Mini Physics
Part of Mini EducationSwitch subject
Search

Simultaneity and Relativity of Time

Key idea: Learn why simultaneity is frame-dependent in special relativity, using the train/boxcar thought experiment and exam-style questions.

  • GCE A-Level H3 Physics 2027
On this page

Learning objectives

  • discuss qualitatively the results of the Michelson–Morley interferometer experiment and its implications on the ether theory (knowledge of the details of the experiment is not required)
  • state the postulates of the special theory of relativity, that in all inertial frames, the laws of physics are the same and the speed of light in free space is the same regardless of the motion of the light source or observer
  • appreciate the failure of Galilean transformation equations when applied to a moving source of light
  • discuss the concept of simultaneity
  • show an understanding of the terms proper time and proper length
  • apply the Lorentz transformation equations to solve one-dimensional problems
  • Derive the time dilation formula and the length contraction formula, making use of the Lorentz factor.
  • apply the time dilation formula and the length contraction formula in related situations (e.g. the lifetime of fast-moving muons) or to solve problems
  • use the one-dimensional relativistic velocity addition formula to calculate velocities in different inertial frames or to solve problems
  • Apply the relativistic energy–momentum relation E² = (pc)² + (mc²)² to solve problems, including selecting its limiting form.
  • Show that E² = (pc)² + (mc²)² reduces to E = pc for massless particles and to E = mc² + ½mv² at low speeds.
Lightning striking both ends of a moving train while ground and train observers compare the arriving light pulses
The ground observer is midway between the ground strike positions; the moving train observer approaches the front pulse and recedes from the rear pulse.

Simultaneity is the key “mind bend” of special relativity: two events can be simultaneous in one inertial frame but not in another.

This maintained page sets up every later derivation, so keep frame labels and simultaneity conditions explicit before moving on.

1. Definitions (Must Know)

  • Event: something that happens at a specific place and time, described by (x,t) in a chosen frame.
  • Simultaneous (in a given frame): two events have the same time coordinate in that frame (t₁ = t₂).
  • Inertial frame: a frame moving at constant velocity where the laws of physics take their standard form.

2. Key Ideas (What Earns Marks)

  • “Simultaneous” is not just “I saw them at the same time.” You must account for signal travel time using a procedure consistent with the postulates.
  • If all inertial observers measure the same speed of light in vacuum (c), then different inertial frames disagree on what events are simultaneous.
  • Relativity of simultaneity is not an “illusion”; it is a consequence of how space and time coordinates relate between frames.

Quick comparison:

StatementCorrect?Why
“I saw them at the same time”Not sufficientLight travel time matters
“They have the same time coordinate in my frame”Definition of simultaneous (in that frame)Depends on synchronization procedure
“If simultaneous in one inertial frame, simultaneous in all”FalseLorentz time mixes t and x

3. Detailed Explanations

A. Why “seeing” is not the same as “happening”

Light takes time to reach you. If you see two flashes simultaneously, you can only conclude the events were simultaneous in your frame if you also know:

  • you are equidistant from the event locations (in your frame), or
  • you have a synchronized clock network and have corrected for signal delays.

B. The train/boxcar lightning thought experiment (core idea)

Two lightning bolts strike the two ends of a moving boxcar. Consider two observers:

  • O on the ground, midway between the ground strike marks.
  • O' on the boxcar, midway between the boxcar ends (in the boxcar’s frame).

In the ground frame, if the light reaches O at the same time and the distances are equal, O concludes the strikes were simultaneous in the ground frame.

But O' is moving toward one flash and away from the other (in the ground frame). Since both observers must measure light moving at speed c, O' concludes the strike at the front end happened first in the boxcar frame.

The conclusion: Events simultaneous in one inertial frame are generally not simultaneous in another inertial frame moving relative to it.

C. Connection to the postulates

This is a direct consequence of:

4. Common Mistakes

  • Saying “they saw different times because of signal delay” and stopping there (each observer corrects for signal travel in their own frame).
  • Assuming simultaneity is absolute, then trying to force Galilean time (t' = t) onto special relativity.
  • Forgetting that “midpoint” depends on frame (lengths and simultaneity are linked).

5. Exam Tips

  • When asked “are they simultaneous?”, always include “in which frame?”
  • Use the safe phrasing: “Simultaneity is frame-dependent because all inertial observers measure the same light speed.”
  • If you use the train thought experiment, state explicitly who is moving toward which flash (this anchors the reasoning).

6. Worked Examples

Modelled example 1

Interpreting “I saw both flashes together”

Core

Problem

You are at rest on the ground exactly midway between two lampposts. You receive their light flashes together. Were the emissions simultaneous in the ground frame?
Study the worked solution

  1. Name the frame

    Method

    Make the conclusion only in the ground frame.

    Reason

    Simultaneity is a relation between time coordinates in a specified frame.

    Working

    Δ t_ground ?

  2. Compare signal paths

    Method

    The two light paths have equal ground-frame length.

    Reason

    You are at the midpoint and both pulses travel at c.

    Working

    t_(travel,A) = d/c = t_(travel,B)

  3. Infer emission times

    Method

    The emissions were simultaneous in the ground frame.

    Reason

    Equal arrival times minus equal travel times give equal emission times.

    Working

    Δ t_(emission,ground) = 0

Guided practice 2

Train lightning (concept)

About 5 min

Problem

In the train-lightning scenario, explain how the ground and train observers can disagree about simultaneity without either observer measuring a different vacuum light speed.

Try this before viewing the solution

Hints

Hint 1: track the train observer
In the ground frame, the train observer moves toward the front pulse and away from the rear pulse.
Hint 2: keep c invariant
The train observer must still assign speed c to both pulses, so unequal reception cannot be explained by unequal light speeds.
View solution step by step

  1. Ground-frame conclusion

    Method

    The ground midpoint observer receives equal-path pulses together and assigns simultaneous ground-frame emission times.

    Reason

    The observer is equidistant from the ground strike positions.

    Working

    Δ t_ground = 0

  2. Train-frame conclusion

    Method

    The train observer assigns the front strike an earlier train-frame time.

    Reason

    The observer approaches the front pulse and recedes from the rear pulse while measuring both at c.

    Working

    t'_front < t'ᵣₑₐᵣ

  3. Reconcile

    Method

    Both descriptions are valid in their own inertial frames.

    Reason

    Lorentz transformations assign frame-dependent time coordinates while preserving the laws and c.

    Working

    Δ t = 0, Δ t' ≠ 0

Common misconception 3

Events simultaneous in S, not in S' (numerical)

Find and correct the mistake

Learner claim

In S, two events are simultaneous and separated by 240 m along + x. Frame S' moves at 0.60c along + x. A learner concludes Δ t' = 0 because the events are simultaneous. Test the claim.

Try this before viewing the solution

Transformed time interval

View solution step by step

  1. Select the equation

    Method

    Use the Lorentz time-difference transformation.

    Reason

    The question compares two spatially separated events across frames.

    Working

    Δ t' = γ(Δ t-(vΔ x)/c²)

  2. Apply the condition

    Method

    Set Δ t = 0 and calculate γ = 1.25.

    Reason

    Simultaneity is given only in S; the spatial term remains.

    Working

    Δ t' = -γ(vΔ x)/c²

  3. Calculate and interpret

    Method

    The events are not simultaneous in S'; event 2 occurs earlier there.

    Reason

    The transformed interval is negative under the stated ordering.

    Working

    Δ t' = -1.25(0.60c)(240)/c² = -6.00 × 10⁻⁷ s

Examiner practice 4

“Saw simultaneously” but not at the midpoint

4 marks

Examination question

Lampposts A and B are 200 m apart. You stand 50 m from A and 150 m from B and receive both flashes together. Determine their ground-frame emission order and explain it. [4 marks]

Try this before viewing the solution

View solution step by step

  1. Compare distances

    1 mark

    Method

    The pulse from A travels 50 m; the pulse from B travels 150 m.

    Reason

    The observer is not at the midpoint.

    Working

    d_A = 50 m, d_B = 150 m

  2. Compare travel times

    1 mark

    Method

    The B pulse travels for 100/c longer.

    Reason

    Both propagate at c in the ground frame.

    Working

    t_(B,travel)-t_(A,travel) = 100/c

  3. Infer order

    1 mark

    Method

    A emitted later than B.

    Reason

    Its shorter travel time compensates for its later start.

    Working

    t_(A,emit)-t_(B,emit) = 100/c > 0

  4. State conclusion

    1 mark

    Method

    The emissions were not simultaneous in the ground frame.

    Reason

    Equal reception times do not cancel unequal signal delays.

    Working

    Δ tₑₘᵢₜ ≠ 0

Challenge 5

What do you need to claim simultaneity in your frame?

Minimal support

Independent transfer

Design a valid procedure for deciding whether two distant detector events are simultaneous in your inertial frame. State what must be synchronized, what is recorded and what is compared.

Try this before viewing the solution

Hints

Hint 1: put clocks at the events
Orient around local time readings, not the time at which one distant observer later receives reports.
View solution step by step

  1. Prepare the frame

    Method

    Place clocks at both detector locations and synchronize them in the chosen inertial frame.

    Reason

    Distant simultaneity requires a frame-specific synchronization convention.

    Working

    C_A ↔ C_B synchronized in S

  2. Record locally

    Method

    Record each event’s local clock reading.

    Reason

    Local records avoid confusing event time with later signal arrival at a third location.

    Working

    t_A, t_B

  3. Compare

    Method

    Call the events simultaneous in S exactly when the recorded readings match.

    Reason

    Equal synchronized-clock coordinates are the operational definition.

    Working

    t_A = t_B ⇒ Δ t_S = 0

7. Mind Stretchers

Mind stretcher 1: What becomes impossible without frame-dependent simultaneity?Extension

If simultaneity were absolute (same for all inertial frames), what would that imply about the speed of light measurements in different inertial frames?

Answer

It would force a Galilean-style velocity addition for light (some observers would measure c± u), contradicting the postulate that all inertial observers measure c in vacuum.

Mind stretcher 2: Why simultaneity and length measurement are linkedExtension

Explain briefly why a “length of a moving object” measurement in your frame requires simultaneity in your frame.

Answer

To measure length you need the positions of both ends at the same time in your frame. If you use different times for the two ends, the object has moved in between and you are not measuring a single-frame length; this is why relativity of simultaneity is tied to length contraction.

8. Optional/Enrichment: Lorentz Transformations and a Formula

Using Lorentz transformations, the time difference between two events in different frames depends on both Δ t and Δ x. This is why spatial separation matters for simultaneity.

Next steps:

Next in the maintained sequence: Lorentz Transformations.

Continue with the next resource in this course.

Course and syllabus information
Course
GCE A-Level H3 Physics
Edition
GCE A-Level H3 Physics 2027