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-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.

In order to perform accurate and stable long-term numerical calculations, we con- struct new sets of ADM and BSSN evolution equations by adjusting constraints to their right-hand-sides. We apply a method suggested by Fiske (2004), which adds functional derivative of the norm of constraints, C 2. We derive their constraint prop- Agation equations (evolution equations of constraints) in at spacetime, which show that C2 itself evolve decaying. We also perform numerical tests with the polarized Gowdy-wave testbed, and show that the constraint-damping appears. The life-times of the standard ADM and BSSN simulations are improved as about twice as longer.

We focus the formulation problem in numerical relativity, propose new sets of evolution equations, and demonstrate some numerical tests. The goal is to construct a robust evolution system against numerical instability during long-term numerical integration in strong gravitational field. One key idea is to adjust the evolution equations with constraint terms, as was systematically formulated by Yoneda and Shinkai. We here apply an adjusting method proposed by Fiske (2004) which uses the norm of constraints, C2, and does not require the background metric for specifying effective Lagrange multipliers. We present sets of evolution equations both in ADM and BSSN formulations and show numerical tests using Gowdy wave propagation. Detail analyses are in progress, but we observe constraint damping effect as expected.

Formulation of the Einstein equations is one of the necessary implements for realizing long-term stable and accurate numerical simulations. We re-examine our formulation scheme, that intends to construct a dynamical system which evolves toward the constraint surface as the attractor, by adjusting evolution equations with constraints. We propose an additional guideline which may delay the final blow-up. The new idea is to avoid multipliers to the evolution equations which produce non-linear growth of the constraints in the later stage. Slight but actual improvement can be seen in our test simulations.

We review recent efforts to re-formulate the Einstein equations for fully relativistic numerical simulations in general relativity. The so-called "numerical relativity" is a promising research field matching with ongoing astrophysical observations such as gravitational wave astronomy. Many trials for longterm stable and accurate simulations of binary compact objects have revealed that mathematically equivalent sets of evolution equations show different numerical stability in free evolution schemes. After reviewing the efforts of the community in the decade, we introduce our idea for understanding all the efforts in a unified way using eigenvalue analysis of the constraint propagation equations. The modifications of (or adjustments to) the evolution equations change the character of constraint propagation, and several particular adjustments using constraints are expected to diminish the constraint-violating modes. We propose several new adjusted evolution equations, and present some numerical demonstrations.

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.

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.

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 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 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.

In order to find a way to have a better formulation for numerical evolution of the Einstein equations, we study the propagation equations of the constraints based on the Arnowitt-Deser-Misner formulation. By adjusting constraint terms in the evolution equations, we try to construct an asymptotically constrained system' which is expected to be robust against violation of the constraints, and to enable a long-term stable and accurate numerical simulation. We first provide useful expressions for analysing constraint propagation in a general spacetime, then apply it to Schwarzschild spacetime. We search when and where the negative real or non-zero imaginary eigenvalues of the homogenized constraint propagation matrix appear, and how they depend on the choice of coordinate system and adjustments. Our analysis includes the proposal of Detweiler (1987 Phys. Rev. D 35 1095), which is still the best one according to our conjecture but has a growing mode of error near the horizon. Some examples are snapshots of a maximally sliced Schwarzschild black hole. The predictions here may help the community to make further improvements.

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 Arnowitt-Deser-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 conformal-traceless ADM systems.

We study asymptotically constrained systems for numerical integration of the Einstein equations, which are intended to be robust against perturbative errors for the free evolution of the initial data. First, we examine the previously proposed 'lambda system', which introduces artificial flows to constraint surfaces based on the symmetric hyperbolic formulation. We show that this system works as expected for the wave propagation problem in the Maxwell system and in general relativity using Ashtekar's connection formulation. Second, we propose a new mechanism to control the stability, which we call the 'adjusted system'. This is simply obtained by adding constraint terms in the dynamical equations and adjusting their multipliers. We explain why a particular choice of multiplier reduces the numerical errors from non-positive or pure-imaginary eigenvalues of the adjusted constraint propagation equations. This 'adjusted system' is also tested in the Maxwell system and in the Ashtekar system. This mechanism affects more than the system's symmetric hyperbolicity.

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. ©2001 The American Physical Society.

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 Ashtekar's 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 Ashtekar's variables. We also describe our procedures in detail.

We present a first-order symmetric hyperbolic system in the Ashtekar<br />
formulation of general relativity for vacuum spacetime. We add terms from<br />
constraint equations to the evolution equations with appropriate combinations,<br />
which is the same technique used by Iriondo, Leguizam\&#039;on and Reula [Phys. Rev.<br />
Lett. 79, 4732 (1997)]. However our system is different from theirs in the<br />
points that we primarily use Hermiticity of a characteristic matrix of the<br />
system to characterize our system &quot;symmetric&quot;, discuss the consistency of this<br />
system with reality condition, and show the characteristic speeds ...

Hyperbolic formulations of the equations of motion are essential technique<br />
for proving the well-posedness of the Cauchy problem of a system, and are also<br />
helpful for implementing stable long time evolution in numerical applications.<br />
We, here, present three kinds of hyperbolic systems in the Ashtekar formulation<br />
of general relativity for Lorentzian vacuum spacetime. We exhibit several (I)<br />
weakly hyperbolic, (II) diagonalizable hyperbolic, and (III) symmetric<br />
hyperbolic systems, with each their eigenvalues. We demonstrate that Ashtekar&#039;s<br />
original equations form a weakly hyperbolic system. We ...

We present a set of dynamical equations based on Ashtekar&#039;s extension of the<br />
Einstein equation. The system forces the space-time to evolve to the manifold<br />
that satisfies the constraint equations or the reality conditions or both as<br />
the attractor against perturbative errors. This is an application of the idea<br />
by Brodbeck, Frittelli, Huebner and Reula who constructed an asymptotically<br />
stable (i.e., constrained) system for the Einstein equation, adding dissipative<br />
forces in the extended space. The obtained systems may be useful for future<br />
numerical studies using Ashtekar&#039;s variables.

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. © 1997 The American Physical Society.

We examine one of the advantages of Ashtekar&#039;s formulation of general<br />
relativity: a tractability of degenerate points from the point of view of<br />
following the dynamics of classical spacetime. Assuming that all dynamical<br />
variables are finite, we conclude that an essential trick for such a continuous<br />
evolution is in complexifying variables. In order to restrict the complex<br />
region locally, we propose some `reality recovering&#039; conditions on spacetime.<br />
Using a degenerate solution derived by pull-back technique, and integrating the<br />
dynamical equations numerically, we show that this idea works in a...

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.

The previous paper I of this series presented four asymptotic conditions of smooth worldlines in Minkowski space and researched the inclusion relations between them. This paper moreover presents three asymptotic conditions that are used in special relativistic particle dynamics. (1) There is no end of the worldline. (2) From any point, the perpendicular to the worldline is drawable. (3) The acceleration vanishes asymptotically. The inclusion relations between seven conditions are researched by showing some inclusion relations and by examining some examples. Then the concluding Venn diagram is obtained. © 1994, THE PHYSICAL SOCIETY OF JAPAN. All rights reserved.

This paper presents four asymptotic conditions of smooth worldlines in Minkowski space that are often used in special relativistic particle dynamics. (A) Permanency with respect to an observer's time. (B) Permanency with respect to the proper time. (C) Finiteness of the gamma factor. (D) Intersection with the light cone of any apex. The conditions (B) and (D) are free from observers and it is verified that (A) and (C) are invariant conditions for all inertial observers. To examine the inclusion relations between (A), (B), (C) and (D), some inclusion relations are proved and some are denied by counter examples. Then the concluding Venn diagram is obtained. Such researches must interest theoretical physicists who unintentionally use one or plural number of these conditions. © 1993, THE PHYSICAL SOCIETY OF JAPAN. All rights reserved.