Numerical relativity has faced the problem that standard 3ϩ1 simulations of black hole spacetimes without singularity excision and with singularity avoiding lapse and vanishing shift fail early on due to the so-called slice stretching. We discuss lapse and shift conditions for the nonexcision case that effectively cure slice stretching and allow run times of 1000M and more.

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We study the stability of three-dimensional numerical evolutions of the Einstein equations, comparing the standard ADM formulation to variations on a family of formulations that separate out the conformal and traceless parts of the system. We develop an implementation of the conformal-traceless ͑CT͒ approach that has improved stability properties in evolving weak and strong gravitational fields, and for both vacuum and spacetimes with active coupling to matter sources. Cases studied include weak and strong gravitational wave packets, black holes, boson stars and neutron stars. We show under what conditions the CT approach gives better results in 3D numerical evolutions compared to the ADM formulation. In particular, we show that our implementation of the CT approach gives more long term stable evolutions than ADM in all the cases studied, but is less accurate in the short term for the range of resolutions used in our 3D simulations.

With the goal of bringing theory, particularly numerical relativity, to bear on an astrophysical problem of critical interest to gravitational wave observers we introduce a model for coalescence radiation from binary black hole systems. We build our model using the Lazarus approach, a technique that bridges far and close limit approaches with full numerical relativity to solve Einstein equations applied in the truly nonlinear dynamical regime. We specifically study the post-orbital radiation from a system of equal-mass non-spinning black holes, deriving waveforms which indicate strongly circularly polarized radiation of roughly 3% of the system's total energy and 12% of its total angular momentum in just a few cycles. Supporting this result we first establish the reliability of the late-time part of our model, including the numerical relativity and close-limit components, with a thorough study of waveforms from a sequence of black hole configurations varying from previously treated head-on collisions to representative target for "ISCO" data corresponding to the end of the inspiral period. We then complete our model with a simple treatment for the early part of the spacetime based on a standard family of initial data for binary black holes in circular orbit. A detailed analysis shows strong robustness in the results as the initial separation of the black holes is increased from 5.0 to 7.8M supporting our waveforms as a suitable basic description of the astrophysical radiation from this system. Finally, a simple fitting of the plunge waveforms is introduced as a first attempt to facilitate the task of analyzing data from gravitational wave detectors.

We present a detailed analysis of binary black hole evolutions in the last orbit, and demonstrate consistent and convergent results for the trajectories of the individual bodies. The gauge choice can significantly affect the overall accuracy of the evolution. It is possible to reconcile certain gauge dependent discrepancies by examining the convergence limit. We illustrate these results using an initial data set recently evolved by Brügmann et al. (Phys. Rev. Lett. 92, 211101). For our highest resolution and most accurate gauge, we estimate the duration of this data set's last orbit to be approximately 59MADM.PACS numbers: 04.25. Dm, 04.30.Db, 04.70.Bw, 95.30.Sf, 97.60.Lf Introduction.Over the course of the next decade, instruments capable of detecting gravitational radiation (such as LIGO, VIRGO, TAMA, GEO600) are expected to open a new observational window on the universe. The collision of binary compact objects such as black holes (BHs) is one of the most promising sources for first generation gravitational wave observatories. The theoretical framework for modelling binary BH systems is the complete set of nonlinear Einstein equations. Intensive efforts to develop numerical codes able to solve these equations using supercomputers, have shown that it is now possible to evolve BHs for periods of an orbit [1,2,3]. If these simulations are to produce waveforms useful for detector searches, high demands are placed on their accuracy [4].The near-zone dynamics of binary BH systems are notoriously difficult to simulate, and to analyse. Using the BSSN formulation and a particular set of gauges, a series of BBH configurations, corresponding to initial data in quasi-circular orbit at successively larger separations [5], were all found to coalesce in slightly more than a half orbit [2]. A similar BSSN evolution carried out using somewhat different gauges and numerical methods for another data set, slightly further out along the orbital sequence, was found to evolve for much more than the estimated orbital timescale 114M ADM without finding a common apparent horizon (AH) [1]. In fact, no common horizon was found long after the BHs would reasonably be expected to have merged.In this paper, we carry out an evolution of the same data set and show that it does indeed carry out a complete orbit before a common AH forms. As the BH separation decreases, a local measure of the angular velocity Ω increases, so that the duration of the final orbit is approximately 59M . The trajectories are convergent for a range of resolutions and within a class of gauge conditions. However, we do find that very high resolutions are required in order to obtain evolutions close to the continuum limit. The resolutions we have applied here are significantly higher than those used in analogous BH evo-

We present a new technique for the numerical simulation of axisymmetric systems. This technique avoids the coordinate singularities which often arise when cylindrical or polar-spherical coordinate finite difference grids are used, particularly in simulating tensor partial differential equations like those of 3 + 1 numerical relativity. For a system axisymmetric about the z axis, the basic idea is to use a 3-dimensional Cartesian (x, y, z) coordinate grid which covers (say) the y = 0 plane, but is only one finite-difference-molecule-width thick in the y direction. The field variables in the central y = 0 grid plane can be updated using normal (x, y, z)-coordinate finite differencing, while those in the y = 0 grid planes can be computed from those in the central plane by using the axisymmetry assumption and interpolation. We demonstrate the effectiveness of the approach on a set of fully nonlinear test computations in 3 + 1 numerical general relativity, involving both black holes and collapsing gravitational waves.

We extend the previous work on 3D black hole excision to the case of distorted black holes, with a variety of dynamic gauge conditions that respond naturally to the spacetime dynamics. We show that in evolutions of highly distorted, rotating black holes, the combination of excision and the gauge conditions we use is able to drive the coordinates to a frame in which the system looks almost static at late times. Further, we show for the first time that one can extract accurate wave forms from these simulations, with the full machinery of excision and dynamic gauge conditions. The evolutions can be carried out for a long time, far exceeding the longevity and accuracy of better resolved 2D codes.