Vacuum Solution (general Relativity) - Examples

Examples

Well known examples of explicit vacuum solutions include:

• Minkowski spacetime (which describes empty space with no cosmological constant)
• Milne model (which is a model developed by E. A. Milne describing an empty universe which has no curvature)
• Schwarzschild vacuum (which describes the spacetime geometry around a spherical mass),
• Kerr vacuum (which describes the geometry around a rotating object),
• Taub-NUT vacuum (a famous counterexample describing the exterior gravitational field of an isolated object with strange properties),
• Kerns–Wild vacuum (Robert M. Kerns and Walter J. Wild 1982) (a Schwarzschild object immersed in an ambient "almost uniform" gravitational field),
• double Kerr vacuum (two Kerr objects sharing the same axis of rotation, but held apart by unphysical zero active mass "cables" going out to suspension points infinitely removed),
• Khan–Penrose vacuum (K. A. Khan and Roger Penrose 1971) (a simple colliding plane wave model),
• Oszváth–Schücking vacuum (the circularly polarized sinusoidal gravitational wave, another famous counterexample).
• Kasner metric

These all belong to one or more general families of solutions:

• the Weyl vacua (Hermann Weyl) (the family of all static vacuum solutions),
• the Beck vacua (Guido Beck 1925) (the family of all cylindrically symmetric nonrotating vacuum solutions),
• the Ernst vacua (Frederick J. Ernst 1968) (the family of all stationary axisymmetric vacuum solutions),
• the Ehlers vacua (Jürgen Ehlers) (the family of all cylindrically symmetric vacuum solutions),
• the Szekeres vacua (George Szekeres) (the family of all colliding gravitational plane wave models),
• the Gowdy vacua (Robert H. Gowdy) (cosmological models constructed using gravitational waves),

Several of the families mentioned here, members of which are obtained by solving an appropriate linear or nonlinear, real or complex partial differential equation, turn out to be very closely related, in perhaps surprising ways.

In addition to these, we also have the vacuum pp-wave spacetimes, which include the gravitational plane waves.