Conformal Embeddings and Ideal Boundaries

A conformal completion represents ideal endpoints by embedding a physical spacetime into a larger, regular Lorentzian manifold whose boundary lies at a finite coordinate location. It preserves light cones but deliberately changes lengths. The construction is therefore a tool for causal geometry—not a claim that infinity consists of ordinary physical events, nor a proof that every spacetime admits a smooth boundary.

Required background. Lorentzian Metrics and Causal Cones supplies the causal classification preserved by conformal rescaling. Helpful background. Geodesics and Jacobi Fields supplies affine parametrization; Causality and Global Structure orients the role of ideal endpoints in global questions; Exact Solutions of Einstein’s Equations supplies the Minkowski, de Sitter, anti-de Sitter, and black-hole test geometries.

Physical and unphysical spacetimes

Write the physical spacetime as $(\widetilde M,\widetilde g)$ and the unphysical spacetime as a manifold-with-boundary $(M,g)$. A conformal completion consists of an embedding $\iota:\widetilde M\to M$ and a function $\Omega$ such that $\iota(\widetilde M)$ is the interior of $M$ and

$$g=\Omega^2\widetilde g \quad\text{on }\widetilde M,$$

with $\Omega>0$ in the physical interior. For a regular conformal boundary $\mathscr I=\partial M$, one further requires

$$\Omega=0, \qquad d\Omega\neq0 \quad\text{on }\mathscr I.$$

The last condition makes $\mathscr I$ a smooth hypersurface by the regular-value theorem. It is not cosmetic: if both $\Omega$ and $d\Omega$ vanished, the zero set could have a corner, self-intersection, or higher-order degeneracy, and the usual boundary geometry would not follow. The physical metric diverges as $\Omega^{-2}$ in this representation, while the unphysical metric remains regular. Frauendiener’s construction and its gauge freedom are developed in Frauendiener 2004, §§2.2–2.3, especially Definition 1.

The notation is not universal: some sources put the tilde on the unphysical metric. What matters is the equation, not the decoration. Here tildes always denote physical quantities.

There is also conformal gauge freedom. If $\omega$ is smooth and strictly positive through the boundary, then

$$\Omega'=\omega\Omega, \qquad g'=\omega^2g$$

defines the same physical metric and the same boundary set. A preferred conformal frame requires extra conditions. Consequently, a component of boundary data that changes under $g\mapsto\omega^2g$ is not by itself an invariant observable.

What the rescaling preserves

For every tangent vector in the physical region,

$$g(v,v)=\Omega^2\widetilde g(v,v).$$

Because $\Omega^2>0$, timelike, null, and spacelike character are unchanged. Chronological and causal curves in the interior are therefore the same unparametrized curves for both metrics. In particular, conformal transformations carry null geodesics to null geodesics as point sets. They do not generally preserve affine parametrization: if $\lambda$ is affine for $g$, a physical affine parameter satisfies $d\widetilde\lambda/d\lambda\propto\Omega^{-2}$. A null ray can reach $\mathscr I$ at finite unphysical parameter while having infinite physical affine length.

Timelike geodesics are not conformally invariant, and neither are proper times, areas, curvature scalars, or the Einstein equation in its original form. A compactified diagram may correctly show that two events are causally related while saying nothing quantitative about their elapsed proper time or areal separation. This is the central bargain of conformal completion: causal accessibility becomes visible because metric scale is suppressed.

The boundary’s causal character is read from its normal

$$n_a=\nabla_a\Omega, \qquad n^2=g^{ab}n_an_b \quad\text{at }\mathscr I.$$

With the site signature, $n^2\lt0$ means that the normal is timelike and the boundary is spacelike; $n^2=0$ gives a null boundary; $n^2\gt0$ means that the normal is spacelike and the boundary is timelike. This test is independent of the conformal gauge. On $\Omega=0$, the transformation above gives $n'_a=\omega n_a$ and $g'^{ab}=\omega^{-2}g^{ab}$, hence

$$g'^{ab}n'_an'_b=g^{ab}n_an_b.$$

For sufficiently regular vacuum Einstein asymptotics, the conformal field equations relate the sign of $n^2$ to the opposite sign of the cosmological constant: positive $\Lambda$ gives spacelike infinity, zero $\Lambda$ gives null infinity, and negative $\Lambda$ gives timelike infinity. Each conclusion still depends on the stated falloff and regularity assumptions.

Compactifying Minkowski space

The standard check begins with radial null coordinates in Minkowski space,

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

Choose an arbitrary length $L\gt0$ and map the infinite ranges to finite, dimensionless ones by

$$U=\arctan\!\left(\frac{u}{L}\right), \qquad V=\arctan\!\left(\frac{v}{L}\right).$$

Set $T=U+V$, $R=V-U$, and choose

$$\Omega=\frac{2}{L}\cos U\cos V =\frac{\cos T+\cos R}{L}.$$

After multiplication by $\Omega^2$, the physical Minkowski metric becomes the restriction of the Einstein-cylinder metric

$$g=-dT^2+dR^2+\sin^2R\,d\omega_2^2$$

to the radial domain $-\pi<T\pm R<\pi$, $R\geq0$. The arbitrary scale $L$ changes no boundary or causal relation. Outgoing future null rays have fixed $U$ and reach $V=\pi/2$, the component called future null infinity $\mathscr I^+$. Inertial timelike worldlines approach the endpoint $i^+$, while spacelike directions approach $i^0$. These pieces are not interchangeable: their normals, generator structure, and limiting curves differ.

The example also exposes two limitations. First, spherical symmetry lets one suppress the two-sphere and draw a triangle, since $R\geq0$; a full diamond belongs instead to signed Cartesian $1+1$ Minkowski coordinates. A generic spacetime need not admit either reduction. Second, the regular-boundary condition $d\Omega\neq0$ applies on the open components $\mathscr I^\pm$, not at their ideal endpoints. At $i^+=(T,R)=(\pi,0)$, $i^-=(-\pi,0)$, and $i^0=(0,\pi)$, the derivatives of $\Omega=(\cos T+\cos R)/L$ vanish. These vertices are not regular hypersurface points of that zero set. More general spatial infinities can require a blow-up, so the boundary construction must be defined before its diagram is trusted.

Regularity, existence, and interpretation

A zero of a chosen conformal factor is not automatically “infinity.” The embedding must identify which physical curves approach it, the unphysical metric must have the claimed regularity, and completeness or asymptotic field equations must supply the intended interpretation. Conversely, a spacetime can have meaningful ideal endpoints even when no smooth conformal completion exists; causal-boundary methods address part of that gap. The distinctions and counterexamples are surveyed in Sánchez 2022, §§4–6, pp. 22–46.

Smoothness is a substantive asymptotic restriction. Radiation, logarithmic terms, massive fields, or unsuitable conformal gauge can lower differentiability at $\mathscr I$. One must not infer peeling, well-defined charges, or completeness merely from having written $g=\Omega^2\widetilde g$. Likewise, the boundary belongs to $M$, not to the physical manifold $\widetilde M$; a physical tensor has a boundary value only after an appropriate conformal rescaling and regularity proof.

Exercise

Consider the half-space $z>0$ with physical metric

$$d\widetilde s^2=\frac{\ell^2}{z^2}\left(-dt^2+dz^2+dx^2+dy^2\right).$$

Use $\Omega=z/\ell$ to classify the surface $z=0$ in the unphysical geometry.

Solution

The unphysical metric is

$$ds^2=-dt^2+dz^2+dx^2+dy^2,$$

which is regular through $z=0$. There $\Omega=0$ and $d\Omega=dz/\ell\neq0$. Its normal has squared norm

$$n^2=g^{zz}\left(\frac{1}{\ell}\right)^2=\frac{1}{\ell^2}>0.$$

The normal is spacelike, so the conformal boundary is timelike. This is the Poincaré patch of anti-de Sitter geometry. The calculation classifies the boundary; it does not yet choose boundary conditions for fields propagating there.

Next, Future and Past Null Infinity studies the null components of a completion and the curves that end on them.

References

• Frauendiener, Jörg, “Conformal Infinity”, Living Reviews in Relativity 7, article 1, §§2.2–2.3 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.