Testing the Cactus code on exact solutions of the Einstein field equations

The article presents a series of numerical simulations of exact solutions of the Einstein equations performed using the Cactus code, a complete 3-dimensional machinery for numerical relativity. We describe an application (``thorn'') for the Cactus code that can be used for evolving a variety of exact solutions, with and without matter, including solutions used in modern cosmology for modeling the early stages of the universe. Our main purpose has been to test the Cactus code on these well-known examples, focusing mainly on the stability and convergence of the code.Comment: 18 pages, 18 figures, Late

Hyperbolic slicings of spacetime: singularity avoidance and gauge shocks

I study the Bona-Masso family of hyperbolic slicing conditions, considering in particular its properties when approaching two different types of singularities: focusing singularities and gauge shocks. For focusing singularities, I extend the original analysis of Bona et. al and show that both marginal and strong singularity avoidance can be obtained for certain types of behavior of the slicing condition as the lapse approaches zero. For the case of gauge shocks, I re-derive a condition found previously that eliminates them. Unfortunately, such a condition limits considerably the type of slicings allowed. However, useful slicing conditions can still be found if one asks for this condition to be satisfied only approximately. Such less restrictive conditions include a particular member of the 1+log family, which in the past has been found empirically to be extremely robust for both Brill wave and black hole simulations.Comment: 11 pages, revtex4. Change in acknowledgment

A hyperbolic slicing condition adapted to Killing fields and densitized lapses

We study the properties of a modified version of the Bona-Masso family of hyperbolic slicing conditions. This modified slicing condition has two very important features: In the first place, it guarantees that if a spacetime is static or stationary, and one starts the evolution in a coordinate system in which the metric coefficients are already time independent, then they will remain time independent during the subsequent evolution, {\em i.e.} the lapse will not evolve and will therefore not drive the time lines away from the Killing direction. Second, the modified condition is naturally adapted to the use of a densitized lapse as a fundamental variable, which in turn makes it a good candidate for a dynamic slicing condition that can be used in conjunction with some recently proposed hyperbolic reformulations of the Einstein evolution equations.Comment: 11 page

Formulations of the 3+1 evolution equations in curvilinear coordinates

Following Brown, in this paper we give an overview of how to modify standard hyperbolic formulations of the 3+1 evolution equations of General Relativity in such a way that all auxiliary quantities are true tensors, thus allowing for these formulations to be used with curvilinear sets of coordinates such as spherical or cylindrical coordinates. After considering the general case for both the Nagy-Ortiz-Reula (NOR) and the Baumgarte-Shapiro-Shibata-Nakamura (BSSN) formulations, we specialize to the case of spherical symmetry and also discuss the issue of regularity at the origin. Finally, we show some numerical examples of the modified BSSN formulation at work in spherical symmetry.Comment: 19 pages, 12 figure

Gauge conditions for long-term numerical black hole evolutions without excision

Numerical relativity has faced the problem that standard 3+1 simulations of black hole spacetimes without singularity excision and with singularity avoiding lapse and vanishing shift fail after an evolution time of around 30-40M due to the so-called slice stretching. We discuss lapse and shift conditions for the non-excision case that effectively cure slice stretching and allow run times of 1000M and more.Comment: 19 pages, 14 figures, REVTeX, Added a missing Acknowledgmen

Advantages of modified ADM formulation: constraint propagation analysis of Baumgarte-Shapiro-Shibata-Nakamura system

Several numerical relativity groups are using a modified ADM formulation for their simulations, which was developed by Nakamura et al (and widely cited as Baumgarte-Shapiro-Shibata-Nakamura system). This so-called BSSN formulation is shown to be more stable than the standard ADM formulation in many cases, and there have been many attempts to explain why this re-formulation has such an advantage. We try to explain the background mechanism of the BSSN equations by using eigenvalue analysis of constraint propagation equations. This analysis has been applied and has succeeded in explaining other systems in our series of works. We derive the full set of the constraint propagation equations, and study it in the flat background space-time. We carefully examine how the replacements and adjustments in the equations change the propagation structure of the constraints, i.e. whether violation of constraints (if it exists) will decay or propagate away. We conclude that the better stability of the BSSN system is obtained by their adjustments in the equations, and that the combination of the adjustments is in a good balance, i.e. a lack of their adjustments might fail to obtain the present stability. We further propose other adjustments to the equations, which may offer more stable features than the current BSSN equations.Comment: 10 pages, RevTeX4, added related discussion to gr-qc/0209106, the version to appear in Phys. Rev.

An alternative approach to solving the Hamiltonian constraint

Solving Einstein's constraint equations for the construction of black hole initial data requires handling the black hole singularity. Typically, this is done either with the excision method, in which the black hole interior is excised from the numerical grid, or with the puncture method, in which the singular part of the conformal factor is expressed in terms of an analytical background solution, and the Hamiltonian constraint is then solved for a correction to the background solution that, usually, is assumed to be regular everywhere. We discuss an alternative approach in which the Hamiltonian constraint is solved for an inverse power of the conformal factor. This new function remains finite everywhere, so that this approach requires neither excision nor a split into background and correction. In particular, this method can be used without modification even when the correction to the conformal factor is singular itself. We demonstrate this feature for rotating black holes in the trumpet topology.Comment: 5 pages, 4 figures, matches version published in PR