Alright, let’s walk through the step-by-step derivation of Penrose diagrams—this is a staple tool in relativity for visualizing the full causal structure of spacetime, including all those tricky "infinity" regions that we can’t see in regular coordinate plots. We’ll start with flat Minkowski spacetime (the simplest case) before touching on how this generalizes to curved spacetimes like black holes.

First, we can ignore the y and z dimensions because Penrose diagrams focus on causal structure, which is rotationally symmetric. So we work with the 2D Minkowski line element:

ds² = -dt² + dx²

To make light paths easier to handle, we switch to null coordinates (aka light coordinates):

u = t - x
v = t + x

Substitute these into the line element, and you’ll get:

ds² = -du dv

Why null coordinates? Because light rays follow paths where u = constant (moving left along the future light cone) or v = constant (moving right along the future light cone)—this makes their trajectories dead simple to track later.

The problem with regular Minkowski coordinates is that u and v range from -∞ to +∞—we can’t plot that on a finite diagram. So we use a conformal transformation: a mathematical trick that scales spacetime (preserving angles and causal relationships) to map infinite ranges into finite ones.

We define new coordinates using arctangent functions (since arctan maps (-∞, ∞) to (-π/2, π/2)):

U = arctan(u)
V = arctan(v)

Now let’s update the line element. First, compute the differentials:
du = sec²(U) dU = (1 + tan²(U)) dU = (1 + u²) dU
dv = sec²(V) dV = (1 + v²) dV
Substitute back into ds² = -du dv:

ds² = -(1 + u²)(1 + v²) dU dV

The term (1 + u²)(1 + v²) is called the conformal factor—it scales the spacetime metric, but crucially, it doesn’t change the causal structure (light still moves along paths where dU/dV = ±1, i.e., 45-degree lines). We can define a "conformal metric" dŝ² that ignores this scaling factor to focus on the causal structure:

dŝ² = -dU dV

Now our coordinates U and V are bounded between -π/2 and π/2—perfect for a finite diagram!

To make this easier to draw, we convert U and V into more intuitive t' (vertical, time) and x' (horizontal, space) coordinates:

t' = (U + V)/2
x' = (V - U)/2

Let’s map the key regions of Minkowski spacetime to this new coordinate system:

• Past timelike infinity (i⁻): When t → -∞ and x is fixed, u = t - x → -∞ and v = t + x → -∞. So U → -π/2, V → -π/2, which gives t' = -π/2, x' = 0 (the bottom midpoint of our diagram).
• Future timelike infinity (i⁺): When t → +∞ and x is fixed, u → +∞, v → +∞. So U → π/2, V → π/2, giving t' = π/2, x' = 0 (the top midpoint).
• Spacelike infinity (i⁰): When x → ±∞ and t is fixed, u → ∓∞, v → ±∞. This maps to t' = 0, x' = ±π/2 (the left and right midpoints).
• Past null infinity (ℐ⁻): This is where incoming light rays originate. For u → -∞ (left-moving light) or v → -∞ (right-moving light), we get the left and bottom edges of the diagram (excluding the timelike infinity points).
• Future null infinity (ℐ⁺): This is where outgoing light rays end up. For u → +∞ or v → +∞, we get the right and top edges (excluding the timelike infinity points).

Now we can sketch the diagram as a square:

• All timelike worldlines (like a particle moving slower than light) start at i⁻ and end at i⁺, curving within the square but never crossing the 45-degree light paths.
• All null (light) worldlines are straight 45-degree lines, running from ℐ⁻ to ℐ⁺.
• All spacelike worldlines connect points on the left and right edges (near i⁰).

The same core idea applies to curved spacetimes—we just need to first rewrite the metric in a form that lets us apply the conformal transformation:

1. For a spacetime like the Schwarzschild black hole, start with its line element and convert to tortoise coordinates (r*) to eliminate the coordinate singularity at the event horizon.
2. Switch to null coordinates (u = t - r*, v = t + r*) to simplify the metric into a form similar to Minkowski’s.
3. Apply a conformal transformation to squash the infinite u/v ranges into finite coordinates.
4. Convert to Cartesian-like coordinates to draw the diagram, which will now include features like the event horizon, inner horizon, and spacetime singularity.

• Penrose diagrams rely on conformal compactization: scaling spacetime to fit infinity into a finite plot while preserving causal relationships (light always moves at 45 degrees).
• Every boundary of the diagram corresponds to a type of infinity, making it easy to see where all particles and light rays originate and end up.
• For curved spacetimes, the process is just a generalization of the Minkowski case—adapt the metric to null coordinates, apply the conformal transformation, and plot.

内容的提问来源于stack exchange，提问作者OkaIki