Asymptotically flat spacetime

An asymptotically flat spacetime is a

Minkowski spacetime

.

While this notion makes sense for any Lorentzian manifold, it is most often applied to a

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).^[1]

Intuitive significance

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.^[citation needed]

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.^{[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.

Formal definitions

A manifold $M$ is asymptotically simple if it admits a

conformal compactification

{\tilde {M}}

such that every null geodesic in

M

has future and past endpoints on the boundary of

{\tilde {M}}

.

Since the latter excludes black holes, one defines a weakly asymptotically simple manifold as a manifold $M$ with an open set $U\subset M$ isometric to a neighbourhood of the boundary of ${\tilde {M}}$ , where ${\tilde {M}}$ 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 ${\tilde {M}}$ .

^[2]

Some examples and nonexamples

Only spacetimes which model an isolated object are asymptotically flat. Many other familiar exact solutions, such as the

FRW models

, are not.

A simple example of an asymptotically flat spacetime is 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.

A coordinate-dependent definition

The simplest (and historically the first) way of defining an asymptotically flat spacetime assumes that we have a coordinate chart, with coordinates $t,x,y,z$ , which far from the origin behaves much like a Cartesian chart on Minkowski spacetime, in the following sense. Write the metric tensor as the sum of a (physically unobservable) Minkowski background plus a perturbation tensor, $g_{ab}=\eta _{ab}+h_{ab}$ , and set $r^{2}=x^{2}+y^{2}+z^{2}$ . Then we require:

$\lim _{r\rightarrow \infty }h_{ab}=O(1/r)$
$\lim _{r\rightarrow \infty }h_{ab,p}=O(1/r^{2})$
$\lim _{r\rightarrow \infty }h_{ab,pq}=O(1/r^{3})$

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 $O(1/r^{4})$ , which would be physically sensible. (In classical electromagnetism, the energy of the electromagnetic field of a point charge decays like $O(1/r^{4})$ .)