[Paper Review] Analytical approach to critical collapse in 2+1 dimensions

Gérard Clément, Alessandro Fabbri|ArXiv.org|Dec 11, 2001

Advancements in Semiconductor Devices and Circuit Design1 references3 citations

TL;DR

This paper presents a family of continuously self-similar (CSS) solutions in 2+1 dimensional gravity with a negative cosmological constant and a massless scalar field, extending known $λ=0$ solutions to $λ<0$. By analyzing linear perturbations and imposing physical boundary conditions, it identifies two growing modes: one corresponding to static singular solutions (including BTZ black holes), and another describing genuine black hole formation with a critical exponent $\gamma = 2/5$ for $c^2 = 1$, consistent with numerical simulations.

ABSTRACT

We present a family of time-dependent solutions to 2+1 gravity with negative cosmological constant and a massless scalar field as source. These solutions are continuously self-similar near the central singularity. We analyze linear perturbations of these solutions, and discuss the subtle question of boundary conditions. We find two growing modes, one of which corresponds to the linearization of static singular solutions, while the other describes black hole formation.

Motivation & Objective

• To construct exact, continuously self-similar (CSS) solutions in 2+1 dimensions with $\Lambda<0$ and a massless scalar field, extending known $\Lambda=0$ solutions.
• To analyze linear perturbations of these threshold solutions to determine whether they describe black hole formation.
• To resolve the subtle issue of boundary conditions for perturbations in asymptotically AdS spacetimes.
• To compute the critical exponent $\gamma$ for black hole formation and compare it with numerical results.

Proposed method

• Derive a new class of $\Lambda=0$ CSS solutions via an infinite boost transformation of Garfinkle's original solutions.
• Construct quasi-CSS solutions for $\Lambda<0$ using a double-null ansatz with $ds^2 = e^{2\nu(x)}dudv - (-u)^{2\alpha}\rho^2(x)d\theta^2$, $\phi = -c\alpha\ln(-u) + \psi(x)$, $x=uv$.
• Solve the resulting system of ODEs for $\rho(x)$, $\nu(x)$, $\psi(x)$ under boundary conditions $\rho(0)=1$, $\nu(0)=0$, $\psi(0)=0$, ensuring asymptotic AdS behavior.
• Perform linear perturbation analysis around the threshold solution, introducing a perturbation parameter $\lambda$ and expanding to first order in $\lambda$.
• Identify two distinct growing modes: one with a static apparent horizon (linearized BTZ), and one with a spacelike singularity and a dynamically forming horizon.
• Impose regularity at the apparent horizon's birth to derive an eigenvalue condition, selecting $c^2 \simeq 1$ as the unique physical solution.

Experimental results

Research questions

• RQ1 Can exact, continuously self-similar solutions with $\Lambda<0$ be constructed that asymptote to AdS spacetime and describe critical collapse in 2+1 dimensions?
• RQ2 Do linear perturbations of these solutions exhibit modes corresponding to black hole formation, and if so, under what conditions?
• RQ3 What is the critical exponent $\gamma$ for black hole formation in this model, and how does it compare to numerical simulations?
• RQ4 How do boundary conditions at spatial infinity and at the apparent horizon constrain the physical validity of perturbation modes?
• RQ5 Is the mode describing black hole formation uniquely selected by regularity conditions at the horizon's formation time?

Key findings

• A one-parameter family of exact, continuously self-similar solutions for $\Lambda<0$ is constructed, with $c^2$ as the key parameter, that are asymptotically AdS and reduce to known $\Lambda=0$ solutions near the singularity.
• Two growing modes are found in the linear perturbation analysis: one corresponds to the linearization of static singular solutions (including BTZ black holes), and the other describes black hole formation with a spacelike singularity and a dynamically forming apparent horizon.
• The condition of regularity at the apparent horizon's birth leads to a unique solution with $c^2 \simeq 1$, which selects a specific physical mode for black hole formation.
• For this mode, the critical exponent is computed as $\gamma = 2/5$, which is of the same order of magnitude as the values $\sim 0.8$ and $\sim 1.2$ reported in numerical simulations.
• The critical exponent for the static BTZ black hole (corresponding to $c^2 = 0$) is found to be $\gamma_{(BTZ)} = 1/2$, consistent with previous analyses.
• The perturbation mode describing black hole formation is conjectured to arise from a nonlinear extension of non-topological soliton singular solutions, with a scaling behavior matching $\gamma = 1/k$ where $k = c^2 + 2$.