Explain how the concept of simultaneity in special relativity leads to the conclusion that two observers in relative motion will each measure the other's clock to be running slower than their own.

IB DP Physics Higher Level (2023 syllabus) — A.5 Galilean and special relativity (HL only) · Explain · 4 marks · View as Markdown

Written & reviewed by James Millett — Biology (Imperial College London), PGCE Science (University of Cambridge).

Two spaceships, A and B, pass each other in empty space at a constant relative velocity of 0.80c. Each ship carries an identical clock. An observer on ship A measures the time interval between two events that both occur at the same location on ship A.

Model answer (4 marks)

The observer on A measures a proper time interval between the two events, because both events occur at the same location in A’s frame – this is the shortest possible interval for those events.

In B’s frame the two events are spatially separated, so B must use synchronised clocks at different positions to record the interval. This gives a longer, dilated time interval.

The argument is symmetric: an observer on B will likewise find that A’s clock runs slow, because events at a fixed location on B are separated in space in A’s frame.

The apparent paradox is resolved by the relativity of simultaneity: the two observers disagree about which events are simultaneous, so there is no single universal ‘now’ to compare the clocks at the same instant. The contradiction only appears if one assumes absolute simultaneity.

Examiner tips

  • Use the term ‘proper time’ for the interval measured in A’s frame. Show that B’s interval is dilated by the Lorentz factor. Explain the symmetry of the situation. Mention that relativity of simultaneity removes the paradox.
  • common_mistakes
  • :
  • Confusing proper time with dilated time. Forgetting that simultaneity is relative and that the comparison of clocks requires a common instant. Using the wrong frame to state which clock runs slow.

Mark scheme (4 marks)

  1. The observer on A measures a proper time interval between the two events (since both events occur at the same location in A's frame), which is the shortest possible time interval for those events.
  2. The observer on B measures a longer (dilated) time interval between the same two events because the events occur at different locations in B's frame, requiring synchronised clocks at different positions.
  3. The situation is symmetric: by the same reasoning, an observer on A measures B's clock to be running slow, because events at a fixed location on B are spatially separated in A's frame.
  4. This apparent paradox is consistent because simultaneity is relative: the two observers disagree on which events are simultaneous, so there is no single universal 'now' to compare the clocks at the same instant — the contradiction only arises if absolute simultaneity is assumed.

Key terms in this question

simultaneity

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