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 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).
Such conditions say that some physical field or mathematical function is asymptotically vanishing in a suitable sense.
[citation needed] In general relativity, an asymptotically flat vacuum solution models the exterior gravitational field of an isolated massive object.
[citation needed] 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.
[2] Only spacetimes which model an isolated object are asymptotically flat.
A simple example of an asymptotically flat spacetime is the Schwarzschild metric solution.
But another well known generalization of the Schwarzschild vacuum, the Taub–NUT space, is not asymptotically flat.
An even simpler generalization, the de Sitter-Schwarzschild metric 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 metrics 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 in terms of an explicit multipole expansion.
, which far from the origin behaves much like a Cartesian chart on Minkowski spacetime, in the following sense.
Then we require: One reason why we require the partial derivatives of the perturbation to decay so quickly is that these conditions turn out to imply that the gravitational field energy density (to the extent that this somewhat nebulous notion makes sense in a metric theory of gravitation) decays like
Around 1962, Hermann Bondi, Rainer K. Sachs, and others began to study the general phenomenon of radiation from a compact source in general relativity, which requires more flexible definitions of asymptotic flatness.
In 1963, Roger Penrose imported from algebraic geometry the essential innovation, now called conformal compactification, and in 1972, Robert Geroch used this to circumvent the tricky problem of suitably defining and evaluating suitable limits in formulating a truly coordinate-free definition of asymptotic flatness.
In the new approach, once everything is properly set up, one need only evaluate functions on a locus in order to verify asymptotic flatness.
The notion of asymptotic flatness is extremely useful as a technical condition in the study of exact solutions in general relativity and allied theories.