(ed.). Gravitational waves (IOP, 2001)(422s).

Tools for analysing the numerical spacetimes 387The theory of black hole perturbations is well developed. One identifiescertain perturbed metric quantities that evolve according to wave equationson the black hole background. These perturbed metric functions are alsodependent of the gauge in which they are computed. We use a gauge-invariantprescription for isolating wave modes on black hole background, developed firstby Moncrief [127]. The basic idea is that although the perturbed metric functionstransform under coordinate transformations (gauge transformations), one canidentify certain linear combinations of these functions that are invariant to firstorder of the perturbation. These gauge-invariant functions are the quantities thatcarry true physics, which does not depend on coordinate systems. They obey thewave equations describing waves propagating on the fixed black hole background.There are two independent wave modes, even- and odd-parity, corresponding tothe two degrees of freedom, or polarization modes, of the waves.A waveform extraction procedure has been developed that allows oneto process the metric and to identify the wave modes. The gravitationalwavefunction (often called the ‘Zerilli function’ for even-parity or the ‘Regge–Wheeler function’ for odd-parity) can be computed by writing the metric as thesum of a background black hole part and a perturbation:g αβ = o g αβ +h αβ (Y l,m ), (18.26)where the perturbation h αβ is expanded in spherical harmonics and theirotensor generalizations and the background part g αβ is spherically symmetric.To compute the elements of h αβ in a numerical simulation, one integratesthe numerically evolved metric components g αβ against appropriate sphericalharmonics over a coordinate 2-sphere surrounding the black hole, makinguse of the orthogonality properties of the tensor harmonics. This process isperformed for each l, m mode for which waveforms are desired. The resultingfunctions h αβ (Y l,m ) can then be combined in a gauge-invariant way, followingthe prescription given by Moncrief [127]. For each l, m mode, this gaugeinvariantgravitational waveform can be extracted when the wave passes through‘detectors’ at some fixed radius in the computational grid. This procedure hasbeen described in detail in [128–130], and more generally in [121, 131, 132]. Itworks amazingly well, allowing extraction of waves that carry very small energies(of order 10 −7 M or less, with M being the mass of the source) away from thesource. The procedure should apply to any isolated source of waves, such ascolliding black holes, neutron stars, etc. The spherical perturbation theory (witha few minor modifications) has also been applied to distorted rotating black holeswith satisfactory results [128–130].