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 [81
], 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 
. Then, the data are
constructed by dividing some quantities by this 
. This second step involves an additional difficulty: 
vanishes at infinity but the ratios are finite and smooth. It is demonstrated in [81] 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 [81] for more
details).
