Conformal Completion, Causal Boundaries, and Penrose Diagrams

Conformal completion brings infinity to a finite boundary while preserving causal cones; causal completion reconstructs ideal endpoints from chronological sets; a Penrose diagram turns selected consequences into a picture. These are related tools, not synonyms. This chapter builds them in that order so that every boundary label has a definition and every drawing comes with an explicit list of information it suppresses.

Helpful background. Lorentzian Metrics and Causal Cones supplies conformal cone geometry; Geodesics and Jacobi Fields supplies affine completeness; Causality and Global Structure gives the volume route; Exact Solutions of Einstein’s Equations supplies the model domains to compactify.

Three boundary languages

A smooth conformal completion starts with an embedding of the physical spacetime $(\widetilde M,\widetilde g)$ into a manifold with boundary and a factor $\Omega$ such that

$$g=\Omega^2\widetilde g, \qquad \Omega=0, \qquad d\Omega\neq0$$

at a regular boundary. It preserves causal type in the physical interior and supplies differentiable boundary geometry when the rescaled metric remains regular. This is the strongest language used here, but its smoothness is a real asymptotic condition.

A causal completion begins instead with $I^+$ and $I^-$. Terminal indecomposable past and future sets encode limiting chronological information of inextendible curves. It uses less metric structure and can attach endpoints when no smooth conformal factor is known, but endpoint pairing and topology require care.

A Penrose diagram is a representation made after one of these global structures and the physical coordinate domain have been understood. It is not itself a completion. The finite picture reliably displays radial causal incidence and labeled ideal boundaries only to the extent justified by the construction behind it.

The sequence of ideas

Begin with Conformal Embeddings and Ideal Boundaries. It fixes physical versus unphysical metrics, regular-boundary conditions, conformal gauge freedom, and the boundary-normal test. The Minkowski compactification shows why infinite physical parameter can end at finite unphysical coordinate.

Future and Past Null Infinity then identifies the null hypersurfaces $\mathscr I^\pm$, their generators and cuts, and the physical curves that approach them. It separates a regular asymptotic end from asymptotic simplicity of the entire spacetime—a distinction needed when black holes remove some null geodesics from $\mathscr I^+$.

Timelike, Spacelike, and Spatial Infinity compares $i^+$, $i^-$, $i^0$, and cosmological spacelike boundaries. It emphasizes that spatial infinity is direction-dependent and often needs to be blown up rather than treated as a regular point identical to the open null boundary.

With those objects defined, Constructing Penrose Diagrams gives a reproducible method: solve radial null curves, extend across coordinate singularities, compactify null coordinates monotonically, impose the transformed domain, rescale, and classify each boundary independently. Causal Boundary Constructions then shows how terminal chronological sets provide ideal endpoints without a smooth conformal embedding.

Finally, What Penrose Diagrams Suppress lists the missing angular, topological, metric, affine, curvature, differentiability, and dynamical data. That final page is part of the method: it sets the boundary between a correct causal inference and an attractive but invalid visual one.

Recurring examples and checks

Minkowski space is the normalization test. Retarded and advanced times $u=t-r$ and $v=t+r$, followed by $U=\arctan(u/L)$ and $V=\arctan(v/L)$ for any length $L\gt0$, produce the compact radial domain on the Einstein cylinder. Outgoing null rays reach $\mathscr I^+$, inertial timelike curves reach $i^+$, and constant-time spacelike directions reach $i^0$. The computation fixes future/past orientation and the causal character of each boundary without assigning physical significance to $L$.

Three variations prevent overgeneralization. Removing a point from Minkowski space creates incomplete curves and a causal ideal endpoint without curvature blowup. Exact de Sitter space has spacelike rather than null conformal infinity. Anti-de Sitter space has a timelike boundary that null rays reach in finite global coordinate time, even though their physical affine parameter is infinite, so bulk evolution needs boundary conditions.

Black-hole examples test the domain. The maximally extended Schwarzschild vacuum contains a white-hole region and two exteriors; a collapse spacetime with the same late exterior does not. A diagram derived only from the static exterior chart cannot decide between them. The manifold, extension, and initial-data history must be specified first.

These examples follow the construction and counterexample strategy surveyed in Frauendiener 2004, §§1–3 and Sánchez 2022, §§4–6, pp. 22–46.

Review the chapter

Suppose a finite diamond has null upper edges labeled $\mathscr I^+$, a vertical center labeled $r=0$, and a horizontal top edge labeled “singularity.” Which claims follow from the picture alone, and which require external justification?

Solution

If the diagram’s construction is accepted, its null slopes and incidence show the represented radial causal relations: which radial null curves reach the upper edges and which encounter the top boundary. The labels do not follow from shape alone. To establish $\mathscr I^+$ one needs a regular asymptotic completion, correct future orientation, and an account of the curves ending there. To establish a regular axis one needs the symmetry-sphere and elementary-flatness behavior at $r=0$. To call the top a singularity one needs geodesic incompleteness, inextendibility in a specified regularity class, curvature behavior, or another precise criterion. The diagram also says nothing by itself about angular topology, affine length, stability, or how the spacetime formed.

What this chapter enables

After this chapter, you should be able to construct the Minkowski compactification, classify a regular boundary from its conformal normal, distinguish null from timelike and spacelike infinity, identify which curves approach each ideal set, outline a causal-boundary construction, and state exactly which conclusions a two-dimensional diagram cannot support.

The chapter deliberately stops before asymptotic field equations, radiation charges, and horizon mechanics. The immediate continuation is Achronal Boundaries and Null Generators, where set boundaries acquire null generators without relying on a picture. Later chapters return to asymptotically flat, de Sitter, and anti-de Sitter boundaries with the falloffs and dynamics appropriate to each regime.

References

• Frauendiener, Jörg, “Conformal Infinity”, Living Reviews in Relativity 7, article 1, §§1–3, 2004.
• Sánchez, Miguel, “Globally Hyperbolic Spacetimes: Slicings, Boundaries and Counterexamples”, General Relativity and Gravitation 54, article 124, §§4–6, pp. 22–46, 2022.