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5.4 Waveform extraction

The prospect of receiving gravitational waves offers exciting opportunities for astrophysics and for testing the dynamical, strong field regime of general relativity (see, e.g., [77]). Black holes are among the most promising sources both for terrestrial and space-based observatories. Extracting gravitational waveform resulting from, say, gravitational collapse [153] or black hole mergers [139] is one of the important goals of numerical relativity. The eventual aim is to provide waveforms which, in conjunction with gravitational wave detectors, can be used to study the astrophysics of these gravitational wave sources. In this section, we summarize a method of waveform extraction based on the isolated horizon formalism.

The theory of gravitational radiation in exact general relativity is based on structures defined at future null infinity ℐ+. In particular, associated with every cross-section of ℐ+ – which represents a retarded ‘instant of time’ – there is a well defined notion of mass, introduced by Bondi [52Jump To The Next Citation Point], which decreases as gravitational radiation flows across ℐ+. On the other hand, except for those based on conformal methods, most simulations only deal with a finite portion of space-time and thus have no direct access to + ℐ. Instead, one usually uses scalars such as the Weyl tensor component Ψ4 to define the radiation waveform. However, Ψ4 depends on the choice of a null tetrad as well as coordinates. While a natural tetrad is available for the perturbation theory on Kerr back ground, this is not true in general. In this section we will sketch an approach to solve both these problems using the isolated horizon framework: One can construct an approximate analog of + ℐ for a suitable, finite portion of space-time, and introduce a geometrically defined null tetrad and coordinates to extract gauge invariant, radiative information from simulations.

View Image

Figure 8: Bondi-like coordinates in a neighborhood of Δ.

Our procedure is analogous to the one used at null infinity to construct the Bondi-type coordinates starting from ℐ+. Let us assume that (Δ, [ℓ]) is an IH and consider its preferred foliation (i.e., the “rest frame” of the horizon) defined in Section 2.1.3. Using terminology from null infinity, we will refer to the leaves of the foliation as the good cuts of Δ. Pick a null normal a ℓ from the a [ℓ ]; this can be done, e.g., by choosing a specific value for surface gravity. Let na be the unique 1-form satisfying ℓana = − 1 and which is orthogonal to the good cuts. Let (v, θ,φ) be coordinates on Δ such that v is an affine parameter along ℓa, and the good cuts are given by surfaces of constant v. Here θ and φ are coordinates on the good cut satisfying ℒ ℓθ = 0 and ℒ ℓφ = 0. Next, consider past null geodesics emanating from the good cuts with a − n as their initial tangent vector. Let the null geodesics be affinely parameterized and let the affine parameter be called r and set r = r0 on Δ. Lie drag the coordinates (v,θ, φ) along null geodesics. This leads to a set of coordinates (r,v,θ,φ ) in a neighborhood of the horizon. The only arbitrariness in this coordinate system is in the choice of (θ,φ ) on one good cut and the choice of a number r0. Using vacuum Einstein’s equations, one can obtain a systematic expansion of the metric in inverse powers of (r − r0) [11].

One can also define a null tetrad in the neighborhood in a similar fashion. Let ma be an arbitrary complex vector tangent to the good cuts such that (ℓ,n,m, ¯m ) is a null tetrad on Δ. Using parallel transport along the null geodesics, we can define a null tetrad in the neighborhood. This tetrad is unique up to the spin rotations of m: m → e2iθm on the fiducial good cut. This construction is shown in Figure 8View Image. The domain of validity of the coordinate system and tetrads is the space-time region in which the null geodesics emanating from good cuts do not cross.

Numerically, it might be possible to adapt existing event horizon finders to locate the outgoing past null cone of a cross-section of the horizon. This is because event horizon trackers also track null surfaces backward in time [81Jump To The Next Citation Point82]; the event horizon is the ingoing past null surface starting from sufficiently close to future timelike infinity while here we are interested in the outgoing past directed null surface starting from an apparent horizon.

In the Schwarzschild solution, it can be shown analytically that this coordinate system covers an entire asymptotic region. In the Kerr space-time, the domain of validity is not explicitly known, but in a numerical implementation, this procedure does not encounter geodesic crossing in the region of interest to the simulation [81]. In general space-times, the extraction of wave forms requires the construction to go through only along the past light cones of good cuts which lie in the distant future, whence problems with geodesic-crossing are unlikely to prevent one from covering a sufficiently large region with these coordinates. This invariantly defined structure provides a new approach to extract waveforms. First, the null tetrad presented above can be used to calculate Ψ4. There is only a phase factor ambiguity (which is a function independent of v and r) inherited from the ambiguity in the choice of ma on the fiducial good cut. Second, the past null cone of a good cut at a sufficiently late time can be used as an approximate null infinity. This should enable one to calculate dynamical quantities such as the analogs of the Bondi mass and the rate of energy loss from the black hole, now on the ‘approximate’ ℐ+. However, a detailed framework to extract the approximate expressions for fluxes of energy and Bondi mass, with sufficient control on the errors, is yet to be developed. This is a very interesting analytical problem since its solution would provide numerical relativists with an algorithm to extract waveforms and fluxes of energy in gravitational waves in an invariant and physically reliable manner. Finally, the invariant coordinates and tetrads also enable one to compare late time results of distinct numerical simulations.


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