Extended Lifetime in Computational Evolution of Isolated Black Holes

TL;DR

This paper investigates the long-term computational evolution of isolated black holes using traditional and constrained methods, revealing limitations in simulation longevity related to boundary features and resolution.

Contribution

It demonstrates that traditional ot gb7b7Kb7 formulations can achieve comparable black hole lifetime results as modern hyperbolic formulations, and explores the effects of constraint solutions on stability.

Findings

Constraint subtraction can substantially stabilize long-term evolutions.

Neither method achieves arbitrarily long simulations in large domains.

Boundary features at the inner excision limit simulation longevity.

Abstract

Solving the 4-d Einstein equations as evolution in time requires solving equations of two types: the four elliptic initial data (constraint) equations, followed by the six second order evolution equations. Analytically the constraint equations remain solved under the action of the evolution, and one approach is to simply monitor them ({\it unconstrained} evolution). The problem of the 3-d computational simulation of even a single isolated vacuum black hole has proven to be remarkably difficult. Recently, we have become aware of two publications that describe very long term evolution, at least for single isolated black holes. An essential feature in each of these results is {\it constraint subtraction}. Additionally, each of these approaches is based on what we call "modern," hyperbolic formulations of the Einstein equations. It is generally assumed, based on computational experience, that the use of such modern formulations is essential for long-term black hole stability. We report here on comparable lifetime results based on the much simpler ("traditional") $\dot g$ - $\dot K$ formulation. We have also carried out a series of {\it constrained} 3-d evolutions of single isolated black holes. We find that constraint solution can produce substantially stabilized long-term single hole evolutions. However, we have found that for large domains, neither constraint-subtracted nor constrained $\dot g$ - $\dot K$ evolutions carried out in Cartesian coordinates admit arbitrarily long-lived simulations. The failure appears to arise from features at the inner excision boundary; the behavior does generally improve with resolution.