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 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 Fourier-component of the constraint propagation equations) are negative or pure-imaginary. We show such a system can be obtained by choosing multipliers of adjusted terms. Our discussion covers Detweiler's proposal (1987) and Frittelli's analysis (1997), and we also mention the so-called conformal-traceless ADM systems.Comment: 11 pages, RevTeX, 2 eps figure

Adjusted ADM systems and their expected stability properties: constraint propagation analysis in Schwarzschild spacetime

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 analyzing 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), 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.Comment: 23 pages, RevTeX4, many figures. Revised version. Added subtitle, reduced figures, rephrased introduction, and a native checked. :-

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 Ashtekar's 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.Comment: 15 pages, RevTeX, minor changes in Introduction. published as Int. J. Mod. Phys. D 9 (2000) 1

Sensitivity of the Fe Kα\alpha Compton shoulder to the geometry and variability of the X-ray illumination of cosmic objects

In an X-ray reflection spectrum, a tail-like spectral feature generated via Compton downscattering, known as a Compton shoulder (CS), appears at the low-energy side of the iron K$\alpha$ line. Despite its great diagnostic potential, its use as a spectral probe of the reflector has been seriously limited due to observational difficulties and modelling complexities. We revisit the basic nature of the CS by systematic investigation into its dependence on spatial and temporal parameters. The calculations are performed by Monte Carlo simulations for sphere and slab geometries. The dependence is obtained in a two-dimensional space of column density and metal abundance, demonstrating that the CS solves parameter degeneration between them which was seen in conventional spectral analysis using photoelectric absorption and fluorescence lines. Unlike the iron line, the CS does not suffer from any observational dependence on the spectral hardness. The CS profile is highly dependent on the inclination angle of the slab geometry unless the slab is Compton-thick, and the time evolution of the CS is shown to be useful to constrain temporal information on the source if the intrinsic radiation is variable. We also discuss how atomic binding of the scattering electrons in cold matter blurs the CS profile, finding that the effect is practically similar to thermal broadening in a plasma with a moderate temperature of ~5 eV. Spectral diagnostics using the CS is demonstrated with grating data of X-ray binary GX 301−2, and will be available in future with high-resolution spectra of active galactic nuclei obtained by microcalorimeters.JSPS KAKENHI (Grant IDs: 24740190, 24105007), Advanced Leading graduate school for Photon Science (ALPS

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.

Neutron wave packet tomography

A tomographic technique is introduced in order to determine the quantum state of the center of mass motion of neutrons. An experiment is proposed and numerically analyzed.Comment: 4 pages, 3 figure