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Excision techniques are used in order to deal with black holes in numerical simulations of Einstein equations and consist in removing a topological sphere containing the physical singularity from the numerical domain, applying instead appropriate boundary conditions at the excised surface. In this work we present recent developments of this technique in the case of constrained formulations of Einstein equations and for spherically symmetric spacetimes. We present a new set of boundary conditions to apply to the elliptic system in the fully-constrained formalism of Bonazzola et al. (2004), at an arbitrary coordinate sphere inside the apparent horizon. Analytical properties of this system of boundary conditions are studied and, under some assumptions, an exponential convergence toward the stationary solution is exhibited for the vacuum spacetime. This is verified in numerical examples, together with the applicability in the case of the accretion of a scalar field onto a Schwarzschild black hole. We also present the successful use of the excision technique in the collapse of a neutron star to a black hole, when excision is switched on during the simulation, after the formation of the apparent horizon. This allows the accretion of matter remaining outside the excision surface and for the stable long-term evolution of the newly formed black hole., Comment: 14 pages, 9 figures. New section added and changes included according to published article

Non-continuous "jumps" of Apparent Horizons occur generically in 3+1 (binary) black hole evolutions. The dynamical trapping horizon framework suggests a spacetime picture in which these "Apparent Horizon jumps" are understood as spatial cuts of a single spacetime hypersurface foliated by (compact) marginally outer trapped surfaces. We present here some work in progress which makes use of uni-parametric sequences of (axisymmetric) binary black hole initial data for exploring the plausibility of this spacetime picture. The modelling of Einstein evolutions by sequences of initial data has proved to be a successful methodological tool in other settings for the understanding of certain qualitative features of evolutions in restricted physical regimes., Comment: Contribution to the proceedings volume of the Spanish Relativity Meeting 2008: Physics and Mathematics of Gravitation, Salamanca, Spain, 15-19 Sep 2008

Using a constrained formalism for Einstein equations in Dirac gauge, we propose to compute excised quasistationary initial data for black hole spacetimes in full general relativity. Vacuum spacetime settings are numerically constructed by using the isolated horizon formalism; we especially tackle the conformal metric part of our equations, assuming global stationarity. We show that a no-boundary treatment can be used on the horizon for the equation related to the conformal metric. We relate this finding to previous suggestions in the literature, and use our results to assess the widely used conformally flat approximation for computing black hole initial data., Comment: 3 pages, 1 figure; to appear in the proceedings of the 12th Marcel Grossmann meeting on general relativity

The wave equation for vectors and symmetric tensors in spherical coordinates is studied under the divergence-free constraint. We describe a numerical method, based on the spectral decomposition of vector/tensor components onto spherical harmonics, that allows for the evolution of only those scalar fields which correspond to the divergence-free degrees of freedom of the vector/tensor. The full vector/tensor field is recovered at each time-step from these two (in the vector case), or three (symmetric tensor case) scalar fields, through the solution of a first-order system of ordinary differential equations (ODE) for each spherical harmonic. The correspondence with the poloidal-toroidal decomposition is shown for the vector case. Numerical tests are presented using an explicit Chebyshev-tau method for the radial coordinate., Comment: 25 pages, 4 figures, to appear in Journal of Computational Physics

We present a numerical work aiming at the computation of excised initial data for black hole spacetimes in full general relativity, using Dirac gauge in the context of a constrained formalism for the Einstein equations. Introducing the isolated horizon formalism for black hole excision, we especially solve the non-conformally flat part of the equations, and assess the boundary condition problem for this part. In the stationary single black hole case, we present and justify a no-boundary treatment on the black hole horizon. We compare the data obtained with the well-known analytic Kerr solution in Kerr-Schild coordinates, and assess the widely used conformally flat approximation for simulating axisymmetric black hole spacetimes. Our method shows good concordance on physical and geometrical issues, with the particular application of the isolated horizon multipolar analysis to confirm that the solution obtained is indeed the Kerr spacetime. Finally, we discuss a previous suggestion in the literature for the boundary conditions for the conformal geometry on the horizon., Comment: 15 pages, 5 figures. Part IV significantly expanded, and additional appendix included. Version coherent with the published paper