Penrose Diagram of Maximally Extended Schwarzschild Spacetime

1. The Art of Conformal Compactification

A Penrose diagram is a two-dimensional map of a spacetime that has been conformally transformed to bring its infinite regions to a finite boundary.

What does this mean? We find a new set of coordinates, say ${\tilde{x}}^{\mu }$, and a scalar function $\Omega (x)$ such that the original metric $g_{\mu \nu }$ is related to a new, simpler metric ${\tilde{g}}_{\mu \nu }$ by:

The key insight is that the paths of light rays (null geodesics) are unchanged by such a transformation. Since $d{s}^2=0$ for light, it follows that $d{\tilde{s}}^2={\Omega }^{2}d{s}^2=0$ as well. This means that while distances and times are distorted, the fundamental causal structure—what can and cannot be influenced by what—is perfectly preserved. In these diagrams, radial light rays always travel at 45-degree angles.

To understand the technique, it's best to start with the simplest case: flat Minkowski spacetime.

As you've just read, the process for Minkowski space yields a diamond shape. The boundaries of this diamond represent the "ends" of spacetime:

• Past Timelike Infinity ($i^{-}$): The point in the infinite past from which all timelike worldlines originate.
• Future Timelike Infinity ($i^{+}$): The point in the infinite future where all timelike worldlines terminate.
• Spacelike Infinity ($i^0$): The point that is infinitely far away in a spatial direction. All spacelike geodesics begin and end here.
• Past Null Infinity (${\mathcal{I}}^{-}$ or "scri minus"): The boundary in the infinite past from which all light rays originate.
• Future Null Infinity (${\mathcal{I}}^{+}$ or "scri plus"): The boundary in the infinite future where all light rays terminate.

Any event in the interior of the diamond is a point in spacetime, and because every point represents a 2-sphere, we have a map of the entire 4D Minkowski spacetime.

2. Constructing the Schwarzschild-Kruskal Penrose Diagram

Now, we will apply this same logic to the maximally extended Schwarzschild spacetime. The good news is that our work in the previous lesson has already put the metric into a form that is ripe for this transformation. Recall the Kruskal-Szekeres metric:

The crucial part is the $(-d{T}^2+d{X}^2)$ term. The radial and time components of the metric are conformally equivalent to 2D Minkowski space. Therefore, we can apply the exact same "squashing" transformation to the $(T,X)$ coordinates that we just saw applied to the $(t,r)$ coordinates of flat space.

The process is a sequence of precise coordinate transformations. While the algebra can be intricate, the logic is straightforward. We want to find a set of coordinates that covers the entire spacetime and brings the infinities to finite coordinate values.

Let's summarize the key results from the paper. The procedure yields a set of compactified coordinates, which we'll call $(T^{\prime },X^{\prime })$ to distinguish them from the Kruskal $(T,X)$, that cover the entire manifold in a finite region. The metric takes the form:

where ${\Omega }^2$ is the new overall conformal factor. Because the $(T^{\prime },X^{\prime })$ part is explicitly the 2D Minkowski metric, we can now draw the spacetime diagram as a diamond, just like we did for flat space.

3. Interpreting the Diagram

With the coordinates established, we can now identify the various regions and boundaries of the complete Schwarzschild spacetime on our finite map. The paper you just read provides all the necessary relations.

Putting everything together, we get the celebrated Penrose diagram for the maximally extended Schwarzschild spacetime.

*This diagram represents the entire maximally extended Schwarzschild spacetime. Each point is a 2-sphere. Radial light rays travel at 45 degrees.

• Regions I and III are the two asymptotically flat "universes".
• Region II is the black hole interior.
• Region IV is the white hole interior.
• The wavy lines at the top and bottom represent the future and past spacelike singularities ($r=0$).
• The diagonal lines separating the regions are the event horizons ($r=2M$).
• The boundaries are labeled with the various infinities: $i^{\pm }$ (timelike), $i^0$ (spacelike), and ${\mathcal{J}}^{\pm }$ (null). Notice that each of the two exterior regions has its own complete set of infinities.*

The power of this diagram is its ability to visualize complex causal relationships. For example:

• An observer starting in Region I who stays outside the black hole has a worldline that begins at $i^{-}$ and ends at $i^{+}$.
• A photon sent from Region I that escapes to infinity has a worldline that begins on ${\mathcal{I}}^{-}$ and ends on ${\mathcal{I}}^{+}$.
• An observer who falls into the black hole has a worldline that starts at $i^{-}$ in Region I, crosses the future event horizon into Region II, and inevitably terminates on the future singularity at $r=0$.
• Nothing can travel from Region I to Region III, as this would require moving outside the local light cone (i.e., faster than light). The connection between them, the Einstein-Rosen bridge, pinches off too quickly to be traversed.

Conclusion

Today we have reached a major milestone. By mastering the Penrose diagram, you have gained a powerful tool for visualizing and understanding the global properties of complex spacetimes.

Key Takeaways:

• A Penrose diagram is a conformal transformation of a spacetime that maps its infinite regions onto a finite diagram, while preserving the causal structure (i.e., light cones remain at 45 degrees).
• The construction proceeds by transforming to coordinates where the metric is conformally flat (like Kruskal's), and then applying a "squashing" transformation (like arctan) to bring infinity to a finite boundary.
• The Penrose diagram for the maximally extended Schwarzschild solution is a diamond shape containing four regions: two asymptotically flat exteriors (a "universe" and a "parallel universe"), a black hole interior, and a white hole interior.
• The diagram makes the causal structure manifest: the spacelike nature of the $r=0$ singularity and the null nature of the $r=2M$ horizons are clear, as are the fates of all possible observers and light rays.

Preview of Next Lesson:

We have now exhaustively analyzed the spacetime of a static, uncharged, non-rotating black hole. What happens if we add electric charge? In the next lesson, we will derive the Reissner-Nordström metric for a charged, non-rotating black hole. We will find that the presence of charge dramatically alters the interior structure, leading to an even more fascinating and complex Penrose diagram.