Asymptotically flat spacetime

An asymptotically flat spacetime is a Lorentzian manifold in which, roughly speaking, the curvature vanishes at large distances from some region, so that at large distances, the geometry becomes indistinguishable from that of Minkowski spacetime. An asymptotically flat spacetime is a Lorentzian manifold in which, roughly speaking, the curvature vanishes at large distances from some region, so that at large distances, the geometry becomes indistinguishable from that of Minkowski spacetime. While this notion makes sense for any Lorentzian manifold, it is most often applied to a spacetime standing as a solution to the field equations of some metric theory of gravitation, particularly general relativity. In this case, we can say that an asymptotically flat spacetime is one in which the gravitational field, as well as any matter or other fields which may be present, become negligible in magnitude at large distances from some region. In particular, in an asymptotically flat vacuum solution, the gravitational field (curvature) becomes negligible at large distances from the source of the field (typically some isolated massive object such as a star). The condition of asymptotic flatness is analogous to similar conditions in mathematics and in other physical theories. Such conditions say that some physical field or mathematical function is asymptotically vanishing in a suitable sense. In general relativity, an asymptotically flat vacuum solution models the exterior gravitational field of an isolated massive object. Therefore, such a spacetime can be considered as an isolated system: a system in which exterior influences can be neglected. Indeed, physicists rarely imagine a universe containing a single star and nothing else when they construct an asymptotically flat model of a star. Rather, they are interested in modeling the interior of the star together with an exterior region in which gravitational effects due to the presence of other objects can be neglected. Since typical distances between astrophysical bodies tend to be much larger than the diameter of each body, we often can get away with this idealization, which usually helps to greatly simplify the construction and analysis of solutions. A manifold M {displaystyle M} is asymptotically simple if it admits a conformal compactification M ~ {displaystyle { ilde {M}}} such that every null geodesic in M {displaystyle M} has future and past endpoints on the boundary of M ~ {displaystyle { ilde {M}}} . Since the latter excludes black holes, one defines a weakly asymptotically simple manifold as a manifold M {displaystyle M} with an open set U ⊂ M {displaystyle Usubset M} isometric to a neighbourhood of the boundary of M ~ {displaystyle { ilde {M}}} , where M ~ {displaystyle { ilde {M}}} is the conformal compactification of some asymptotically simple manifold. A manifold is asymptotically flat if it is weakly asymptotically simple and asymptotically empty in the sense that its Ricci tensor vanishes in a neighbourhood of the boundary of M ~ {displaystyle { ilde {M}}} . 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.