Self-Similar Gravitational Dynamics, Singularities and Criticality in 2D

TL;DR

This paper explores self-similar gravitational dynamics in two dimensions, revealing how such models can produce singularities, relate to string theory and Liouville gravity, and exhibit critical phenomena similar to Choptuik scaling.

Contribution

It systematically studies CSS gravitational models in 2D, deriving their form, analyzing static solutions, and investigating matter collapse and singularities with analytical and numerical methods.

Findings

Self-similar spacetimes often lead to singularities.

Static solutions exhibit uncommon asymptotic behaviors.

Hints of Choptuik-like scaling laws in matter collapse.

Abstract

We initiate a systematic study of continuously self-similar (CSS) gravitational dynamics in two dimensions, motivated by critical phenomena observed in higher dimensional gravitational theories. We consider CSS spacetimes admitting a homothetic Killing vector (HKV) field. For a general two-dimensional gravitational theory coupled to a dilaton field and Maxwell field, we find that the assumption of continuous self-similarity determines the form of the dilaton coupling to the curvature. Certain limits produce two important classes of models, one of which is closely related to two-dimensional target space string theory and the other being Liouville gravity. The gauge field is shown to produce a shift in the dilaton potential strength. We consider static black hole solutions and find spacetimes with uncommon asymptotic behaviour. We show the vacuum self-similar spacetimes to be special limits of the static solutions. We add matter fields consistent with self-similarity (including a certain model of semi-classical gravity) and write down the autonomous ordinary differential equations governing the gravitational dynamics. Based on the phenomenon of finite-time blow-up in ODEs, we argue that spacetime singularities are generic in our models. We present qualitatively diverse results from analytical and numerical investigations regarding matter field collapse and singularities. We find interesting hints of a Choptuik-like scaling law.