Gen Yoneda

Constraint propagation in the family of ADM systems

The current important issue in numerical relativity is to determine which formulation of the Einstein equations provides us with stable and accurate simulations. Based on our previous work on ‘‘asymptotically constrained’’ systems, we here present constraint propagation equations and their eigenvalues for the ArnowittDeser-Misner ~ADM! evolution equations with additional constraint terms ~adjusted terms! on the right-hand side. We conjecture that the system is robust against violation of constraints if the amplification factors ~eigenvalues of the Fourier component of the constraint propagation equations! are negative or purely imaginary. We show that such a system can be obtained by choosing multipliers of the adjusted terms. Our discussion covers Detweiler’s proposal and Frittelli’s analysis, and we also mention the so-called conformaltraceless ADM systems.

Advantages of a modified ADM formulation: Constraint propagation analysis of the Baumgarte-Shapiro-Shibata-Nakamura system

Several numerical relativity groups are using a modified Arnowitt-Deser-Misner ~ADM! formulation for their simulations, which was developed by Nakamura and co-workers ~and widely cited as the BaumgarteShapiro-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 reformulation has such an advantage. We try to explain the background mechanism of the BSSN equations by using an 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.

Trick for passing degenerate points in the Ashtekar formulation

We examine one of the advantages of Ashtekar’s formulation of general relativity: a tractability of degenerate points from the point of view of following the dynamics of classical spacetime. Assuming that all dynamical variables are finite, we conclude that an essential trick for such a continuous evolution is in complexifying variables. In order to restrict the complex region locally, we propose some ‘‘reality recovering’’ conditions on spacetime. Using a degenerate solution derived by a pullback technique, and integrating the dynamical equations numerically, we show that this idea works in an actual dynamical problem. We also discuss some features of these applications. @S0556-2821~97!06916-6#

Diagonalizability of constraint propagation matrices

In order to obtain stable and accurate general relativistic simulations, reformulations of the Einstein equations are necessary. In a series of our works, we have proposed using eigenvalue analysis of constraint propagation equations for evaluating violation behaviour of constraints. In this letter, we classify asymptotical behaviours of constraint violation into three types (asymptotically constrained, asymptotically bounded and diverge), and give their necessary and sufficient conditions. We find that degeneracy of eigenvalues sometimes leads constraint evolution to diverge (even if its real part is not positive) and conclude that it is quite useful to check the diagonalizability of constraint propagation matrices. The discussion is general and can be applied to any numerical treatments of constrained dynamics.

Hyperbolic formulations and numerical relativity: Experiments using Ashtekar's connection variables

In order to perform accurate and stable long-time numerical integration of the Einstein equation, several hyperbolic systems have been proposed. Here we present a numerical comparison between weakly hyperbolic, strongly hyperbolic and symmetric hyperbolic systems based on Ashtekars connection variables. The primary advantage for using this connection formulation in this experiment is that we can keep using the same dynamical variables for all levels of hyperbolicity. Our numerical code demonstrates gravitational wave propagation in plane-symmetric spacetimes, and we compare the accuracy of the simulation by monitoring the violation of the constraints. By comparing with results obtained from the weakly hyperbolic system, we observe that the strongly and symmetric hyperbolic system show better numerical performance (yield less constraint violation), but not so much difference between the latter two. Rather, we find that the symmetric hyperbolic system is not always the best in terms of numerical performance. This study is the first to present full numerical simulations using Ashtekars variables. We also describe our procedures in detail. (Some figures in this article are in colour only in the electronic version; see www.iop.org) PACS numbers: 0420C, 0425, 0425D

Constructing hyperbolic systems in the Ashtekar formulation of general relativity

Hyperbolic formulations of the equations of motion are essential technique for proving the well-posedness of the Cauchy problem of a system, and are also helpful for implementing stable long time evolution in numerical applications. We, here, present three kinds of hyperbolic systems in the Ashtekar formulation of general relativity for Lorentzian vacuum spacetime. We exhibit several (I) weakly hyperbolic, (II) diagonalizable hyperbolic, and (III) symmetric hyperbolic systems, with each their eigenvalues. We demonstrate that Ashtekars original equations form a weakly hyperbolic system. We discuss how gauge conditions and reality conditions are constrained during each step toward constructing a symmetric hyperbolic system.

Constraints and reality conditions in the Ashtekar formulation of general relativity

We show how to treat the constraints and reality conditions in the SO(3) - ADM (Ashtekar) formulation of general relativity, for the case of a vacuum spacetime with a cosmological constant. We clarify the difference between the reality conditions on the metric and on the triad. Assuming the triad reality condition, we find a new variable, allowing us to solve the gauge constraint equations and the reality conditions simultaneously.

Constraint Propagation in (N + 1)-Dimensional Space-Time

Higher dimensional space-time models provide us an alternative interpretation of nature, and give us different dynamical aspects than the traditional four-dimensional space-time models. Motivated by such recent interests, especially for future numerical research of higher-dimensional space-time, we study the dimensional dependence of constraint propagation behavior. The N+1 Arnowitt-Deser-Misner evolution equation has matter terms which depend on N, but the constraints and constraint propagation equations remain the same. This indicates that there would be problems with accuracy and stability when we directly apply the N+1 ADM formulation to numerical simulations as we have experienced in four-dimensional cases. However, we also conclude that previous efforts in re-formulating the Einstein equations can be applied if they are based on constraint propagation analysis.

Constraint propagation of C2-adjusted formulation. II. Another recipe for robust Baumgarte-Shapiro-Shibata-Nakamura evolution system

was proposed by Fiske (2004) and was applied to the ADM formulation in our previous study. We derive the constraint propagation equations, discuss the behavior of constraint damping, and present the results of numerical tests using the gauge-wave and polarized Gowdy wave spacetimes. The construction of the C 2 -adjusted system is straightforward. However, in BSSN, there are two kinetic constraints and three algebraic constraints; thus, the definition of C 2 is a matter of concern. By analyzing constraint propagation equations, we conclude that C 2 should include all the constraints, which is also confirmed numerically. By tuning the parameters, the lifetime of the simulations can be increased 2‐10 times longer than those of the standard Baumgarte-Shapiro-Shibata-Nakamura evolutions.

Constraint propagation of C2-adjusted formulation: Another recipe for robust ADM evolution system

With a purpose of constructing a robust evolution system against numerical instability for integrating the Einstein equations, we propose a new formulation by adjusting the ADM evolution equations with constraints. We apply an adjusting method proposed by Fiske (2004) which uses the norm of the constraints, C{sup 2}. One of the advantages of this method is that the effective signature of adjusted terms (Lagrange multipliers) for constraint-damping evolution is predetermined. We demonstrate this fact by showing the eigenvalues of constraint propagation equations. We also perform numerical tests of this adjusted evolution system using polarized Gowdy-wave propagation, which show robust evolutions against the violation of the constraints than that of the standard ADM formulation.