Domain wall interacting with a black hole: A new example of critical phenomena

TL;DR

The paper analyzes a stationary Dirac-Nambu-Goto membrane in static black-hole spacetimes to reveal a second-order phase transition between Minkowski- and black-hole-topology membranes. By focusing on near-horizon (Rindler) dynamics and exploiting symmetries, it derives exact mass-scaling laws with universal exponents $\gamma = \frac{2}{3}$ and, in Schwarzschild backgrounds, a wiggle period $\omega = \frac{3\pi}{\sqrt{7}}$, including explicit expressions for the wiggles. The results show that critical phenomena characteristic of gravitational collapse—universality, self-similarity, and sub-/super-critical symmetry—arise in this simple, analytically tractable system, suggesting a broader generality of black hole formation as a critical phenomenon beyond dynamical GR. The work provides analytic footing for features previously observed numerically and highlights the central role of near-horizon geometry in governing critical behavior.

Abstract

We study a simple system that comprises all main features of critical gravitational collapse, originally discovered by Choptuik and discussed in many subsequent publications. These features include universality of phenomena, mass-scaling relations, self-similarity, symmetry between super-critical and sub-critical solutions, etc. The system we consider is a stationary membrane (representing a domain wall) in a static gravitational field of a black hole. For a membrane that spreads to infinity, the induced 2+1 geometry is asymptotically flat. Besides solutions with Minkowski topology there exists also solutions with the induced metric and topology of a 2+1 dimensional black hole. By changing boundary conditions at infinity, one finds that there is a transition between these two families. This transition is critical and it possesses all the above-mentioned properties of critical gravitational collapse. It is remarkable that characteristics of this transition can be obtained analytically. In particular, we find exact analytical expressions for scaling exponents and wiggle-periods. Our results imply that black hole formation as a critical phenomenon is far more general than one might expect.

Figures (5)

• Figure 1: Stationary and axially symmetric cosmic membranes in Rindler space fall into two families: Those of 2+1 dimensional Minkowski spacetime topology and those of 2+1 dimensional black hole spacetime topology. The limiting membrane $R=Z$ has a curvature singularity at $(0,0)$. To obtain the full spatial structure of the membranes, the curves must be rotated around the $Z$-axis.
• Figure 2: The Minkowski topology and black hole topology solutions oscillate around the critical solution with a fixed period in a logarithmic plot. Notice the symmetry relating the two topologies.
• Figure 3: Conformal magnification of the parameter space near the critical solution. This illustrates how the membranes of the two topologies approach the critical solution along logarithmic spiral arms.
• Figure 4: Phase-portrait of the system of equations (\ref{['auto']}). It has a saddle point at $(0,1/2)$ and a focus point at $(1,1)$. The curve connecting these two points represents all the black hole topology membranes.
• Figure 5: A simple fit of the non-inverted mass-scaling relation (\ref{['fitscale']}) for the black hole topology membranes in the Schwarzschild background indicates that the constant $C_2$ is indeed quite large. Here the values $C_1 \approx 0.0188$, $C_2=0.858$ and $\varphi=4.69$ have been used.