In this section, we will look at Cauchy initial data that represent one or more black holes in an asymptotically flat spacetime. The majority of these will be either vacuum solutions or solutions of the Einstein-Maxwell equations. With no matter to support the gravitational field, we find that we must usually use a spacetime with a non-trivial topology, although a black hole can be supported by a compact gravitational wave [2]. It is certainly possible to construct black-hole solutions supported by matter [90, 91, 5], but it is often desirable to avoid the complications of matter sources.

This raises a point about solutions of Einstein’s equations which we have not yet mentioned. When constructing solutions of Einstein’s initial-value equations, we are free to specify the topology of the initial-data hypersurface. Einstein’s equations of general relativity place no constraints on the topology of the spacetime they describe or of spacelike hypersurfaces that foliate it. For astrophysical black holes (i.e., black holes in an asymptotically flat spacetime), the freedom in the choice of the topology has relatively minor consequences. The primary effects of different topology choices are hidden within the black hole’s event horizon.

In the sections below, we will explore many of the existing black-hole solutions and the schemes for generating them.

3.1 Classic solutions

3.1.1 Schwarzschild

3.1.2 Time-symmetric multi-hole solutions

3.2 General multi-hole solutions

3.2.1 Bowen–York data

3.2.2 Puncture data

3.2.3 Problems with these data

3.3 Horizon-penetrating solutions

3.3.1 Kerr–Schild coordinates

3.3.2 Harmonic coordinates

3.3.3 Generalized Painlevé–Gullstrand coordinates

3.4 Quasicircular binary data

3.1.1 Schwarzschild

3.1.2 Time-symmetric multi-hole solutions

3.2 General multi-hole solutions

3.2.1 Bowen–York data

3.2.2 Puncture data

3.2.3 Problems with these data

3.3 Horizon-penetrating solutions

3.3.1 Kerr–Schild coordinates

3.3.2 Harmonic coordinates

3.3.3 Generalized Painlevé–Gullstrand coordinates

3.4 Quasicircular binary data

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