5 Applications in Numerical
Relativity
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, 177]), black hole excision
techniques have been successfully implemented [73
, 3
], and the stability
of numerical codes has steadily improved [64]. 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.