Explain how the Lorentz transformation for time differs from the Galilean transformation for time, and describe the physical consequence this difference has for the synchronisation of clocks in two inertial frames moving relative to each other.
Written & reviewed by James Millett — Biology (Imperial College London), PGCE Science (University of Cambridge).
In Galilean relativity, time is assumed to be absolute — the same for all observers regardless of their state of motion. Special relativity replaces this assumption with a transformation that couples time to spatial displacement.
Model answer (4 marks)
In the Galilean transformation time is absolute: t′ = t, so all observers agree on the time of any event irrespective of their relative velocity.
In the Lorentz transformation time is coupled to space: t′ = γ(t – vx/c²). The time of an event in the moving frame depends on both its time coordinate and its spatial position x in the stationary frame.
Because of the spatial term, two events that are simultaneous in one frame (t₁ = t₂) will in general have different times in the other frame (t′₁ ≠ t′₂). Thus clocks that are synchronised in one inertial frame are not synchronised in a frame moving relative to it.
The synchronisation error grows with the spatial separation of the clocks (∝ vx/c²), showing that this is a fundamental, not mechanical, effect of relativity.
In the Lorentz transformation time is coupled to space: t′ = γ(t – vx/c²). The time of an event in the moving frame depends on both its time coordinate and its spatial position x in the stationary frame.
Because of the spatial term, two events that are simultaneous in one frame (t₁ = t₂) will in general have different times in the other frame (t′₁ ≠ t′₂). Thus clocks that are synchronised in one inertial frame are not synchronised in a frame moving relative to it.
The synchronisation error grows with the spatial separation of the clocks (∝ vx/c²), showing that this is a fundamental, not mechanical, effect of relativity.
Examiner tips
- State the two equations explicitly; use γ and the correct sign. Show the consequence for simultaneity and clock synchronisation. Mention the dependence on spatial separation to justify the 4‑mark answer.
Common mistakes
- Confusing the sign in the Lorentz time equation (t′ = γ(t + vx/c²)). Forgetting that the Galilean time is t′ = t, not t′ = t + constant. Failing to explain that the synchronisation discrepancy is proportional to the spatial separation, not a mechanical limitation.
Mark scheme (4 marks)
- In the Galilean transformation, time is absolute: t′ = t, meaning all observers agree on the time of any event regardless of their relative velocity.
- The Lorentz transformation for time includes a spatial term: t′ = γ(t − vx/c²), so the time of an event depends on both its time coordinate and its spatial position x in the original frame.
- Because the Lorentz time transformation depends on position, two spatially separated events that are simultaneous (t₁ = t₂) in one frame will in general have different time coordinates in the other frame, so clocks synchronised in one frame are not synchronised in the other.
- The greater the spatial separation of the clocks, the larger the synchronisation discrepancy (proportional to vx/c²), showing that this is not a mechanical or practical limitation but a fundamental feature of spacetime.
Key terms in this question
Galilean transformation · Lorentz transformation · synchronisation
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