Spacetime Intervals and Four-Vectors · seed 1 · A4, ink-friendly. The answer key prints on its own page for grown-ups.
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The interval between two events

Science · Quantum & Modern Physics · ages 19-20
Name ______________________   Date ____________

  1. Two distinct events have s squared = 0. What kind of separation is this?

    • Timelike: connected below light speed
    • Spacelike: connected above light speed
    • Lightlike: connected at light speed

  2. Two inertial frames measure different time gaps for the same event pair. Which quantity must agree?

    • The measured time separation
    • The spacetime interval s squared
    • The measured spatial separation

  3. Which ordered set is the position four-vector?

    • The components (ct, x, y, z)
    • The components (t, speed, mass, energy)
    • The components (x, y, z, acceleration)

  4. Event separations are ct = 5, x = 3, y = z = 0. A boost along positive x has beta = 0.6 and gamma = 1.25. What are ct', x' and s squared?

    • ct' = 4, x' = 0, s squared = 25
    • ct' = 4, x' = 0, s squared = 16
    • ct' = 0, x' = 4, s squared = 16

  5. Two events are 6 seconds apart and 2 light-seconds apart in space. Give s squared in light-seconds squared, using time squared minus space squared.

    Answer: ______________

  6. Why can a frame-dependent event order coexist with causality in special relativity?

    • Causal signals can outrun light and reverse event order
    • Every event pair has the same order in every frame
    • Only spacelike pairs can reverse their event order

  7. Two events are 2 seconds apart and 5 light-seconds apart in space. Which conclusion follows from their interval?

    • s squared = -21: no causal signal can connect them
    • s squared = 21: a slower-than-light signal can connect them
    • s squared = -21: a light-speed signal can connect them

  8. A timelike interval proves that the earlier event actually caused the later event.

    Circle one:   True   False

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Answer key

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The interval between two events W1-mt_CYdi7Qpvrf-s1

  1. Lightlike: connected at light speed · Zero interval for distinct events means lightlike separation.
  2. The spacetime interval s squared · Lorentz transformations preserve s squared, even when the time and spatial separations change.
  3. The components (ct, x, y, z) · The position four-vector uses time multiplied by c and three spatial coordinates.
  4. ct' = 4, x' = 0, s squared = 16 · ct' = 1.25(5 - 1.8) = 4 and x' = 1.25(3 - 3) = 0. Both frames give s squared = 16.
  5. 32 · c delta t is 6 light-seconds, so s squared = 36 - 4 = 32 light-seconds squared.
  6. Only spacelike pairs can reverse their event order · Spacelike pairs cannot exchange a signal at speed c or below. Causally connectable pairs retain their order.
  7. s squared = -21: no causal signal can connect them · s squared = 4 - 25 = -21 light-seconds squared. This spacelike separation would require a signal faster than light.
  8. False · Timelike separation allows a causal signal but does not establish that a causal interaction occurred.
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