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research38 min studyLesson 821 in Physics

Penrose diagrams and maximal extensions

Read the causal content of Schwarzschild extensions without confusing coordinate boundaries, singularities, and a collapse spacetime.

Before you begin
On this page

The question for today

What you’ll understand

  • Extend Schwarzschild coordinates through a regular horizon.
  • Use compactified null coordinates to classify causal directions.
  • Distinguish the eternal maximal extension from a black hole formed by collapse.

A coordinate chart can fail where the spacetime itself is smooth. A causal diagram separates that failure from a genuine endpoint of the geometry by displaying light propagation rather than coordinate distances. For Schwarzschild, extending through the horizon reveals an interior in which decreasing areal radius is a future direction. Extending the eternal vacuum solution further produces additional regions. Those regions belong to that mathematical spacetime and must not automatically be attributed to a black hole created by the collapse of one star.

Remove the horizon coordinate singularity

Use signature (−,+,+,+) and geometrized units G=c=1 throughout the classical geometry in this module; t, r, and the Schwarzschild mass parameter M then have length units. The Schwarzschild chart has a metric coefficient that diverges at r=2M, but curvature is finite there. Introducing tortoise and null coordinates, followed by exponential Kruskal coordinates, gives a metric regular across both horizon branches. The curvature singularity at r=0 is a separate issue.

In the right exterior, U is negative and V positive. Extending the metric to positive U and positive V gives the future black-hole interior, while both negative signs give the past white-hole region. Positive U with negative V gives another exterior. These sign statements fix the regions without depending on a particular textbook's Roman numerals. The future orientation has nondecreasing U and V along radial causal curves. Constant U or constant V describes a radial null direction.

u=t−r∗,v=t+r∗,r∗=r+2Mln⁡∣r2M−1∣,U=−e−u/(4M),V=ev/(4M),UV=−(r2M−1)er/(2M)u=t-r_*,\quad v=t+r_*,\quad r_*=r+2M\ln\left|\frac r{2M}-1\right|,\quad U=-e^{-u/(4M)},\quad V=e^{v/(4M)},\qquad UV=-\left(\frac r{2M}-1\right)e^{r/(2M)}
The exponential definitions initially hold in the right exterior; the final relation and regular metric define their extension. U,V are dimensionless. The horizon has UV=0 and the excluded r=0 curvature boundary has UV=1.↔ Scroll wide equations horizontally.

Compactification preserves light cones, not distances

A positive conformal rescaling preserves null directions and timelike versus spacelike character while changing proper times, lengths, and affine parameters. Monotone functions of null coordinates can bring infinite coordinate ranges to a finite diagram. A convenient compactification applies arctangent to U and V, then takes their sum and difference. Radial light rays remain diagonal, and the future horizon branch U=0 becomes a diagonal boundary separating the right exterior from the future interior.

Each point of the two-dimensional diagram represents a symmetry sphere except where that sphere degenerates. The horizontal separation between two plotted points is not a physical distance, and the apparent slope of a timelike worldline is meaningful only relative to the displayed light directions. A horizon is crossed in finite proper time by a suitable infalling observer even though the original Schwarzschild time diverges. The diagram removes that chart-dependent obstruction without making the interior accessible again to an outgoing signal.

P=arctan⁡U,Q=arctan⁡V,T=P+Q2,X=Q−P2,U=0⟹T=XP=\arctan U,\qquad Q=\arctan V,\qquad T=\frac{P+Q}{2},\qquad X=\frac{Q-P}{2},\qquad U=0\Longrightarrow T=X
P,Q,T,X are dimensionless diagram coordinates. This compactification is for the radial causal structure; it is not a coordinate system for computing physical distances from Euclidean lengths on the page.↔ Scroll wide equations horizontally.

Separate global extensions from the history of formation

The eternal vacuum extension includes a white-hole region and a second asymptotic exterior, but it is not a traversable shortcut between the two exteriors. Future-directed causal paths entering the black-hole interior reach its future boundary rather than emerging into the other exterior. Inside that future region, the product UV increases along future radial causal curves, corresponding to decreasing r. Thinking of the singularity simply as a stationary place that one might steer around gives the wrong causal picture.

A collapse diagram instead joins an exterior geometry to the collapsing matter and its regular pre-collapse interior. It does not require an eternal white hole or the second vacuum exterior. An event horizon is defined relative to the causal past of future null infinity, so locating it requires global information about the future. Other horizons defined on spatial slices need not coincide with it during evolution. These distinctions matter before applying stationary mechanics or choosing a quantum state on an idealized eternal background.

Work it through

Locate an exterior point and a horizon point

Use the compactification above for point A with (U,V)=(−1,1) and point B with (U,V)=(0,1). Find their diagram coordinates, classify them, and check the slope of the line from their displayed coordinates. Do not interpret that line as a physical trajectory without further information.

  1. For A, the compact null coordinates are P=−π/4P=-\pi/4 and Q=π/4Q=\pi/4. Thus (T,X)=(0,π/4)(T,X)=(0,\pi/4). Its signs place it in the right exterior.

  2. For B, P=0 and Q=π/4, giving (T,X)=(π/8,π/8)(T,X)=(\pi/8,\pi/8). It lies on the future horizon branch U=0,V>0 and satisfies T=X.

  3. The coordinate displacement from A to B is (ΔT,ΔX)=(π/8,−π/8)(\Delta T,\Delta X)=(\pi/8,-\pi/8), which is null in the radial conformal diagram. A specific null curve connecting them has V fixed; the diagram statement does not assign a finite proper length to the segment.

The result

A is an exterior point at (0,π/4), and B is a future-horizon point at (π/8,π/8); their connecting constant-V radial direction is null.

The living notebook

Explore the model

Interactive lab

Interactive lab · spacetime

A light-cone relation survives a coordinate change

For each fresh comparison, Reset and set BOTH Event time ct=1 and Relative velocity β. Compare β=0 and β=0.60 for the fixed event x=0.60. Predict transformed (ct,x)=(1,0.60) and (0.80,0), with invariant interval 0.64. The instrument performs a Lorentz transformation in flat spacetime; it does not compactify coordinates, extend Schwarzschild, or locate a horizon.

A light-cone relation survives a coordinate changeA fixed x = 0.6 event is Lorentz transformed with c = 1. Playback highlights the relation between coordinate descriptions; it does not move the event. Current readings: Lorentz factor γ: 1.12; Event x′: 0.168; Event ct′: 0.817. All parameters are adjustable with the labeled controls below.−4.69−2.3502.354.69−2.2−1.101.12.2x or x′ · c = 1ct or ct′EE′ct² − x² = 0.64
Original eventTransformed eventLight cone
Lorentz factor γ
1.12
Event x′
0.168
Event ct′
0.817
t = 0 s

A fixed x = 0.6 event is Lorentz transformed with c = 1. Playback highlights the relation between coordinate descriptions; it does not move the event. Parameter changes pause playback and recompute the model at the current elapsed time. Reset restores both parameters and returns time to zero. Playback starts paused. Adjust either parameter, then choose Play or Step.

Look forCausal classification can survive a coordinate change even when coordinate values change. Penrose diagrams use a different, conformal construction.

Pause & reason

Check your understanding

What does the white-hole region in the maximal vacuum Schwarzschild extension establish about stellar collapse?

Your turn

Try it yourself

A radial point has (U,V)=(0.50,0.50). Compare proposed future steps to C=(0.60,0.60) and D=(0.40,0.60). Which is compatible with the stated future causal orientation, and what happens to r along the compatible step?

Take a moment to make a prediction or work it out on paper. Open a hint when you need one.

Give me a hint

In the future interior both U and V must be nondecreasing along a radial future causal path. For r>0, the right side of the UV relation is a decreasing function of r.

Compare with the solution

The starting point is in the future black-hole interior and has UV=0.25. The step to C increases both null coordinates and can be timelike; its product becomes 0.36, so r decreases. The proposed step to D decreases U while increasing V and is spacelike in the radial conformal metric, so it is not a future causal escape. A coordinate choice that makes one component look outgoing does not reverse the future orientation of the light cones inside the horizon.

Keep the idea

Take it with you

  • Regular horizon charts distinguish a coordinate failure from a curvature boundary.
  • Conformal diagrams preserve causal structure while discarding physical distance information.
  • The maximal eternal extension and a collapse spacetime are different global solutions.

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