On Perturbation Theory and Critical Exponents for Self-Similar Systems

TL;DR

This paper investigates gravitational critical collapse in the Einstein-axion-dilaton system, computing perturbations and critical exponents for new self-similar solutions, revealing dependence on spacetime dimensions and initial ansatz.

Contribution

It extends the analysis of critical exponents by computing perturbations for parabolic and hyperbolic self-similar solutions in the Einstein-axion-dilaton system, highlighting their dependence on initial conditions.

Findings

Computed linear perturbation equations for parabolic solutions.

Solved perturbation equations for hyperbolic solutions to find the Choptuik exponent.

Found that the critical exponent depends on spacetime dimensions and initial ansatz.

Abstract

Gravitational critical collapse in the Einstein-axion-dilaton system is known to lead to continuous self-similar solutions characterized by the Choptuik critical exponent $\gamma$. We complete the existing literature on the subject by computing the linear perturbation equations in the case where the axion-dilaton system assumes a parabolic form. Next, we solve the perturbation equations in a newly discovered self-similar solution in the hyperbolic case, which allows us to extract the Choptuik exponent. Our main result is that this exponent depends not only on the dimensions of spacetime but also the particular ansatz and the critical solutions that one started with.