Asymptotically flat spacetime

An asymptotically flat spacetime approaches Minkowski spacetime sufficiently far from isolated sources. At null infinity this is stated invariantly by requiring a smooth conformal completion whose boundary contains future and past null infinity.

Asymptotic simplicity requires a suitable smooth conformal completion with $\tilde{g}={\Omega }^{2}g$, $\Omega >0$ in the physical spacetime, and $\Omega =0$, $d\Omega \ne 0$ on null infinity, such that every inextendible null geodesic has both past and future endpoints there. It is stronger than merely possessing an asymptotically flat exterior. Null geodesics ending on singularities or on omitted finite-affine-distance horizons prevent this global condition.

A conformal completion embeds a physical spacetime $(\mathcal{M},g)$ into a manifold with boundary carrying a regular metric $\bar{g}={\Omega }^{2}g$. Infinity is represented by $\Omega =0$ with $d\Omega \ne 0$; the causal character of each boundary component is determined by ${\bar{g}}^{ab}{\bar{\nabla }}_a\Omega {\bar{\nabla }}_b\Omega$.

This spacetime boundary construction is a conformal compactification, not an arbitrary topological compactification.

For four-dimensional Minkowski spacetime, set $p=\arctan ((t-r)/L)$ and $q=\arctan ((t+r)/L)$, with $L>0$, then $T=p+q$, $R=q-p$. Multiplication by the square of the conformal factor $2\cos p\cos q/L$ yields the displayed metric on $R\ge 0$, $|T|+R<\pi$. Its radial Penrose diagram is a triangle: the timelike edge $R=0$ is the ordinary centre, while the two sloping edges are future null infinity and past null infinity. A signed Cartesian spatial coordinate in two dimensions gives a full diamond instead.

A complete inertial timelike geodesic in Minkowski spacetime has $\mathbf{x}=\mathbf{b}+\mathbf{v}t$ with $|\mathbf{v}|<1$. Under $p=\arctan (t-r)$, $q=\arctan (t+r)$, both $p,q$ tend to the same endpoint, giving past timelike infinity and future timelike infinity. A complete null geodesic has one finite limiting null coordinate at each end and reaches past null infinity and future null infinity. Complete spacelike geodesics have $r>|t|$ asymptotically and both ends approach spacelike infinity. The endpoints have finite compactified coordinates but occur at infinite physical affine parameter. A conformal transformation preserves null paths as unparametrized geodesics; it does not generally preserve timelike or spacelike geodesic paths.

The Lorentzian real form of the Klein quadric is the projectivized real null cone of signature $(2,4)$. Scaling a nonzero null vector makes its positive two-dimensional and negative four-dimensional parts both unit vectors. Real projectivization then identifies $(u,v)$ with $(-u,-v)$, yielding the displayed quotient. Its universal cover is $\mathbb{R}\times S^3$, and its fundamental group is infinite cyclic. An affine Minkowski chart omits a null cone at infinity. This projective compact space is distinct from an unwrapped causal cover used for a Penrose diagram.

Future null infinity is the boundary reached by future-directed outgoing null geodesics in an asymptotically flat spacetime. In four dimensions it has topology $\mathbb{R}\times S^2$ under the standard asymptotic-flatness assumptions.

It is the future component of null infinity; the past component is ${\mathcal{I}}^{-}$.