Constraint-preserving boundary conditions in the 3+1 first-order approach

A set of energy-momentum constraint-preserving boundary conditions is proposed for the first-order Z4 case. The stability of a simple numerical implementation is tested in the linear regime (robust stability test), both with the standard corner and vertex treatment and with a modified finite-differences stencil for boundary points which avoids corners and vertices even in cartesian-like grids. Moreover, the proposed boundary conditions are tested in a strong field scenario, the Gowdy waves metric, showing the expected rate of convergence. The accumulated amount of energy-momentum constraint violations is similar or even smaller than the one generated by either periodic or reflection conditions, which are exact in the Gowdy waves case. As a side theoretical result, a new symmetrizer is explicitly given, which extends the parametric domain of symmetric hyperbolicity for the Z4 formalism. The application of these results to first-order BSSN-like formalisms is also considered.Comment: Revised version, with conclusive numerical evidence. 23 pages, 12 figure

New Formalism for Numerical Relativity

We present a new formulation of the Einstein equations that casts them in an explicitly first order, flux-conservative, hyperbolic form. We show that this now can be done for a wide class of time slicing conditions, including maximal slicing, making it potentially very useful for numerical relativity. This development permits the application to the Einstein equations of advanced numerical methods developed to solve the fluid dynamic equations, {\em without} overly restricting the time slicing, for the first time. The full set of characteristic fields and speeds is explicitly given.Comment: uucompresed PS file. 4 pages including 1 figure. Revised version adds a figure showing a comparison between the standard ADM approach and the new formulation. Also available at http://jean-luc.ncsa.uiuc.edu/Papers/ Appeared in Physical Review Letters 75, 600 (1995

Three dimensional numerical relativity: the evolution of black holes

We report on a new 3D numerical code designed to solve the Einstein equations for general vacuum spacetimes. This code is based on the standard 3+1 approach using cartesian coordinates. We discuss the numerical techniques used in developing this code, and its performance on massively parallel and vector supercomputers. As a test case, we present evolutions for the first 3D black hole spacetimes. We identify a number of difficulties in evolving 3D black holes and suggest approaches to overcome them. We show how special treatment of the conformal factor can lead to more accurate evolution, and discuss techniques we developed to handle black hole spacetimes in the absence of symmetries. Many different slicing conditions are tested, including geodesic, maximal, and various algebraic conditions on the lapse. With current resolutions, limited by computer memory sizes, we show that with certain lapse conditions we can evolve the black hole to about $t=50M$, where $M$ is the black hole mass. Comparisons are made with results obtained by evolving spherical initial black hole data sets with a 1D spherically symmetric code. We also demonstrate that an ``apparent horizon locking shift'' can be used to prevent the development of large gradients in the metric functions that result from singularity avoiding time slicings. We compute the mass of the apparent horizon in these spacetimes, and find that in many cases it can be conserved to within about 5\% throughout the evolution with our techniques and current resolution.Comment: 35 pages, LaTeX with RevTeX 3.0 macros. 27 postscript figures taking 7 MB of space, uuencoded and gz-compressed into a 2MB uufile. Also available at http://jean-luc.ncsa.uiuc.edu/Papers/ and mpeg simulations at http://jean-luc.ncsa.uiuc.edu/Movies/ Submitted to Physical Review

Improved Determination of the CKM Angle alpha from B to pi pi decays

Motivated by a recent paper that compares the results of the analysis of the CKM angle alpha in the frequentist and in the Bayesian approaches, we have reconsidered the information on the hadronic amplitudes, which helps constraining the value of alpha in the Standard Model. We find that the Bayesian method gives consistent results irrespective of the parametrisation of the hadronic amplitudes and that the results of the frequentist and Bayesian approaches are equivalent when comparing meaningful probability ranges or confidence levels. We also find that from B to pi pi decays alone the 95% probability region for alpha is the interval [80^o,170^o], well consistent with recent analyses of the unitarity triangle where, by using all the available experimental and theoretical information, one gets alpha = (93 +- 4)^o. Last but not least, by using simple arguments on the hadronic matrix elements, we show that the unphysical region alpha ~ 0, present in several experimental analyses, can be eliminated.Comment: 16 pages, 7 figure

First order hyperbolic formalism for Numerical Relativity

The causal structure of Einstein's evolution equations is considered. We show that in general they can be written as a first order system of balance laws for any choice of slicing or shift. We also show how certain terms in the evolution equations, that can lead to numerical inaccuracies, can be eliminated by using the Hamiltonian constraint. Furthermore, we show that the entire system is hyperbolic when the time coordinate is chosen in an invariant algebraic way, and for any fixed choice of the shift. This is achieved by using the momentum constraints in such as way that no additional space or time derivatives of the equations need to be computed. The slicings that allow hyperbolicity in this formulation belong to a large class, including harmonic, maximal, and many others that have been commonly used in numerical relativity. We provide details of some of the advanced numerical methods that this formulation of the equations allows, and we also discuss certain advantages that a hyperbolic formulation provides when treating boundary conditions.Comment: To appear in Phys. Rev.

Exploiting gauge and constraint freedom in hyperbolic formulations of Einstein's equations

We present new many-parameter families of strongly and symmetric hyperbolic formulations of Einstein's equations that include quite general algebraic and live gauge conditions for the lapse. The first system that we present has 30 variables and incorporates an algebraic relationship between the lapse and the determinant of the three metric that generalizes the densitized lapse prescription. The second system has 34 variables and uses a family of live gauges that generalizes the Bona-Masso slicing conditions. These systems have free parameters even after imposing hyperbolicity and are expected to be useful in 3D numerical evolutions. We discuss under what conditions there are no superluminal characteristic speeds

Robust evolution system for Numerical Relativity

The paper combines theoretical and applied ideas which have been previously considered separately into a single set of evolution equations for Numerical Relativity. New numerical ingredients are presented which avoid gauge pathologies and allow one to perform robust 3D calculations. The potential of the resulting numerical code is demonstrated by using the Schwarzschild black hole as a test-bed. Its evolution can be followed up to times greater than one hundred black hole masses.Comment: 11 pages, 4 figures; figure correcte

Harmonic coordinate method for simulating generic singularities

This paper presents both a numerical method for general relativity and an application of that method. The method involves the use of harmonic coordinates in a 3+1 code to evolve the Einstein equations with scalar field matter. In such coordinates, the terms in Einstein's equations with the highest number of derivatives take a form similar to that of the wave equation. The application is an exploration of the generic approach to the singularity for this type of matter. The preliminary results indicate that the dynamics as one approaches the singularity is locally the dynamics of the Kasner spacetimes.Comment: 5 pages, 4 figures, Revtex, discussion expanded, references adde

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