[Paper Review] Self-Similar Solutions, Critical Behavior and Convergence to Attractor in Gravitational Collapse

Tomohiro Harada|ArXiv.org|Feb 3, 2003

Earth Systems and Cosmic Evolution3 citations

TL;DR

This paper establishes that self-similar solutions in spherically symmetric gravitational collapse—particularly critical and attractor solutions—unify the understanding of two key phenomena: critical behavior near black hole threshold formation and generic convergence to an attractor in non-fine-tuned collapse. Using stability analysis of self-similar solutions in both general relativity and Newtonian gravity, the authors show that the critical solution is unstable (saddle-type), while the attractor solution is stable, explaining the observed scaling laws and universal dynamics in collapse simulations.

ABSTRACT

General relativity as well as Newtonian gravity admits self-similar solutions due to its scale-invariance. This is a review on these self-similar solutions and their relevance to gravitational collapse. In particular, our attention is mainly paid on the crucial role of self-similar solutions in the critical behavior and attraction in gravitational collapse.

Motivation & Objective

• To unify the understanding of critical behavior and attractor convergence in gravitational collapse through the lens of self-similar solutions.
• To investigate the role of self-similarity in simplifying the dynamics of inhomogeneous, spherically symmetric gravitational collapse.
• To compare the stability properties of self-similar solutions in general relativity and Newtonian gravity to explain observed numerical behaviors.
• To clarify the distinction between critical solutions (unstable, requiring fine-tuning) and attractor solutions (stable, generic) in collapse dynamics.
• To validate the self-similar hypothesis in gravitational collapse by demonstrating its predictive power for scaling laws and asymptotic behavior.

Proposed method

• Employing continuous self-similarity via a homothetic vector field satisfying the homothetic Killing equation $\mathcal{L}_{\xi}g_{\mu\nu} = 2g_{\mu\nu}$, reducing the Einstein equations to ordinary differential equations.
• Analyzing spherically symmetric perfect fluid spacetimes with self-similar ansatz, where metric functions and density depend only on $\xi = t/r$, ensuring scale invariance.
• Applying linear stability analysis to self-similar solutions, particularly focusing on the kink mode, to determine stability against perturbations.
• Using eigenvalue analysis to compute critical exponents and compare them with known values in general relativity and Newtonian gravity.
• Performing numerical simulations of the spherically symmetric, isothermal gas system governed by PDEs to observe convergence to attractor and critical behavior.
• Comparing results across Newtonian and general relativistic frameworks to test universality of self-similar dynamics in gravitational collapse.

Experimental results

Research questions

• RQ1 How do self-similar solutions explain the critical behavior observed in gravitational collapse near the black hole threshold?
• RQ2 What distinguishes the stability properties of critical solutions from attractor solutions in spherically symmetric gravitational collapse?
• RQ3 To what extent do self-similar solutions serve as asymptotic attractors or critical points in the phase space of gravitational collapse?
• RQ4 How do the scaling laws for black hole mass and core density in near-critical collapse emerge from the stability of self-similar solutions?
• RQ5 Can the same self-similar framework explain both critical behavior and generic convergence in both Newtonian and general relativistic collapse?

Key findings

• The Hunter (a) solution in Newtonian gravity is identified as the critical solution due to its single unstable mode, with a critical exponent of approximately 0.10567.
• The Larson-Penston solution is stable and acts as an attractor in Newtonian collapse, explaining the generic convergence of isothermal gas collapse to this self-similar profile.
• Numerical simulations confirm a scaling law for the mass of the collapsed core in supercritical collapse with a critical exponent of ≈0.11, consistent with the expected value of ≈0.10567.
• For subcritical collapse, the maximum central density scales with a critical exponent of ≈-0.22, matching the theoretical prediction of ≈-0.21134.
• The Evans-Coleman solution in general relativity is identified as the relativistic counterpart of the Newtonian Hunter (a) solution, sharing similar dynamical features such as central contraction and surrounding expansion.
• The stability analysis via the kink mode confirms that the Hunter (a) solution is unstable (critical), while the Larson-Penston solution is stable (attractor), providing a unified explanation for both critical and generic collapse behaviors.