The first application of spectral methods to black-hole–neutron-star binaries can be found in . The main approximation is to consider that the black hole is not influenced by the neutron star. Technically, this means that Einstein’s equations are split into two parts (i.e. as for neutron star binaries 5.5.2). However, the part of the fields associated with the black hole is fixed to its analytical value. As the fields are not solved for the black-hole part, the results should depend on the actual splitting, the equations being nonlinear. The part of the fields associated with the neutron star are solved using the standard setting for the Meudon group. Of course, this whole procedure is only valid if the black hole is much more massive than the neutron star and this is why  is limited to mass ratios of 10. In this particular case, it is shown that the results depend, to the level of a few percent, on the choice of splitting, which is a measure of the reached accuracy. It is also shown that the state of rotation of the star (i.e. synchronized or irrotational) has little influence on the results.
In  the equations of the extended thin-sandwich formulation are solved consistently. As for the neutron-star–binary case, two sets of spherical coordinates are used, one centered around each object. The freely specifiable variables are derived from the Kerr–Schild approach. Configurations are obtained with a moderate mass ratio of five. However, the agreement with post-Newtonian results is not very good and the data seem to be rather noisy (especially the deformation of the star).
Quasiequilibrium configurations based on a helical Killing vector and conformal flatness have been obtained independently by  and . Both codes are based on the Lorene library  and use two sets of spherical coordinates. They differ mainly in their choice of boundary conditions for the black hole. However, it is shown in the erratum of  that the results match pretty well and are in very good agreement with post-Newtonian results. Mass ratios ranging from 1 to 10 are obtained in  and the emitted energy spectra are estimated. The work of  has been extended in , where the parameter space of the binary is extensively explored. In particular, the authors determine whether the end point of the sequences is due to an instability or to the mass-shedding limit. It turns out that the star is more likely to reach the mass-shedding limit if it is less compact and if the mass ratio between the black hole and the star is important, as expected.
More recently, the Caltech/Cornell group has applied the spectral solver of [171, 167] in order to compute black-hole–neutron-star configurations . Some extensions have been made to enable the code to deal with matter by making use of surface-fitting coordinates. Thanks to the domain decomposition used (analogous to the one of [171, 167]), the authors of  can reach an estimated accuracy , which is better than the precision of previous works (by roughly an order of magnitude). Configurations with one spinning black hole and configurations with reduced eccentricity are also presented, along the lines of .
Living Rev. Relativity 12, (2009), 1
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