Black hole critical behaviour with the generalized BSSN formulation

TL;DR

This paper demonstrates the effectiveness of a generalized BSSN hyperbolic formulation in simulating type II critical collapse of a scalar field, successfully capturing key features like self-similarity and mass scaling.

Contribution

It introduces a generalized BSSN formulation suitable for curvilinear coordinates and applies it to critical collapse, achieving stable evolutions near the black hole threshold.

Findings

Successfully identified discrete self-similarity of the critical solution

Measured a mass scaling exponent consistent with previous results

Demonstrated stable evolutions near the black hole threshold

Abstract

The development of hyperbolic formulations of Einstein's equations has revolutionized our ability to perform long-time, stable, accurate numerical simulations of strong field gravitational phenomena. However, hyperbolic methods have seen relatively little application in one area of interest, type II critical collapse, where the challenges for a numerical code are particularly severe. Using the critical collapse of a massless scalar field in spherical symmetry as a test case, we study a generalization of the Baumgarte-Shapiro-Shibata-Nakamura (BSSN) formulation due to Brown that is suited for use with curvilinear coordinates. We adopt standard dynamical gauge choices, including 1+log slicing and a shift that is either zero or evolved by a Gamma-driver condition. With both choices of shift we are able to evolve sufficiently close to the black hole threshold to (1) unambiguously identify the discrete self-similarity of the critical solution (2) determine an echoing exponent consistent with previous calculations and (3) measure a mass scaling exponent, also in accord with prior computations. Our results can be viewed as an encouraging first step towards the use of hyperbolic formulations in more generic type II scenarios, including the as yet unresolved problem of critical collapse of axisymmetric gravitational waves.