Timelike, Spacelike, and Spatial Infinity

The phrase “go to infinity” hides several inequivalent limiting processes. A future-complete massive observer, a past-complete observer, a spacelike ray on an initial slice, and a universe expanding toward a spacelike conformal boundary do not approach the same kind of ideal endpoint. Their causal characters determine what data can be prescribed there and what signals can reach them.

Required background. Lorentzian Metrics and Causal Cones supplies the classification of boundary normals. Helpful background. Geodesics and Jacobi Fields supplies completeness and limiting geodesics; Causality and Global Structure orients the boundary constructions; Exact Solutions of Einstein’s Equations provides Minkowski and cosmological examples.

Timelike endpoints in asymptotically flat spacetime

In the conformal completion of Minkowski spacetime, future timelike infinity $i^+$ is the ideal endpoint of future-inextendible timelike geodesics that remain at finite asymptotic velocity. Past timelike infinity $i^-$ is the corresponding past endpoint. For an inertial worldline $x=vt+x_0$ with $\lvert v\rvert<1$,

$$u=t-r\longrightarrow +\infty, \qquad v_a=t+r\longrightarrow +\infty$$

as $t\to+\infty$, so the worldline reaches $i^+$ rather than future null infinity. The subscript on $v_a$ here only distinguishes advanced time from the velocity $v$.

Timelikeness alone does not determine the ideal endpoint. The complete uniformly accelerated worldline, for $a\gt0$, $t=a^{-1}\sinh(a\tau)$, $x=a^{-1}\cosh(a\tau)$ has $u=t-x\to0$ and $v_a=t+x\to+\infty$ as $\tau\to+\infty$, so it approaches an open point of $\mathscr I^+$ rather than $i^+$. The $i^\pm$ statement is therefore about the specified geodesic or finite-subluminal-asymptotic class, not every complete timelike curve.

The label timelike infinity describes the causal direction of curves approaching the endpoint. It does not mean that $i^+$ is a timelike hypersurface: in the idealized Penrose compactification it is a point where boundary components meet. Nor is it a physical event through which the observer continues. Infinite physical proper time can be represented by a finite unphysical coordinate interval.

Stationary massive fields and late-time tails are often analyzed near $i^+$, but a smooth point structure is not automatic in a general asymptotically flat spacetime. Mass, radiation, and asymptotic regularity influence how the conformal geometry behaves at the endpoints of $\mathscr I$. The Minkowski picture is a model to be justified, not a definition copied unchanged.

Spatial infinity is a direction-dependent limit

Spatial infinity $i^0$ is approached along spacelike directions—for example, $r\to\infty$ with $t$ fixed in Minkowski space. In null coordinates this sends

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

It lies where $\mathscr I^-$ and $\mathscr I^+$ meet in the compactified two-dimensional picture. Treating it as a single dot suppresses an entire sphere of spatial directions and, more subtly, information about boosts. For gravitational fields with nonzero total mass, the conformal metric is generally not regular at $i^0$ in the same simple way as on the open null boundary.

One remedy replaces the dot by richer geometry: a hyperboloid of spatial directions, a cylinder at spatial infinity, or another blow-up adapted to the chosen framework. This makes it possible to compare limits toward $i^0$ along different directions and to formulate parity, regularity, and matching conditions. The need for such structure is why ADM data at spatial infinity cannot simply be read as Bondi data on an adjacent cut of $\mathscr I^+$. A matching theorem requires explicit falloffs and control of the corner.

Frauendiener’s discussion of the Minkowski completion and the special behavior of $i^0$ gives the standard geometric starting point Frauendiener 2004, §2.2, Eqs. (2)–(7), and §3.5. Sánchez emphasizes that different completion procedures can attach different ideal-boundary structures Sánchez 2022, §§4–6, pp. 22–46.

Spacelike conformal infinity

A conformal boundary is spacelike when its normal is timelike. If $n_a=\nabla_a\Omega$, then in the site signature

$$g^{ab}n_an_b<0$$

at the boundary. Exact de Sitter spacetime supplies the canonical example. In global conformal coordinates,

$$d\widetilde s^2=\frac{\ell^2}{\cos^2\eta} \left(-d\eta^2+d\Omega_3^2\right), \qquad -\frac{\pi}{2}<\eta<\frac{\pi}{2}.$$

Choosing $\Omega=\cos\eta/\ell$ gives the Einstein-cylinder metric as $g$. The surfaces $\eta=\pm\pi/2$ are spacelike three-spheres because their normal $d\eta$ is timelike. Future-directed timelike and null curves can end on the future component. Unlike null infinity, that component is not generated by null curves lying within it.

This changes the characteristic problem. A null boundary naturally separates directions along and transverse to generators; a spacelike boundary resembles a final asymptotic slice. Yet it is not generally a Cauchy surface inside the physical spacetime, and prescribing arbitrary data there need not define a physical evolution. Field equations, conformal constraints, and asymptotic conditions decide which boundary data are admissible.

Four ideal limits compared

The safest comparison names both the limiting curves and the boundary geometry:

Ideal set	Typical approach	Boundary character	Information suppressed by a dot or line
$i^+$	Future timelike geodesics with finite subluminal asymptotic velocity	Endpoint, not an open hypersurface	velocities, late-time directions, endpoint regularity
$i^-$	Past timelike geodesics with finite subluminal asymptotic velocity	Endpoint, not an open hypersurface	incoming massive histories and regularity
$i^0$	Spacelike directions	Corner or blown-up set	angles, boosts, parity, null–spatial matching
de Sitter future infinity	Timelike or null curves in an expanding spacetime	Spacelike hypersurface	conformal-frame and asymptotic-data constraints

One cannot determine physical distance from any of these compactified locations. A finite separation on the unphysical cylinder may represent infinite physical distance or duration. Nor does a common symbol $i$ imply identical differentiable structure. The completion’s hypotheses, rather than typography, determine which tensors and limits exist.

Exercise

In the de Sitter metric above, verify the causal character of $\eta=\pi/2$ and contrast it with Minkowski future null infinity.

Solution

For $g=-d\eta^2+d\Omega_3^2$ and $\Omega=\cos\eta/\ell$,

$$n_a=\nabla_a\Omega=-\frac{\sin\eta}{\ell}(d\eta)_a, \qquad n^2=-\frac{\sin^2\eta}{\ell^2}.$$

At $\eta=\pi/2$, $n^2=-1/\ell^2<0$, so the normal is timelike and the boundary is spacelike. At regular Minkowski $\mathscr I^+$ the conformal normal is null, $n^2=0$, and is tangent to the boundary generators. The two future boundaries therefore have different intrinsic causal structures even though both receive escaping future-directed light rays.

The coordinate construction that places all these sets in one finite picture is developed in Constructing Penrose Diagrams.

References

• Frauendiener, Jörg, “Conformal Infinity”, Living Reviews in Relativity 7, article 1, §2.2 and §3.5, 2004.
• Sánchez, Miguel, “Globally Hyperbolic Spacetimes: Slicings, Boundaries and Counterexamples”, General Relativity and Gravitation 54, article 124, §§4–6, pp. 22–46, 2022.