TY - JOUR

T1 - Extended lifetime in computational evolution of isolated black holes

AU - Anderson, Matthew

AU - Matzner, Richard A.

N1 - Funding Information:
Computations were performed at the Texas Advanced Computing Center at the University of Texas. This work was supported by NSF grants PHY 0102204 and PHY 0354842, and by NASA grant NNG04GL3 7G. Additionally, portions of this work were conducted at the Kavli Institute for Theoretical Physics, The University of California at Santa Barbara, under NSF grant PHY99 07947, and at the Laboratory for High Energy Astrophysics, NASA/Goddard Space flight Center, Greenbelt Maryland, with support from the University Space Research Association. M. Anderson acknowledges support from a Department of Energy Computational Science Graduate Fellowship administered by the Krell Institute. A preliminary version of this work was presented at the meeting of the International Association of Relativistic Dynamics, summer 2003, and RM thanks the organizers for their hospitality.

PY - 2005/9

Y1 - 2005/9

N2 - 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 (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 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") ġ - K̇ formulation. With specific subtraction of constraints, with a simple analytic gauge, with very simple boundary conditions, and for moderately large domains with moderately fine resolution, we find computational evolutions of isolated non-spinning black holes for times exceeding 1000 GM/c2. We have also carried out a series of 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 ġ - K̇ 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.

AB - 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 (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 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") ġ - K̇ formulation. With specific subtraction of constraints, with a simple analytic gauge, with very simple boundary conditions, and for moderately large domains with moderately fine resolution, we find computational evolutions of isolated non-spinning black holes for times exceeding 1000 GM/c2. We have also carried out a series of 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 ġ - K̇ 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.

KW - Constrained evolution

KW - Constraint subtraction

KW - Numerical relativity

UR - http://www.scopus.com/inward/record.url?scp=28944455399&partnerID=8YFLogxK

U2 - 10.1007/s10701-005-6477-x

DO - 10.1007/s10701-005-6477-x

M3 - Article

AN - SCOPUS:28944455399

SN - 0015-9018

VL - 35

SP - 1477

EP - 1495

JO - Foundations of Physics

JF - Foundations of Physics

IS - 9

ER -