This is the second in a series of papers on the construction and validation of a three-dimensional code for the solution of the coupled system of the Einstein equations and of the general relativistic hydrodynamic equations, and on the application of this code to problems in general relativistic astrophysics. In particular, we report on the accuracy of our code in the long-term dynamical evolution of relativistic stars and on some new physics results obtained in the process of code testing. The following aspects of our code have been validated: the generation of initial data representing perturbed general relativistic polytropic models ͑both rotating and nonrotating͒, the long-term evolution of relativistic stellar models, and the coupling of our evolution code to analysis modules providing, for instance, the detection of apparent horizons or the extraction of gravitational waveforms. The tests involve single nonrotating stars in stable equilibrium, nonrotating stars undergoing radial and quadrupolar oscillations, nonrotating stars on the unstable branch of the equilibrium configurations migrating to the stable branch, nonrotating stars undergoing gravitational collapse to a black hole, and rapidly rotating stars in stable equilibrium and undergoing quasiradial oscillations. We have carried out evolutions in full general relativity and compared the results to those obtained either with perturbation techniques, or with lower dimensional numerical codes, or in the Cowling approximation ͑in which all the perturbations of the spacetime are neglected͒. In all cases an excellent agreement has been found. The numerical evolutions have been carried out using different types of polytropic equations of state using either the rest-mass density only, or the rest-mass density and the internal energy as independent variables. New variants of the spacetime evolution and new high resolution shock capturing treatments based on Riemann solvers and slope limiters have been implemented and the results compared with those obtained from previous methods. In particular, we have found the ''monotonized central differencing'' limiter to be particularly effective in evolving the relativistic stellar models considered. Finally, we have obtained the first eigenfrequencies of rotating stars in full general relativity and rapid rotation. A long standing problem, such frequencies have not been obtained by other methods. Overall, and to the best of our knowledge, the results presented in this paper represent the most accurate long-term three-dimensional evolutions of relativistic stars available to date.

This is the first in a series of papers on the construction and validation of a three-dimensional code for general relativistic hydrodynamics, and its application to general relativistic astrophysics. This paper studies the consistency and convergence of our general relativistic hydrodynamic treatment and its coupling to the spacetime evolutions described by the full set of Einstein equations with a perfect fluid source, complimenting a similar study of the ͑vacuum͒ spacetime part of the code. The numerical treatment of the general relativistic hydrodynamic equations is based on high resolution shock capturing schemes, specifically designed to solve non-linear hyperbolic systems of conservation laws. These schemes rely on the characteristic information of the system. A spectral decomposition for general relativistic hydrodynamics suitable for a general spacetime metric is presented. Evolutions based on different approximate Riemann solvers ͑flux-splitting, Roe, and Marquina͒ are studied and compared. The coupling between the hydrodynamics and the spacetime ͑the right and left hand side of the Einstein equations͒ is carried out in a treatment which is second order accurate in both space and time. The spacetime evolution allows for a choice of different formulations of the Einstein equations, and different numerical methods for each formulation. Together with the different hydrodynamical methods, there are twelve different combinations of spacetime and hydrodynamical evolutions. Convergence tests for all twelve combinations with a variety of test beds are studied, showing consistency with the differential equations and correct convergence properties. The test-beds examined include shock tubes, Friedmann-RobertsonWalker cosmology tests, evolutions of self-gravitating compact ͑TOV͒ stars, and evolutions of relativistically boosted TOV stars. Special attention is paid to the numerical evolution of strongly gravitating objects, e.g., neutron stars, in the full theory of general relativity, including a simple, yet effective treatment for the surface region of the star ͑where the rest mass density is abruptly dropping to zero͒.PACS number͑s͒: 04.25. Dm, 47.75.ϩf, 95.30.Sf, 97.60.Jd

This paper reports on our effort in modeling realistic astrophysical neutron star binaries in general relativity. We analyze under what conditions the conformally flat quasiequilibrium (CFQE) approach can generate "astrophysically relevant" initial data, by developing an analysis that determines the violation of the CFQE approximation in the evolution of the binary described by the full Einstein theory. We show that the CFQE assumptions significantly violate the Einstein field equations for corotating neutron stars at orbital separations nearly double that of the innermost stable circular orbit (ISCO) separation, thus calling into question the astrophysical relevance of the ISCO determined in the CFQE approach.With the need to start numerical simulations at large orbital separation in mind, we push for stable and long term integrations of the full Einstein equations for the binary neutron star system. We demonstrate the stability of our numerical treatment and analyze the stringent requirements on resolution and size of the computational domain for an accurate simulation of the system.

We describe a new numerical 3D relativistic hydrodynamical code and present the results of three validation tests. A comparison of an axisymmetric jet simulation using the 3D code, with corresponding results from an earlier 2D code, reveal that a) the enforcement of axisymmetry in the 2D case had no significant influence on the global morphology and dynamics; b) although 3D studies typically have lower resolution than those using 2D, limiting their ability to fully capture internal jet structure, such 3D studies can provide a reliable model of global morphology and dynamics.The 3D code has been used to study the deflection and precession of relativistic flows. We find that even quite fast jets (γ ∼ 10) can be significantly influenced by impinging on an oblique density gradient, exhibiting a rotation of the Mach disk in the jet's head. The flow is bent via a potentially strong, oblique internal shock that arises due to asymmetric perturbation of the flow by its cocoon. In extreme cases this cocoon can form a marginally relativistic flow orthogonal to the jet, leading to large scale dynamics quite unlike that normally associated with astrophysical jets. Exploration of a γ = 5 flow subject to a

We propose a reformulation of the Einstein evolution equations that cleanly separates the conformal degrees of freedom and the nonconformal degrees of freedom with the latter satisfying a first order strongly hyperbolic system. The conformal degrees of freedom are taken to be determined by the choice of slicing and the initial data, and are regarded as given functions ͑along with the lapse and the shift͒ in the hyperbolic part of the evolution. We find that there is a two parameter family of hyperbolic systems for the nonconformal degrees of freedom for a given set of trace free variables. The two parameters are uniquely fixed if we require the system to be ''consistently trace-free,'' i.e., the time derivatives of the trace free variables remain trace-free to the principal part, even in the presence of constraint violations due to numerical truncation error. We show that by forming linear combinations of the trace free variables a conformal hyperbolic system with only physical characteristic speeds can also be constructed. ͓S0556-2821͑99͒05516-2͔ PACS number͑s͒: 04.25.Dm

Abstract. We perform fully relativistic hydrodynamic simulations of the deceleration and lateral expansion of a relativistic jet as it expands into an ambient medium. The hydrodynamic calculations use a 2D adaptive mesh refinement (AMR) code, which provides adequate resolution of the thin shell of matter behind the shock. We find that the sideways propagation is different than predicted by simple analytic models. The physical conditions at the sides of the jet are found to be significantly different than at the front of the jet, and most of the emission occurs within the initial opening angle of the jet. The light curves, as seen by observers at different viewing angles with respect to the jet axis, are then calculated assuming synchrotron emission. For an observer along the jet axis, we find a sharp achromatic 'jet break' in the light curve at frequencies above the typical synchrotron frequency, at tjet ≈ 5.8(E52/n1) 1/3 (θ0/0.2) 8/3 days, while the temporal decay index α (Fν ∝ t α ) after the break is steeper than −p (α = −2.85 for p = 2.5). At larger viewing angles tjet increases and the jet break becomes smoother. The HydrodynamicsThe hydrodynamic simulations are performed using a fully relativistic 2D AMR code. The jet propagates into a cold and homogeneous ambient medium of number density n 1 cm −3 . We use n 1 = 1, typical of the ISM. We assume an adiabatic evolution (i.e. that radiation losses do not influence the dynamics). For the initial conditions, we use a wedge of opening angle θ 0 = 0.2 taken out of the Blandford McKee [1] self similar spherical solution, with an isotropic equivalent energy of E 52 10 52 ergs, where we use E 52 = 1. Since the lateral expansion is expected to become significant only when the Lorentz factor of the flow drops to γ ∼ 1/θ 0 = 5, the initial Lorentz factor of the shock was chosen to be Γ = √ 2γ = 23.7. Figure (1) shows a 3D view of the jet at the last time step of the simulation, with color-maps of the proper number density, n ′ , and proper emissivity. Primed quantities are measured in the local rest frame of the fluid. While the number density does not change significantly between the front and the sides of the jet, the emissivity is significant only at the front of the jet, and drops sharply at angles larger than the initial opening angle, θ 0 . The overall egg-shaped structure is very different from the quasi-spherical structure assumed in 1D analytic models. The Light CurvesThe synchrotron spectral emissivity, P ′ ν ′ , is calculated using the physical conditions determined by the hydrodynamic simulation. The electrons and the magnetic field are assumed to hold fractions ǫ e and ǫ B , respectively, of the internal

The convergence properties of numerical Regge calculus as an approximation to continuum vacuum General Relativity is studied, both analytically and numerically. The Regge equations are evaluated on continuum spacetimes by assigning squared geodesic distances in the continuum manifold to the squared edge lengths in the simplicial manifold. It is found analytically that, individually, the Regge equations converge to zero as the second power of the lattice spacing, but that an average over local Regge equations converges to zero as (at the very least) the third power of the lattice spacing. Numerical studies using analytic solutions to the Einstein equations show that these averages actually converge to zero as the fourth power of the lattice spacing.

We study the question of prompt vs. delayed collapse in the head-on collision of two neutron stars. We show that the prompt formation of a black hole is possible, contrary to a conjecture of Shapiro which claims that collapse is delayed until after neutrino cooling. We discuss the insight provided by Shapiro's conjecture and its limitation. An understanding of the limitation of the conjecture is provided in terms of the many time scales involved in the problem. General relativistic simulations in the Einstein theory with the full set of Einstein equations coupled to the general relativistic hydrodynamic equations are carried out in our study.