Critical Phenomena in the Einstein-Massless-Dirac System

TL;DR

This paper explores the critical phenomena in gravitational collapse of massless Dirac fields, revealing a Type II critical solution with specific self-similarity properties and deriving an analytic solution consistent with numerical findings.

Contribution

It introduces a new model of massless Dirac fields in gravitational collapse and identifies a critical solution with unique self-similarity features, supported by both numerical and analytical methods.

Findings

Identification of a Type II critical solution with a mass scaling exponent ~0.26.

Discovery of a continuously self-similar geometry with matter fields exhibiting discrete self-similarity.

Derivation of an analytic solution to the reduced equations matching numerical critical solutions.

Abstract

We investigate the general relativistic collapse of spherically symmetric, massless spin-1/2 fields at the threshold of black hole formation. A spherically symmetric system is constructed from two spin-1/2 fields by forming a spin singlet with no net spin-angular momentum. We study the system numerically and find strong evidence for a Type II critical solution at the threshold between dispersal and black hole formation, with an associated mass scaling exponent $\gamma ~ 0.26$. Although the critical solution is characterized by a continuously self-similar (CSS) geometry, the matter fields exhibit discrete self-similarity with an echoing exponent $\Delta ~ 1.34$. We then adopt a CSS ansatz and reduce the equations of motion to a set of ODEs. We find a solution of the ODEs that is analytic throughout the solution domain, and show that it corresponds to the critical solution found via dynamical evolutions.