5.9 Hyperboloidal initial data

If the 3+1 decomposition is the most widely used for numerical relativity, some other possibilities have been proposed, with possibly better features. In particular, one can vary the foliation of spacetime to get hyperboloidal data. With such a setting, at infinity spacetime is foliated by light cones instead of spatial hypersurfaces, which makes the extraction of gravitational waves, in principle, easier.

In [81Jump To The Next Citation Point], Frauendiener is interested in generating hyperboloidal initial-data sets from data in physical space. The technique proceeds in two steps. First a nonlinear partial differential equation (the Yamabe equation) must be solved to determine the appropriate conformal factor ω. Then, the data are constructed by dividing some quantities by this ω. This second step involves an additional difficulty: ω vanishes at infinity but the ratios are finite and smooth. It is demonstrated in [81Jump To The Next Citation Point] that spectral methods can deal with these two steps. Some symmetry is assumed so that the problem reduces to a two-dimensional one. The first variable is periodic and expanded in terms of a Fourier series, whereas Chebyshev polynomials are used for the other. The Yamabe equation is solved using an iterative scheme based on Richardson’s iteration procedure. The construction of the fields, and hence the division by a field vanishing at infinity, is then handled by making use of the nonlocal nature of the spectral expansion (i.e. by working in the coefficient space; see Section 4 of [81Jump To The Next Citation Point] for more details).

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