On a Penrose diagram, light rays always travel at 45-degree angles. What property of the conformal transformation guarantees this?
AThe transformation preserves all distances and angles
BThe transformation preserves the causal structure (null cones) — a conformal rescaling g_μν → Ω²g_μν does not change which curves are null
CThe transformation maps all geodesics to straight lines
DThe transformation preserves the Riemann curvature tensor
A conformal transformation multiplies the metric by a positive scalar function Ω²(x). Since null curves are defined by ds² = g_μν dx^μ dx^ν = 0, and Ω² > 0 does not change whether this expression vanishes, all null curves remain null under conformal transformations. Light cones are therefore preserved, which is why light rays can always be drawn at 45 degrees. The transformation does change distances, curvature, and the distinction between timelike curves of different proper lengths — but not the causal structure.
Question 2 True / False
On the Penrose diagram for the maximally extended Schwarzschild spacetime, the singularity at r = 0 appears as a horizontal (spacelike) line.
TTrue
FFalse
Answer: True
The Schwarzschild singularity at r = 0 is spacelike — it lies in the future (or past) of observers who cross the horizon, not at a fixed spatial location. On the Penrose diagram, it appears as a horizontal line at the top of region II (the black hole interior) and at the bottom of region IV (the white hole). Its spacelike character means it cannot be avoided once the horizon is crossed: all future-directed causal paths from inside the horizon terminate at the singularity. This visual representation makes the inevitability of hitting the singularity geometrically obvious.
Question 3 Short Answer
Explain how the Penrose diagram of a collapsing star differs from the maximally extended Schwarzschild Penrose diagram.
Think about your answer, then reveal below.
Model answer: The maximally extended Schwarzschild diagram has four regions: exterior (I), black hole interior (II), second exterior (III), and white hole (IV). For a realistic collapsing star, the spacetime before collapse is not vacuum — the star's interior replaces part of the diagram. The left half of the diagram (regions III and IV) is replaced by the interior of the collapsing star, which is described by a different metric (e.g., Oppenheimer-Snyder for a dust collapse). The result is a diagram with only the right exterior (I) and the black hole interior (II), with the star's surface worldline appearing as a timelike curve that crosses the horizon and hits the singularity. The white hole and second exterior are absent because they require the spacetime to have been vacuum for all past time.
This distinction is physically important: the wormhole (Einstein-Rosen bridge) connecting regions I and III in the eternal solution does not exist for a physically formed black hole. Penrose diagrams make this immediately clear by showing which regions of the maximal extension are actually realized.
Question 4 Short Answer
Identify and explain the five types of 'infinity' that appear on the boundary of a Penrose diagram for Minkowski spacetime.
Think about your answer, then reveal below.
Model answer: The five types are: (1) Future timelike infinity i⁺ — where all timelike geodesics end (the endpoint of the worldline of any massive particle that exists forever). (2) Past timelike infinity i⁻ — where all timelike geodesics begin. (3) Future null infinity J⁺ (scri-plus) — where all outgoing light rays end. (4) Past null infinity J⁻ (scri-minus) — where all incoming light rays originate. (5) Spatial infinity i⁰ — the single point representing all spatial directions at infinite distance at any given time. On the Penrose diagram for Minkowski space, these form the boundary of a diamond shape: i⁺ at the top vertex, i⁻ at the bottom, i⁰ at the right vertex, J⁺ as the upper-right diagonal edge, and J⁻ as the lower-right diagonal edge.
These five infinities provide a precise language for discussing the global behavior of fields and particles. For example, gravitational radiation is defined by the behavior of the gravitational field at J⁺ (future null infinity), and the Bondi mass measures the total energy remaining after radiation has escaped to J⁺.