Asymptotically Flat Spacetime - Some Examples and Nonexamples

Some Examples and Nonexamples

Only spacetimes which model an isolated object are asymptotically flat. Many other familiar exact solutions, such as the FRW dust models (which are homogeneous spacetimes and therefore in a sense at the opposite end of the spectrum from asymptotically flat spacetimes), are not.

A simple example of an asymptotically flat spacetime is the Schwarzschild vacuum solution. More generally, the Kerr vacuum is also asymptotically flat. But another well known generalization of the Schwarzschild vacuum, the NUT vacuum, is not asymptotically flat. An even simpler generalization, the Schwarzschild-de Sitter lambdavacuum solution (sometimes called the Köttler solution), which models a spherically symmetric massive object immersed in a de Sitter universe, is an example of an asymptotically simple spacetime which is not asymptotically flat.

On the other hand, there are important large families of solutions which are asymptotically flat, such as the AF Weyl vacuums and their rotating generalizations, the AF Ernst vacuums (the family of all stationary axisymmetric and asymptotically flat vacuum solutions). These families are given by the solution space of a much simplified family of partial differential equations, and their metric tensors can be written down (say in a prolate spheroidal chart) in terms of an explicit multipole expansion.