Questions: Spacetime Diagrams and Minkowski Geometry

Question 1

Observer A measures two events separated by Δt = 5 s and Δx = 0 m. Observer B moves relative to A and measures Δt' = 8 s and Δx' ≠ 0. Which quantity is the same for both observers?

• A. Δt = 5 s — time intervals are absolute in special relativity
• B. Δx = 0 m — spatial separations are invariant under boosts
• C. c²(Δt)² − (Δx)² — the spacetime interval
• D. Δt + Δx/c — the coordinate sum of time and space

Answer: (C)

The spacetime interval s² = c²(Δt)² − (Δx)² is the Lorentz-invariant quantity — all inertial observers agree on it, even though they disagree on Δt and Δx separately. For A: s² = c²(25) − 0 = 25c². For B: s² = c²(64) − (Δx')². Setting these equal gives (Δx')² = c²(64 − 25) = 39c², consistent with length contraction and time dilation being two aspects of the same invariant geometry. This is the relativistic analogue of how spatial rotations preserve r² = x² + y² even while changing x and y individually.

Question 2

Two events are simultaneous in frame S — they lie on the same horizontal line (constant t) in the Minkowski diagram. In frame S' moving relative to S, what is true?

• A. The events are also simultaneous in S', because simultaneity is an objective fact about events
• B. The events are not simultaneous in S'; lines of constant t' are tilted relative to horizontal lines in the S diagram
• C. The events are simultaneous in S' only if they are also co-located
• D. Whether the events are simultaneous in S' depends on whether they lie inside or outside the light cone

Answer: (B)

This is the geometric content of the relativity of simultaneity. In a Minkowski diagram for S, the x-axis (t = 0) is horizontal. For a frame S' moving at velocity v, the line of constant t' = 0 is tilted: it makes an angle arctan(v/c) with the horizontal. Two events on the same horizontal line (simultaneous in S) generally lie on different t' = const lines (not simultaneous in S'). Option C is wrong: co-location (Δx = 0) and simultaneity (Δt = 0) are distinct conditions. Option D describes causal structure, which is separate from simultaneity.

Question 3

On a Minkowski diagram with ct on the vertical axis and x on the horizontal axis, a light ray always traces a 45° line.

• T. True
• F. False

Answer: True

True, and this is exactly why we use ct rather than t on the vertical axis. A light ray satisfies x = ct (or x = −ct), so dx/(d(ct)) = ±1, which is a slope of ±1 — a 45° line. The choice of ct normalizes the light speed to 1 in diagram units, making the light cone's geometry visually clean and universal. Every inertial frame's light cone has 45° boundaries. The constraint that no worldline tilts more than 45° from vertical is the geometric statement that nothing travels faster than light.

Question 4

Two events that are simultaneous in one inertial reference frame are simultaneous in most inertial reference frames.

• T. True
• F. False

Answer: False

False. Simultaneity is relative — this is one of the most fundamental (and counterintuitive) consequences of special relativity. Two spatially separated events that occur at the same time in one frame occur at different times in a frame moving relative to the first. On a Minkowski diagram, this is visible: the tilted 'horizontal' lines (t' = const) of a moving frame cut across the untilted lines (t = const) of the rest frame. Only events at the same location (Δx = 0) are necessarily simultaneous in all frames if they are simultaneous in any.

Question 5

Explain why the spacetime interval s² = c²t² − x² is invariant under Lorentz boosts, and what this invariance reveals about the geometry of spacetime.

Model answer: A Lorentz boost mixing time and space is a hyperbolic rotation in spacetime — it changes t and x but preserves c²t² − x², just as an ordinary spatial rotation changes x and y but preserves x² + y². The sign difference (+ vs −) between the time and space terms reflects the Minkowski (non-Euclidean) signature of spacetime. Physical meaning: all inertial observers agree on whether two events are timelike-separated (s² > 0, causal connection possible), lightlike-separated (s² = 0, connected by a light signal), or spacelike-separated (s² < 0, no causal connection possible). The invariance is not an accident but encodes the physical requirement that the speed of light is the same in all inertial frames — it is the algebraic consequence of the postulates.

One way to see it: the Lorentz transformation is defined precisely as the linear transformation that preserves c²t² − x² (and its generalization to higher dimensions: c²t² − x² − y² − z²). The geometry of spacetime is Minkowski geometry, characterized by this pseudo-Riemannian metric. The invariant interval plays the same foundational role in relativistic physics that the invariant distance r² = x² + y² + z² plays in Euclidean geometry.