Choptuik Scaling and The Merger Transition

TL;DR

This work establishes a precise link between Choptuik critical collapse and the merger transition of black strings through double analytic continuation and boundary-condition mapping, showing that the two systems share the same action after dimensional reduction. It identifies a single complex exponent framework that unifies the Choptuik scaling constants γ and Δ, and predicts a critical dimension D* = 10 for the merger, below which off-critical oscillations (GHP-like) arise and above which they vanish. The analysis provides explicit expressions for γ across dimensions and forecasts merger-specific oscillatory behavior that can be tested numerically, while suggesting broader connections to conformal/large-N dynamics and holography. Overall, the paper offers a unifying perspective on two seemingly distinct gravitational critical phenomena and proposes concrete avenues for cross-system verification and deeper theoretical interpretation.

Abstract

The critical solution in Choptuik scaling is shown to be closely related to the critical solution in the black-string black-hole transition (the merger), through double analytic continuation, and a change of a boundary condition. The interest in studying various space-time dimensions D for both systems is stressed. Gundlach-Hod-Piran off-critical oscillations, familiar in the Choptuik set-up, are predicted for the merger system and are predicted to disappear above a critical dimension D*=10. The scaling constants, Delta(D), gamma(D), are shown to combine naturally to a single complex number.

Figures (4)

• Figure 1: An illustration of a continuously self-similar geometry (CSS), which we often call a cone, and a discretely self-similar geometry (DSS).
• Figure 2: Definition of coordinates for the merger system. For backgrounds with a single compact dimension the essential geometry is 2d and Euclidean after suppressing the time $t$ and angular coordinates in the extended dimensions. The cylindrical coordinates $(r,z)$ are defined such that $z \sim z+L$ is the coordinate along the compact dimension and $r$ is the radial coordinate in the extended spatial directions. Locally at the "pinching singularity" we define another set of local coordinates $(\rho,\chi)$ (defined only for $\rho \le L/2$), which are radial coordinates in the 2d plane with origin at the singularity. We shall sometimes call $\rho$ a "scaling coordinate" and $\chi$ "tangential".
• Figure 3: The merger transition. A black string (left) turns into a black hole (right) as the waist pinches. Shaded regions are inside the horizon and the dashed line is a boundary far away. The singular configuration is a cone over ${\bf S}^2 \times {\bf S}^{D-3}$ -- the double-cone.
• Figure 4: Definition of coordinates for spherical collapse system and Choptuik scaling (based on Gundlach96GarciaGundlachGlobal). The essential coordinates $(\tilde{r},\tilde{t})$ parametrize a 2d Lorentzian plane after suppressing the angular coordinates. The scaling direction is along lines of fixed $\tilde{t}/\tilde{r}$ and may be parametrized by ${\tilde{\rho}}^2:=\tilde{r}^2-\tilde{t}^2$. $\tilde{z}$ parametrizes an additional dimension, the dimensional uplift of the scalar field $\Phi$. The domain is made out of three patches: the past patch, bounded by the $\tilde{r}=0$ axis and the past horizon, the outer patch bounded by the past and future horizons and the future patch bounded by the future (Cauchy) horizon and the axis. The critical solutions is periodic on smaller and smaller scales as the singularity is approached. One period is denoted by the line-filled (blue) region and a second one is denoted by a shaded (green) regions. The pattern continues towards the singularity.