5.3 Single black holes

Compared to the compact star case, single black holes have not been studied very much. This is probably because the structure of a stationary black hole is somewhat simpler than the one of a compact star. However, as will be seen, there are still properties that must be investigated.

Spacetimes containing a single black hole constitute a good benchmark for numerical methods, a lot of results being known analytically. In [125Jump To The Next Citation Point], the authors have implemented a spectral solver and applied it to various test problems. The solver itself is two dimensional and thus applicable only to axisymmetric systems. There is a single domain that consists of the whole space outside a sphere of given radius (i.e. the black hole). Space is compactified by using the standard variable 1∕r. The two physical variables (r,𝜃) are mapped onto squares in ℝ2 and then expanded in terms of Chebyshev polynomials. The equations are written using a 2D collocation method (see Section 2.5.3) and the resulting system is solved by an iterative algorithm (here Richardson’s method with preconditioning). This solver is applied to solve Einstein’s constraint equations for three different systems: i) a single black hole ii) a single black hole with angular momentum iii) a black hole plus Brill waves. In all three cases, spectral convergence is achieved and accuracy on the order of 10–10 is reached with 30 points in each dimension.

A black hole is somewhat simpler than a neutron star, in the sense that there is no need for a description of matter (no equation of state, for instance). However, in some formalisms, the presence of a black hole is enforced by imposing a nontrivial solution on a surface (typically a sphere). The basic idea is to demand that the surface be a trapped surface. Such surfaces are known to lie inside event horizons and so are consistent with the presence of a black hole. Discussions about such boundary conditions can be found in [62Jump To The Next Citation Point]. However, in nonstationary cases, the set of equations to be used is not easy to derive. The authors of [123] implement various sets of boundary conditions to investigate their properties. Two different and independent spectral codes are used. Both codes are very close to those used in the case of neutron stars, one of them based on Lorene library [99Jump To The Next Citation Point] (see Section 5.2.1) and the other one developed by Ansorg and sharing a lot a features with [9Jump To The Next Citation Point, 10Jump To The Next Citation Point]. Such numerical tools have proved useful in clarifying the properties of some sets of boundary conditions that could be imposed on black hole horizons.

The reverse problem is also important in the context of numerical relativity. In some cases one needs to know if a given configuration contains a trapped surface and if it can be located at each timestep. Several algorithms have been proposed in the past to find the locus at which the expansion of the outgoing light rays vanishes (thus defining the trapped surface). Even if the term is not used explicitly, the first application of spectral expansions to this problem is detailed in [23]. The various fields are expanded in a basis of symmetric trace-free tensors. The algorithm is applied to spacetimes containing one or two black holes. However, results seem to indicate that high order expansions are required to locate horizons with a sufficient precision.

More recently, another code [135] using spectral methods has been used to locate apparent horizons. It is based on the Lorene library with its standard setting, i.e. a multidomain decomposition of space and spherical coordinates (see Section 5.2.1 for more details). The horizon finder has been successfully tested on known configurations, like Kerr–Schild black holes. The use of spectral methods makes it both fast and accurate. Even if the code uses only one set of spherical coordinates (hence its presentation in this section), it can be applied to situations with more than one black hole, like the well-known Brill–Lindquist data [50Jump To The Next Citation Point].

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