Future and Past Null Infinity

Future and past null infinity, written $\mathscr I^+$ and $\mathscr I^-$, are the ideal endpoints of light rays that escape to or arrive from arbitrarily large distance in an asymptotically flat spacetime. They are null hypersurfaces of a conformal completion, not distant timelike walls. Their existence, smoothness, topology, and completeness are assumptions to be checked—not consequences of the symbol $r\to\infty$.

Required background. Lorentzian Metrics and Causal Cones supplies null hypersurfaces and their normals. Helpful background. Geodesics and Jacobi Fields supplies affine completeness; Causality and Global Structure gives the volume-wide role of ideal boundaries; Exact Solutions of Einstein’s Equations supplies asymptotically flat examples and counterexamples.

Null components of a conformal boundary

Let $(\widetilde M,\widetilde g)$ have a regular conformal completion $(M,g,\Omega)$, so that $g=\Omega^2\widetilde g$ in the physical region, $\Omega=0$ at the boundary, and $d\Omega\neq0$ there. A boundary component is null when

$$g^{ab}\nabla_a\Omega\nabla_b\Omega=0 \quad\text{on the component}.$$

Its normal $n^a=g^{ab}\nabla_b\Omega$ is then also tangent to the boundary. The integral curves of $n^a$, after a possible rescaling, are its null generators. In the standard asymptotically flat setting, the future component reached by outgoing null curves is $\mathscr I^+$, while the past component from which incoming null curves originate is $\mathscr I^-$. The conformal geometry and orientation are reviewed in Frauendiener 2004, §§2.2–2.3.

For four-dimensional Minkowski spacetime, each component has topology

$$\mathscr I^\pm\simeq\mathbb R\times S^2.$$

The real coordinate labels position along a generator, and the sphere labels its asymptotic direction. A cut is a spacelike $S^2$ meeting each generator once. This topology is characteristic of one regular asymptotically flat end; multiple ends or weaker asymptotic structures require separate statements.

The superscript records causal orientation, not the sign of the radial coordinate. Future null infinity receives future-directed outgoing light. Past null infinity emits future-directed incoming light. A future-directed timelike worldline of an inertial massive observer instead approaches timelike infinity $i^+$, because it remains inside the light cone rather than becoming asymptotically null. A spacelike ray on a constant-time slice approaches spatial infinity $i^0$.

Which curves end where

In Minkowski space use retarded and advanced time,

$$u=t-r, \qquad v=t+r.$$

Along a future-directed outgoing radial null ray, $u$ is constant and $v\to+\infty$; after compactification it ends on $\mathscr I^+$. Along a future-directed incoming ray, $v$ is constant, so its past endpoint as $u\to-\infty$ lies on $\mathscr I^-$. A static massive observer has fixed $r$ and $t\to+\infty$, sending both $u$ and $v$ to $+\infty$; its endpoint is $i^+$ rather than a point of $\mathscr I^+$.

This classification is conformally invariant as an unparametrized causal statement, but affine length is not. If $\lambda$ is an unphysical affine parameter on a null geodesic and $\widetilde\lambda$ a physical one, then, up to constant normalization,

$$\frac{d\widetilde\lambda}{d\lambda}=\Omega^{-2}.$$

For a null ray meeting a regular boundary transversely at $\lambda=\lambda_*$, smoothness gives $\Omega=C(\lambda_*-\lambda)+O((\lambda_*-\lambda)^2)$ with $C\gt0$. Thus

$$\widetilde\lambda\sim\frac{1}{C^2(\lambda_*-\lambda)}\longrightarrow+\infty.$$

The finite unphysical endpoint then represents infinite physical affine length. This is a check for rays reaching the regular boundary; it makes no claim about other rays ending at a singularity or failing to reach that boundary. One must also distinguish physical null-geodesic completeness from completeness of the boundary generators in a chosen conformal gauge, or from the stronger global condition below.

Asymptotic flatness and black holes

In the asymptotically flat setting discussed here, asymptotic simplicity requires every inextendible physical null geodesic to acquire past and future endpoints on the regular conformal boundary. The boundary is null under the asymptotically flat hypotheses; asymptotic simplicity as a general completion condition does not itself require zero cosmological constant or null boundary character Frauendiener 2004, §2.3.1, Definition 1. Minkowski space qualifies. A spacetime containing a black hole does not satisfy this global endpoint requirement: future-directed null geodesics entering the black-hole region fail to end on $\mathscr I^+$. Black-hole applications therefore use weakened conditions—often asymptotic simplicity near infinity, weak asymptotic simplicity, or a suitably regular asymptotic end—rather than declaring the entire spacetime asymptotically simple.

This distinction is essential for the global event-horizon definition. Given a specified component of future null infinity, the black-hole region is

$$\mathcal B=M_{\rm phys}\setminus J^-(\mathscr I^+),$$

and its future event horizon is the boundary of that set. Changing which asymptotic end is included can change the causal past and hence the horizon. The construction presupposes that $\mathscr I^+$ has already been identified with enough regularity for $J^-(\mathscr I^+)$ to make sense. Null infinity is therefore prior to, not a decorative addition after, the event horizon.

Radiative data also live naturally at $\mathscr I^+$. Yet its null character alone does not supply Bondi coordinates, a universal structure, news, flux, or charges. Those require field-equation falloffs and conformal-frame choices. Smoothness may fail through logarithmic or polyhomogeneous terms even when a weaker boundary exists. The later chapters separate these additional layers.

Checks, endpoints, and limitations

The Einstein-cylinder compactification gives a direct orientation check. For any chosen length $L\gt0$, set $U=\arctan(u/L)$ and $V=\arctan(v/L)$. Future null infinity is $V=\pi/2$ with $-\pi/2<U<\pi/2$, while past null infinity is $U=-\pi/2$ with $-\pi/2<V<\pi/2$. Their endpoints meet ideal points in the completed picture, but $i^0$ and $i^\pm$ are not ordinary smooth points of the same type as the open null components.

Three cautions control interpretation. First, a cosmological spacetime with positive cosmological constant typically has spacelike conformal infinity, so calling its distant boundary $\mathscr I^+$ does not make that boundary null. Second, a timelike artificial boundary at finite radius is not null infinity, even if outgoing radiation conditions are imposed there. Third, the absence of a smooth $\mathscr I$ does not prove that radiation or causal escape is meaningless; it means a weaker asymptotic framework is required. Comparisons between conformal and causal completions, including non-equivalences, are given in Sánchez 2022, §§4–6, pp. 22–46.

Exercise

In $1+1$ Minkowski spacetime, classify the ideal endpoint approached in each limit, and state the curve’s causal character: (a) $x=0$, $t\to+\infty$; (b) $x=t-u_0$, $t\to+\infty$; (c) $t=0$, $x\to+\infty$.

Solution

For (a), $u=t$ and $v=t$ both diverge to $+\infty$, so the timelike worldline approaches $i^+$. For (b), $u=t-x=u_0$ is fixed while $v=t+x\to+\infty$; the curve is outgoing null and approaches $\mathscr I^+$. Curve (c) is spacelike rather than future-directed causal: $u=-x\to-\infty$ and $v=x\to+\infty$, so it approaches spatial infinity $i^0$. The classification depends on the limiting causal direction, not merely on coordinates becoming large.

The next page compares these null components with Timelike, Spacelike, and Spatial Infinity.

References

• Frauendiener, Jörg, “Conformal Infinity”, Living Reviews in Relativity 7, article 1, §§2.2–2.3, 2004.
• Sánchez, Miguel, “Globally Hyperbolic Spacetimes: Slicings, Boundaries and Counterexamples”, General Relativity and Gravitation 54, article 124, §§4–6, pp. 22–46, 2022.