Why Solve the Hamiltonian Constraint in Numerical Relativity?

TL;DR

This paper discusses the challenges of solving the Hamiltonian constraint in numerical relativity, emphasizing the importance of maintaining the delicate balance to prevent qualitatively incorrect results in simulations.

Contribution

It provides an analysis of the impact of Hamiltonian constraint violations and compares the behavior of simulations with and without constraint enforcement.

Findings

Constraint violations can lead to qualitatively wrong behavior.

Stable simulations can be achieved with proper constraint enforcement.

The sign-indefinite nature of the Hamiltonian constraint complicates numerical solutions.

Abstract

The indefinite sign of the Hamiltonian constraint means that solutions to Einstein's equations must achieve a delicate balance--often among numerically large terms that nearly cancel. If numerical errors cause a violation of the Hamiltonian constraint, the failure of the delicate balance could lead to qualitatively wrong behavior rather than just decreased accuracy. This issue is different from instabilities caused by constraint-violating modes. Examples of stable numerical simulations of collapsing cosmological spacetimes exhibiting local mixmaster dynamics with and without Hamiltonian constraint enforcement are presented.