BSSN formulation

BSSN formulation
Name	BSSN formulation
Introduced	1995
Developers	Masaaki Shibata; Thomas W. Baumgarte
Field	Numerical relativity

BSSN formulation The BSSN formulation is a reformulation of the Einstein field equations used in numerical relativity for stable time evolution of black hole and neutron star spacetimes. It was developed by Masaaki Shibata and Thomas W. Baumgarte and built on earlier work by James M. Bardeen, Kip S. Thorne, and others, and it became widely used after breakthroughs connected to the Breakthrough Prize era of gravitational wave modeling and LIGO Scientific Collaboration detections.

Introduction

The BSSN formulation recasts the Einstein field equations from general relativity into a system of evolution equations for conformally related variables, combining ideas from the ADM formalism, conformal decompositions used in studies by Yvonne Choquet-Bruhat, and hyperbolic formulations influenced by researchers at Albert Einstein Institute and Caltech. It achieves improved numerical behavior in simulations of binary black hole mergers studied by teams such as NASA Goddard Space Flight Center and the Max Planck Institute for Gravitational Physics.

Mathematical formulation

The system introduces a conformal metric, conformal factor, trace-free extrinsic curvature, and a conformal connection function, extending the Arnowitt–Deser–Misner approach used in formulations by Richard Arnowitt, Stanley Deser, and Charles W. Misner. BSSN variables include the conformal metric ḡ_ij, conformal factor φ, trace of the extrinsic curvature K, trace-free part Ā_ij, and the contracted Christoffel symbols Γ̄^i, following methods akin to conformal techniques used by Ludwig Faddeev and mathematical structures studied by Élie Cartan. The evolution equations incorporate lapse and shift gauge conditions often chosen from families such as 1+log slicing and Gamma-driver shift inspired by gauge choices applied in work at Princeton University and Yale University. Constraint equations stem from the Hamiltonian constraint and momentum constraints as formulated in the ADM context and solved using techniques from computational groups at CERN and Los Alamos National Laboratory.

Numerical implementation

Practical implementations use finite-difference, finite-volume, and spectral methods developed in codes like the Einstein Toolkit, SpEC code from Caltech and Cornell University, and codebases from RIT and SXS Collaboration. Time integration commonly uses Runge–Kutta schemes promoted by numerical analysts from Courant Institute and University of Illinois at Urbana-Champaign, while mesh refinement is handled via adaptive algorithms from teams at Argonne National Laboratory and Flatiron Institute. Boundary conditions and excision techniques leverage methods tested by researchers at Max Planck Institute for Gravitational Physics and University of Maryland, and parallelization employs frameworks developed at Lawrence Livermore National Laboratory and Oak Ridge National Laboratory.

Applications in numerical relativity

BSSN underlies simulations of binary neutron star and binary black hole mergers used to produce gravitational waveforms for the LIGO Scientific Collaboration, Virgo Collaboration, and KAGRA analyses. It supports studies of phenomena such as gravitational recoil investigated by groups at University of Texas at Brownsville and relativistic hydrodynamics coupling developed at University of Illinois. Cosmological scenarios and critical collapse problems explored at Perimeter Institute and University of British Columbia have also used BSSN-based codes for insight into nonlinear dynamics first highlighted by researchers at Princeton University and Caltech.

Stability and constraint damping

The BSSN formulation improves stability relative to plain ADM by using conformal rescaling and introducing Γ̄^i as independent variables, an approach that echoes stabilization techniques developed by Lars Hörmander in hyperbolic PDE theory and constraint damping methods pioneered by teams at Max Planck Institute for Gravitational Physics and Cornell University. Constraint-damping terms similar to those used in generalized harmonic formulations employed by SXS Collaboration can be adapted to BSSN to control violations of the Hamiltonian and momentum constraints, as demonstrated by numerical experiments at Albert Einstein Institute and NASA Goddard Space Flight Center.

Comparison with other formulations

Compared with the generalized harmonic formulation used by groups at Caltech and Princeton University, BSSN is often simpler to implement in finite-difference codes and became popular in community toolkits such as the Einstein Toolkit, while generalized harmonic systems provide alternate hyperbolic structures exploited in the SpEC code developed by Caltech and Cornell University. The Z4c formulation developed by researchers at University of Valencia and AEI introduces explicit constraint propagation similar to damping methods used in BSSN variants, and the conformal covariant formulations from Bonn University explore different trade-offs between constraint control and computational cost.

Extensions and variants

Variants include conformal and covariant extensions developed by teams at University of Pisa and University of Valencia, damping-enhanced versions inspired by work at Max Planck Institute for Gravitational Physics, and formulations coupling BSSN to hydrodynamics modules produced by groups at University of Tokyo and Northwestern University. Research into high-order numerical methods and multi-domain spectral extensions connects to developments at Caltech, Cornell University, and Princeton University for improved waveform accuracy in LIGO-era astrophysical inference.

Category:Numerical relativity