By their very nature, numerical simulations of space-times are invariably tied to choices of coordinates, gauge conditions, dynamical variables, etc. Therefore, it is non-trivial to extract from them gauge invariant physics, especially in the strong curvature regions. Traditionally, the analytical infrastructure available for this purpose has been based on properties of the Kerr solution and its perturbations. However, a priori it is not clear if this intuition is reliable in the fully dynamical, strong curvature regime. On the numerical side, a number of significant advances have occurred in this area over the past few years. In particular, efficient algorithms have been introduced to find apparent horizons (see, e.g., [6, 168, 178]), black hole excision techniques have been successfully implemented [71, 3], and the stability of numerical codes has steadily improved [62]. To take full advantage of these ongoing improvements, one must correspondingly ‘upgrade’ the analytical infra-structure so that one can extract physics more reliably and with greater accuracy.

These considerations provided stimulus for a significant body of research at the interface of numerical relativity and the dynamical and isolated horizon frameworks. In this section, we will review the most important of these developments. Section 5.1 summarizes calculations of mass and angular momentum of black holes. Section 5.2 discusses applications to problems involving initial data. Specifically, we discuss the issue of constructing the ‘quasi-equilibrium initial data’ and the calculation of the gravitational binding energy for a binary black hole problem. Section 5.3 describes how one can calculate the source multipole moments for black holes, and Section 5.4 presents a ‘practical’ approach for extracting gauge invariant waveforms. Throughout this section we assume that vacuum equations hold near horizons.

5.1 Numerical computation of black hole mass and angular momentum

5.2 Initial data

5.2.1 Boundary conditions at the inner boundary

5.2.2 Binding energy

5.3 Black hole multipole moments

5.4 Waveform extraction

5.2 Initial data

5.2.1 Boundary conditions at the inner boundary

5.2.2 Binding energy

5.3 Black hole multipole moments

5.4 Waveform extraction

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