Critical Collapse of an Ultrarelativistic Fluid in the $\Gamma\to 1$ Limit

TL;DR

This study explores the critical gravitational collapse of an ultrarelativistic fluid as the equation of state approaches the limit of a cosmological constant, combining analytical solutions with high-precision numerical simulations to examine naked singularity formation.

Contribution

It provides the first detailed calculation of the CSS solution and scaling exponent in the $ o 1$ limit, and compares these with comprehensive numerical simulations across a range of parameters.

Findings

Numerical results agree well with CSS solutions.

Scaling exponents are calculated with high precision.

Evidence supports naked singularity formation in supercritical collapse.

Abstract

In this paper we investigate the critical collapse of an ultrarelativistic perfect fluid with the equation of state $P=(\Gamma-1)\rho$ in the limit of $\Gamma\to 1$. We calculate the limiting continuously self similar (CSS) solution and the limiting scaling exponent by exploiting self-similarity of the solution. We also solve the complete set of equations governing the gravitational collapse numerically for $(\Gamma-1) = 10^{-2},...,10^{-6}$ and compare them with the CSS solutions. We also investigate the supercritical regime and discuss the hypothesis of naked singularity formation in a generic gravitational collapse. The numerical calculations make use of advanced methods such as high resolution shock capturing evolution scheme for the matter evolution, adaptive mesh refinement, and quadruple precision arithmetic. The treatment of vacuum is also non standard. We were able to tune the critical parameter up to 30 significant digits and to calculate the scaling exponents accurately. The numerical results agree very well with those calculated using the CSS ansatz. The analysis of the collapse in the supercritical regime supports the hypothesis of the existence of naked singularities formed during a generic gravitational collapse.