Conformal completion

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 ${\mathcal{I}}^{-}$.