We present a new formulation of the Einstein equations that casts them in an explicitly rst order, ux-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 uid dynamic equations, without overly restricting the time slicing, for the rst time. The full set of characteristic elds and speeds is explicitly given.PACS numbers: 04.25.Dm Introduction. For decades the standard approach 1] to numerical relativity has been based on a direct application of the 3+1 formulation of the Einstein equations by Arnowitt, Deser, and Misner 2]. This important contribution laid the foundation for most numerical work in the eld. As convenient as this formulation is for numerical relativity, the structure of the equations is extremely complicated and most of the work consisted in developing \ad hoc" numerical codes for every case considered. This is in contrast with the situation of modern Computational Fluid Dynamics, which deals with rst order, ux conservative, hyperbolic (FOFCH) systems of equations, for which many advanced numerical methods have been developed 3] based on the particular mathematical structure of the equations 4].After many pioneering attempts 5{7] it was shown that the full set of 3D Einstein equations could be put in the FOFCH form 8], similar to hydrodynamics. This development allowed the same advanced numerical methods used in hydrodynamics, such as conservative schemes and modern shock capturing methods, to be applied to the Einstein equations for the rst time. In a number of tests involving spherical black holes 9,10], 1D general relativistic hydrodynamics 11] and 3D gravitational waves 12], this formulation showed some of its strengths over the standard approach. However, the price to pay for this original formulation was that it required the restrictive assumption of harmonic time slicing. Although this slicing condition is singularity avoiding, making it potentially useful for studies of strongly gravitating systems, it is just barely so 13]. For this reason metric and curvature components can grow without bound near the singularity and, without some sort of horizon boundary condition 10,14,15], this slicing is not well suited for black hole spacetimes.In this Letter, we will show that this FOFCH formal-

We present a new formulation of the Einstein equations based on a conformal and traceless decomposition of the covariant form of the Z4 system. This formulation combines the advantages of a conformal decomposition, such as the one used in the BSSNOK formulation (i.e. well-tested hyperbolic gauges, no need for excision, robustness to imperfect boundary conditions) with the advantages of a constraint-damped formulation, such as the generalized harmonic one (i.e. exponential decay of constraint violations when these are produced). We validate the new set of equations through standard tests and by evolving binary black hole systems. Overall, the new conformal formulation leads to a better behaviour of the constraint equations and a rapid suppression of the violations when they occur. The changes necessary to implement the new conformal formulation in standard BSSNOK codes are very small as are the additional computational costs.PACS numbers: 04.40.-b,95.35.+d

A general covariant extension of Einstein´s field equations is considered with a view to Numerical Relativity applications. The basic variables are taken to be the metric tensor and an additional four-vector Zµ. Einstein's solutions are recovered when the additional four-vector vanishes, so that the energy and momentum constraints amount to the covariant algebraic condition Zµ = 0. The extended field equations can be supplemented by suitable coordinate conditions in order to provide symmetric hyperbolic evolution systems: this is actually the case for either harmonic coordinates or normal coordinates with harmonic slicing.

In recent years, many different numerical evolution schemes for Einstein's equations have been proposed to address stability and accuracy problems that have plagued the numerical relativity community for decades. Some of these approaches have been tested on different spacetimes, and conclusions have been drawn based on these tests. However, differences in results originate from many sources, including not only formulations of the equations, but also gauges, boundary conditions, numerical methods, and so on. We propose to build up a suite of standardized testbeds for comparing approaches to the numerical evolution of Einstein's equations that are designed to both probe their strengths and weaknesses and to separate out different effects, and their causes, seen in the results. We discuss general design principles of suitable testbeds, and we present an initial round of simple tests with periodic boundary conditions. This is a pivotal first step toward building a suite of testbeds to serve the numerical relativists and researchers from related fields who wish to assess the capabilities of numerical relativity codes. We present some examples of how these tests can be quite effective in revealing various limitations of different approaches, and illustrating their differences. The tests are presently limited to vacuum spacetimes, can be run on modest computational resources, and can be used with many different approaches used in the relativity community.

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, which 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 a 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.

Boson stars, self-gravitating objects made of a complex scalar field, have been proposed as simple models for very different scenarios, ranging from galaxy dark matter to black hole mimickers. Here we focus on a very compact type of boson stars to study binary mergers by varying different parameters, namely the phase shift, the direction of rotation and the angular momentum. Our aim is to investigate the properties of the object resulting from the merger in these different scenarios by means of numerical evolutions. These simulations, performed by using a modification of the covariant conformal Z4 (CCZ4) formalism of the Einstein Equations that does not require the algebraic enforcing of any constraint, indicate that the final state after a head-on collision of low mass boson stars is another boson star. However, almost complete annihilation of the stars occurs during the merger of a bosonantiboson pair. The merger of orbiting boson stars form a rotating bar that quickly relaxes to a non-rotating boson star.

Einstein evolution equations are written as a hyperbolic system of balance laws. A harmonic time coordinate is used with zero shift vector (harmonic slicing). The principal part of the evolution system reduces to a set of uncoupled wave equations in first-order form. The relevance for three-dimensional numerical relativity of both the harmonic slicing and the resulting evolution system is stressed.PACS numbers: 04.20.CvIn this paper, we write down the evolution system of Einstein field equations as a hyperbolic first-order system of balance laws in terms of densities, fluxes, and sources. This is a structure well known to those who numerically solve the Navier-Stokes equations for hydrodynamics and this fact should provide numerical relativists with the opportunity to employ many of the powerful methods of computational fluid dynamics in simulations of general relativistic systems.The numerical techniques so far used in general relativity have been developed in a somewhat ad hoc manner simply because the Einstein equations have not been put in the form of hyperbolic balance law equations. Therefore a large body of numerical methods useful for such equations were not applicable to general relativity. The results of this paper should change that situation and hopefully will stimulate rapid progress in the development of numerical simulations of relativistic astrophysical systems.The main concern in numerical relativity is to follow the space-time evolution from given initial data. This sets a strong connection with the general relativistic Cauchy problem: One can expect to be able to develop consistent stable numerical algorithms only when the existence and uniqueness of the solution is ensured by a well-posed initial-value problem. As is well known, the Einstein field equations must be supplemented with some coordinate conditions and this determines the mathematical structure of the evolution system.Existence and uniqueness problems have been proven for hyperbolic systems and these theorems have been applied to the Einstein field equations with different coordinate conditions which lead to a hyperbolic evolution system. The most popular ones are the harmonic coordinates: =0,1,2,3),where the symbol D stands for the d'Alembert differential operator acting on functions, namely,•/s^8i/-ra f /,and we have used V a =g h Tl.^-nx\ (3) where IX are the connection coefficients associated with the space-time metric g a f,. These conditions have recently been generalized to the case where the V a are given in terms of suitably regular conditions [l] (but not necessarily equal to zero). Other authors impose the harmonicity conditions (1) by using the d'Alembert operator (2) associated with some background metric [2], or keep only the harmonicity condition for the time coordinate, r°=-njc°=0, and choose the space coordinates such that S°'=0 0 = 1,2,3), (4) (5) leading to the "harmonic slicing" [3]. Harmonic slicing.-Allowing for (5), the line element can be decomposed in the form ds : : -« 2 a l dt 1^g lj dx l dx J (6) and this...

The general-covariant Z4 formalism is further analyzed. The gauge conditions are generalized with a view to Numerical Relativity applications and the conditions for obtaining strongly hyperbolic evolution systems are given both at the first and the second order levels. A symmetry-breaking mechanism is proposed that allows one, when applied in a partial way, to recover previously proposed strongly hyperbolic formalisms, like the BSSN and the Bona-Massó ones. When applied in its full form, the symmetry breaking mechanism allows one to recover the full five-parameter family of first order KST systems. Numerical codes based in the proposed formalisms are tested. A robust stability test is provided by evolving random noise data around Minkowski space-time. A strong field test is provided by the collapse of a periodic background of plane gravitational waves, as described by the Gowdy metric.