Characteristic Critical Collapse of a Yang-Mills Field With Null Infinity

TL;DR

This work demonstrates that a purely magnetic SU(2) Yang–Mills field undergoing gravitational collapse exhibits discrete self-similarity near the threshold of black-hole formation, with an echoing period $Δ\approx0.7388$ and a universal critical exponent $γ\approx0.198$ governing mass scaling. Using a characteristic evolution in compactified Bondi coordinates, the authors access global quantities at future null infinity, showing that the Bondi mass and the news function inherit the same DSS structure as the local fields. The study confirms universality across initial data families and provides robust measurements of radiative quantities (Bondi mass decay and news) radiating the critical dynamics to $\mathscr{I}^+$, along with a precise quantification of black-hole formation and curvature scaling. Overall, the approach yields a computationally efficient, high-precision validation of critical phenomena in YM collapse with direct connection to observables at null infinity.

Abstract

Solutions to the Einstein equations near the threshold of black hole formation exhibit remarkable behavior known as critical phenomena gravitational collapse. In this work we perform characteristic evolution in compactified Bondi coordinates in order to study the critical collapse of a Yang-Mills field, allowing for the extraction of global quantities such as the Bondi mass and news function. Our numerical approach is fourth-order accurate. First, we demonstrate that the collapsing field exhibits local DSS behavior, characterized by an echoing period of~$Δ\simeq 0.7388$, agreeing with previous works up to the second decimal place. We find that global quantities such as the Bondi mass and news function display the same DSS behavior. We furthermore show that the mass of the black holes formed during near-threshold evolutions scales as a function of the distance to the critical parameter, with a critical exponent of approximately~$γ=0.1977\pm0.0009$. Finally, our findings indicate that these results are universal, irrespective of the initial data.

Figures (11)

• Figure 1: Convergence factor $Q(u)$ throughout the evolution, using a code that is fourth-order accurate. The dotted and solid lines correspond to a grid with $100$ and $200$ points at the lower resolution, respectively. We see that as we increase resolution, $Q$ gets closer to 2. The initial data is constructed using Equation \ref{['eq:initial-dataym']}, and setting $A=0.01$, $r_0=0.3$ and $\sigma=0.08$. We stop the convergence test when the magnitude of all fields is smaller than $10^{-7}$. We use a lower resolution here than in our near-critical runs, but as we increase resolution and data strength, the solution converges in a similar manner.
• Figure 2: $\xi=W-1$ throughout the evolution. The top panel shows a subcritical evolution with initial amplitude $\mathrm{A}=0.001$ and the lower a supercritical evolution with $\mathrm{A}=0.089$ for $N=400$. For clarity only a subset of the data points are shown.
• Figure 3: $\Tilde{\chi}$ at the origin as a function of proper time $u$ and similarity time $T$ for a run of N=$6\cdot10^3$, tuned to $9$ decimal places, with $\mathrm{A}_*=0.088640995$.
• Figure 4: News function at $\mathscr{I^+}$ as a function of Bondi time $u$ and similarity time $T$ for a run of N=$6\cdot10^3$, tuned to 9 decimal places, with $\mathrm{A}=0.088640995$.
• Figure 5: Redshift factor $H(T)=\beta(u_c,\infty$ as a function of Bondi time $u$ and similarity time $T$ for a run of N=$6\cdot10^3$, tuned to 9 decimal places, with $\mathrm{A}_*=0.088640995$.