The BSSN formulation is a partially constrained evolution system

TL;DR

This paper reveals that the BSSN formulation of Einstein's equations explicitly incorporates momentum constraints as evolution equations, which may explain its effectiveness in numerical relativity simulations.

Contribution

It demonstrates that the BSSN formulation is a partially constrained evolution system by rewriting momentum constraints as evolution equations, providing insight into its success.

Findings

BSSN explicitly uses momentum constraints as evolution equations

Rewriting constraints as evolution equations can improve numerical stability

Supports the effectiveness of BSSN in relativistic simulations

Abstract

Relativistic simulations in 3+1 dimensions typically monitor the Hamiltonian and momentum constraints during evolution, with significant violations of these constraints indicating the presence of instabilities. In this paper we rewrite the momentum constraints as first-order evolution equations, and show that the popular BSSN formulation of the Einstein equations explicitly uses the momentum constraints as evolution equations. We conjecture that this feature is a key reason for the relative success of the BSSN formulation in numerical relativity.