zero modes and a non-linear dynamical system study confirming its role as a critical merger solution.

Findings

Double cone is a co-dimension 1 attractor for smooth perturbations.

Spectral analysis of zero modes supports the criticality of the double cone.

Non-linear analysis confirms the existence of a smoothed cone solution.

Abstract

The double cone, a cone over a product of a pair of spheres, is known to play a role in the black-hole black-string phase diagram, and like all cones it is continuously self similar (CSS). Its zero modes spectrum (in a certain sector) is determined in detail, and it implies that the double cone is a co-dimension 1 attractor in the space of those perturbations which are smooth at the tip. This is interpreted as strong evidence for the double cone being the critical merger solution. For the non-symmetry-breaking perturbations we proceed to perform a fully non-linear analysis of the dynamical system. The scaling symmetry is used to reduce the dynamical system from a 3d phase space to 2d, and obtain the qualitative form of the phase space, including a non-perturbative confirmation of the existence of the "smoothed cone".