Scalar field critical collapse in 2+1 dimensions

TL;DR

This paper investigates the critical collapse of a scalar field in 2+1 dimensions with negative cosmological constant, proposing a new approximate solution and predicting critical exponents consistent with numerical results.

Contribution

The authors introduce a new theoretical approximation for the critical solution in scalar field collapse in 2+1 dimensions, matching numerical experiments and predicting critical exponents.

Findings

Approximate critical solution matches numerical data.

Predicted Ricci-scaling exponent γ=8/7.

Predicted mass-scaling exponent δ=16/23.

Abstract

We carry out numerical experiments in the critical collapse of a spherically symmetric massless scalar field in 2+1 spacetime dimensions in the presence of a negative cosmological constant and compare them against a new theoretical model. We approximate the true critical solution as the $n=4$ Garfinkle solution, matched at the lightcone to a Vaidya-like solution, and corrected to leading order for the effect of $\Lambda<0$. This approximation is only $C^3$ at the lightcone and has three growing modes. We {\em conjecture} that pointwise it is a good approximation to a yet unknown true critical solution that is analytic with only one growing mode (itself approximated by the top mode of our amended Garfinkle solution). With this conjecture, we predict a Ricci-scaling exponent of $\gamma=8/7$ and a mass-scaling exponent of $\delta=16/23$, compatible with our numerical experiments.