Analytic Evidence for Continuous Self Similarity of the Critical Merger Solution

TL;DR

The paper investigates the critical merger point in the black-hole/black-string transition by focusing on the continuously self-similar double-cone geometry as the local near-pinch metric. It employs a gauge-fixed perturbation analysis to compute a zero-mode spectrum, showing two mode families with a single relevant (s_+) direction, which supports the double cone as a co-dimension-1 attractor for asymptotic perturbations. It also carries out a non-linear spherical analysis, revealing non-perturbative smoothed-cone solutions and a reduced 2d phase-space dynamics that captures the essential self-similar structure. A detailed phase-space analysis confirms the existence and stability of smoothed cones and clarifies the role of scaling symmetry in organizing the dynamics, thus strengthening CSS as the mechanism governing the critical merger.

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".

Figures (2)

• Figure 1: The merger metric. $r$ is the radial coordinate in the extended directions, $z$ is periodically compactified, while time and angular coordinates are suppressed. The heavy lines denote the horizon of a static black object which is at threshold between being a black-hole and being a black-string. The naked singularity is at the $\times$-shaped pinching (horizon crossing) point. Upon zooming onto the encircled singularity it is convenient to replace $(r,z)$ by radial coordinates $(\rho,\chi)$ radial coordinates. We shall be mostly interested in the "critical merger solution" -- the local metric near the singularity, namely the encircled portion of the metric (in the limit that the circle's size is infinitesimal).
• Figure 2: The phase space (for $m=n=2$). There is one interior equilibrium point at $(0,0)$ which represents the double cone. For $D<10$ it is focal repulsive but the spirals cannot be seen in the figure since their log-period is too large, while for $D\ge 10$ it is nodal repulsive, and hence the figure represents the whole range of dimensions. There are two finite equilibria on the boundary at $(1/m,\theta_1)$ and $(-1/n,\theta_2)$ which are saddle points. Each one has a critical curve which approaches it (heavy line). These two special trajectories denote the smoothed cones, where either ${\bf S}^n$ or ${\bf S}^m$ shrinks smoothly while the other one stays finite. Finally there are two attractive equilibria at infinity at $(1/m,-\infty)$ and $(-1/n,+\infty)$, and the thin lines trajectories represent generic trajectories which end at these infinite equilibrium points.