What Penrose Diagrams Suppress

A Penrose diagram can encode an enormous amount of causal information in a small space, but its visual economy comes from discarding most metric and often most topological information. A correct reading begins by asking what reduction and completion produced the picture. Only causal incidence and explicitly justified boundary labels survive automatically; lengths, curvature, dynamics, and even point multiplicity usually do not.

Required background. Lorentzian Metrics and Causal Cones supplies the conformally preserved structure. Helpful background. Geodesics and Jacobi Fields supplies affine and tidal data that diagrams omit; Causality and Global Structure supplies the causal questions; Exact Solutions of Einstein’s Equations supplies coordinate domains and maximal extensions.

Angular and topological information

In a spherically symmetric Penrose diagram, an ordinary interior point with areal radius $r>0$ represents an entire symmetry sphere. A point on the regular axis $r=0$ represents a single spacetime event. A point on null infinity often labels one symmetry sphere of endpoints, while the diagram has suppressed all angular directions. These are three different multiplicities drawn with the same graphic primitive.

The reduction also hides nonradial causal curves. Spherical symmetry often makes radial null curves sufficient to locate causal boundaries, but it does not imply that every geodesic or lensing path is represented. Angular momentum barriers, caustics, multiple images, and frame dragging require information outside a two-dimensional orbit space. In a nonsymmetric spacetime, a familiar diamond may be only schematic, and its causal completeness must come from a theorem rather than the sketch.

Topology is similarly compressed. Identifying two edges of a coordinate domain can create a cylinder or torus without changing the local metric. A two-dimensional diagram whose edges are not marked may look identical before and after the quotient. Conversely, replacing a spatial-infinity point by a sphere or hyperboloid of directions changes the completion’s topology while leaving a simplified picture visually unchanged.

Metric, affine, and curvature data

Conformal rescaling preserves null cones because

$$g_{ab}=\Omega^2\widetilde g_{ab}$$

has the same sign on every nonzero tangent where $\Omega>0$. It does not preserve proper time, proper distance, area, expansion, surface gravity, or curvature. The displayed coordinate distance from an event to infinity can be finite while the physical affine or proper parameter is infinite. Two regions of equal apparent size can contain radically different physical volumes.

Null geodesics remain null geodesics as unparametrized curves, but their affine parameters change. Consequently, a diagram can show a null curve reaching a boundary without deciding whether the physical geodesic is complete. That decision requires the conformal factor’s behavior or direct integration of the physical geodesic equation.

Curvature is even less visible. The Rindler horizon in flat spacetime, the Schwarzschild event horizon, and a regular null coordinate surface can all be drawn as null lines. Their Riemann tensors and physical interpretations differ. A jagged spacelike line is conventional notation for a singular boundary, but it neither proves curvature blowup nor fixes extension regularity. Flat Minkowski space with a point removed supplies the sharp counterexample: the diagram can contain an incomplete endpoint while all curvature invariants vanish.

Global definitions are not visual labels

Several boundaries share the same local null appearance. An event horizon is defined relative to a chosen future null infinity by

$$\mathcal H^+=\partial J^-(\mathscr I^+),$$

whereas a Cauchy horizon is the boundary of a domain of dependence. An acceleration horizon is tied to a restricted observer family, and a Killing horizon to a symmetry generator becoming null. A diagram can display each after the corresponding set or field has been identified, but the line’s slope does not establish which definition holds.

Teleology makes this especially important. Two collapse spacetimes may agree on a large early region and diverge later because one star disperses while the other forms a black hole. The event horizon can pass through the common region in only the second spacetime. No measurement confined to the early patch decides the global label; the completed future does.

Likewise, drawing a Cauchy surface across a picture does not prove global hyperbolicity. One must show that every inextendible causal curve intersects it exactly once, or verify an equivalent compact-diamond criterion. Omitted boundaries, identifications, or holes may defeat the claim without being visually conspicuous.

Dynamics, regularity, and coordinate choices

A Penrose diagram records a spacetime after a solution and its domain have been selected. It does not derive the Einstein equation, energy conditions, stability, or formation scenario. An eternal black-hole diagram includes a white-hole region and a second exterior because of its maximal analytic extension; a collapse diagram generally does not. Neither picture alone says which initial data are generic or stable.

Differentiability is also suppressed. A boundary drawn as smooth may support only a continuous or polyhomogeneous conformal structure. Peeling, asymptotic charges, and field limits can depend on precisely that missing regularity. Sánchez’s catalog of embeddings, causal boundaries, slicings, and counterexamples shows why boundary appearance cannot substitute for the construction’s hypotheses Sánchez 2022, §§4–6, pp. 22–46.

Coordinates can distort the silhouette while preserving causal relations. Any increasing reparametrization of the compactified null coordinates changes angles between nonnull curves and relative line lengths. Only null directions are conventionally fixed at $45^\circ$. A timelike curve drawn nearly null need not have ultrarelativistic physical speed, and the slope of a spacelike slice does not quantify its extrinsic curvature.

A disciplined reading checklist

Before using a diagram in an argument, identify:

• the physical manifold and whether the picture is maximal, exterior-only, collapse-based, or a quotient;
• what each interior and boundary point represents before symmetry reduction;
• the conformal factor or causal-boundary construction and its regularity;
• the time orientation, coordinate domain, identifications, and omitted regions;
• the definitions establishing every horizon or singular label;
• whether the claim concerns causal incidence, metric scale, affine completeness, topology, or dynamics; and
• which theorem or calculation supplies information not preserved by conformal rescaling.

Frauendiener’s treatment of conformal infinity states the differentiable and conformal ingredients behind the standard diagrams Frauendiener 2004, §§2.2–2.3. A diagram is strongest when it summarizes those checked ingredients and weakest when its conventional shape is treated as evidence.

Exercise

Two radial metrics satisfy $d\widetilde s_2^2=A^2(t,r)d\widetilde s_1^2$ with smooth $A>0$ on the same domain. List three conclusions their common Penrose diagram supports and three quantities it cannot show to be equal.

Solution

The metrics have the same pointwise causal cones, the same unparametrized null curves, and the same chronological/causal relations on the shared domain. Hence a correctly constructed common diagram can support causal reachability, boundary incidence, and time orientation.

It cannot show equal proper time along a timelike curve, equal affine length along a null geodesic, or equal curvature. It also cannot by itself establish equal areas, expansions, surface gravities, or stress-energy. Those quantities change under a nonconstant conformal factor or require the suppressed angular metric.

The next chapter develops Achronal Boundaries and Null Generators without relying on a diagram as proof.

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.