Mini Physics
Part of Mini EducationSwitch subject
Search

Confirming Time Dilation, Length Contraction and Simultaneity

Key idea: Use the Lorentz transformation difference form to derive time dilation, length contraction, and relativity of simultaneity as special cases.

  • 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.

This lesson shows how time dilation, length contraction, and relativity of simultaneity all drop out of the same Lorentz transformation equations (in difference form).

This page is the maintained consolidation step that ties the whole special-relativity cluster together before practice.

1. Definitions (Must Know)

  • Standard setup: frame S' moves at speed v along + x relative to frame S.

  • Lorentz factor: γ = 1/(square root of (1-v²/c²))

  • Difference form (two events): Δ x' = γ(Δ x-vΔ t)

    Δ t' = γ(Δ t-(vΔ x)/c²)

  • Inverse difference form: Δ x = γ(Δ x' + vΔ t')

    Δ t = γ(Δ t' + (vΔ x')/c²)

  • Proper time, Δτ: time between two events measured in the frame where Δ x' = 0 (events at the same place).

  • Proper length, L₀: length of an object measured in its rest frame.

  • Symbols used in this lesson: Δ x,Δ x' (m), Δ t,Δ t' (s), v (m s⁻¹), c (m s⁻¹), γ (dimensionless), Δτ (s), L,L₀ (m).

2. Key Ideas (What Earns Marks)

  • Pick the right “special case” by setting the correct condition:
    • simultaneity in a frame: Δ t = 0 (or Δ t' = 0),
    • same place in a frame: Δ x = 0 (or Δ x' = 0).
  • Relativity of simultaneity comes from the vΔ x/c² mixing term in Δ t'.
  • Length contraction depends on measuring endpoints simultaneously in the measuring frame.

3. Detailed Explanations

A. Relativity of simultaneity (derived)

Assume two events are simultaneous in S': Δ t' = 0, Δ x' ≠ 0 Use the inverse time transformation: Δ t = γ(Δ t' + (vΔ x')/c²) = γ(vΔ x')/c² So Δ t ≠ 0 in general. Simultaneity is frame-dependent.

B. Time dilation (derived)

Assume two events occur at the same place in S': Δ x' = 0 Then: Δ t = γ(Δ t' + vΔ x'/c²) = γΔ t' Since Δ x' = 0, the interval Δ t' is a proper time Δτ: Δ t = γΔτ

C. Length contraction (derived)

Let a rod be at rest in S', so its proper length is: L₀ = Δ x' (measured at Δ t' = 0)

To measure its length L in S, you must measure endpoints simultaneously in S: Δ t = 0, L = Δ x

Use Δ x' = γ(Δ x-vΔ t) with Δ t = 0: L₀ = Δ x' = γΔ x = γ L ⇒ L = L₀/γ

4. Common Mistakes

  • Mixing up which frame’s simultaneity condition you’re using (Δ t = 0 vs Δ t' = 0).
  • Forgetting that a “length measurement” requires simultaneity in the measuring frame.
  • Treating Δ t' as proper time even when Δ x' ≠ 0 (then it’s not one-clock time).

5. Exam Tips

  • Write the difference-form Lorentz equations first, then set conditions.
  • Use words to avoid sign errors: “S' moves at + v relative to S”.
  • Quick checks:
    • if v = 0, you should get Δ x' = Δ x and Δ t' = Δ t.
    • if Δ x' = 0, you should recover time dilation.

6. Worked Examples

Modelled example 1

Simultaneity difference from separation

Core

Problem

Two events are simultaneous in S' and separated by Δ x' = 3.00 × 10² m. At v = 0.60c, find Δ t in S.
Study the worked solution

  1. Translate the condition

    Method

    Simultaneous in S' means Δ t' = 0.

    Reason

    The prime belongs to the named frame.

    Working

    Δ t' = 0

  2. Choose and simplify

    Method

    Use the inverse time-difference transformation.

    Reason

    The known separation and condition are primed.

    Working

    Δ t = γ(0 + (vΔ x')/c²)

  3. Calculate

    Method

    Use γ = 1.25 at 0.60c.

    Reason

    The spatially separated events acquire a non-zero time separation in S.

    Working

    Δ t = 7.50 × 10⁻⁷ s

Guided practice 2

Length contraction from Δ t = 0

About 5 min

Problem

A rod at rest in S' has L₀ = 2.0 m and moves at 0.80c relative to S. Derive the condition used in S and find L.

Try this before viewing the solution

Hints

Hint 1: measure both endpoints together
A length in S uses endpoint events with Δ t = 0.
Hint 2: use the forward spatial difference
Then Δ x' = γ(Δ x-vΔ t) simplifies directly.
View solution step by step

  1. Apply simultaneity

    Method

    Set Δ t = 0 in the measuring frame S.

    Reason

    Both moving endpoints must be located at the same S time.

    Working

    Δ t = 0

  2. Map lengths

    Method

    Identify Δ x' = L₀ and Δ x = L.

    Reason

    The rod is at rest in S'.

    Working

    L₀ = γ L ⇒ L = L₀/γ

  3. Calculate

    Method

    Use γ = 1.67.

    Reason

    v = 0.80c.

    Working

    L = 2.0/1.67 = 1.2 m

Common misconception 3

Time dilation from Δ x' = 0

Find and correct the mistake

Learner claim

Two events have Δ x' = 0, Δ t' = 2.0 s and v = 0.80c. A learner says the longer S interval is proper because it is measured by the laboratory. Explain the proper-time error and calculate Δ t.

Try this before viewing the solution

Proper-time condition

View solution step by step

  1. Identify proper time

    Method

    Δ t' = Δτ.

    Reason

    The two events are co-located in S'.

    Working

    Δ x' = 0

  2. Simplify the inverse time equation

    Method

    The spatial term vanishes.

    Reason

    vΔ x'/c² = 0.

    Working

    Δ t = γΔ t'

  3. Calculate

    Method

    Use γ = 1.67.

    Reason

    The non-proper interval is dilated.

    Working

    Δ t = (1.67)(2.0) = 3.3 s

Examiner practice 4

Simultaneity in S (compute Δ t')

4 marks

Examination question

In S, simultaneous events are separated by 3.00 × 10² m. Frame S' moves at 0.60c along + x. Find and interpret Δ t'. [4 marks]

Try this before viewing the solution

View solution step by step

  1. Select equation

    1 mark

    Method

    Use the forward time-difference transform.

    Reason

    The given quantities are unprimed.

    Working

    Δ t' = γ(Δ t-(vΔ x)/c²)

  2. Apply condition

    1 mark

    Method

    Set Δ t = 0 and use γ = 1.25.

    Reason

    The events are simultaneous only in S.

    Working

    Δ t' = -γ vΔ x/c²

  3. Calculate

    1 mark

    Method

    Substitute signed values.

    Reason

    v and Δ x are positive.

    Working

    Δ t' = -7.50 × 10⁻⁷ s

  4. Interpret

    1 mark

    Method

    The events are not simultaneous in S' and event 2 is earlier.

    Reason

    Δ t' = t'₂-t'₁ < 0.

    Working

    t'₂ < t'₁

Challenge 5

Which condition corresponds to which effect?

Minimal support

Independent transfer

Match and justify: (1) Δ x' = 0, (2) Δ t = 0, (3) Δ t' = 0 with time dilation, a moving-length measurement in S, and events simultaneous in S' but generally not S.

Try this before viewing the solution

Hints

Hint 1: translate symbols into events
Orient by saying “same place” for Δ x = 0 and “same time” for Δ t = 0 in the matching frame.
View solution step by step

  1. Co-location

    Method

    Δ x' = 0 identifies proper time in S' and leads to time dilation.

    Reason

    One primed-frame clock records both events.

    Working

    Δ t = γΔ t'

  2. Unprimed simultaneity

    Method

    Δ t = 0 is the endpoint condition for measuring a moving length in S.

    Reason

    Both endpoints must be located together in S time.

    Working

    L = L₀/γ

  3. Primed simultaneity

    Method

    Δ t' = 0 defines simultaneity in S' but generally gives Δ t ≠ 0.

    Reason

    The inverse time transform contains vΔ x'/c².

    Working

    Δ t = γ vΔ x'/c²

7. Mind Stretchers

Mind stretcher 1: Why the three results are linkedExtension

If you accept time dilation and insist that c is invariant, why must simultaneity become frame-dependent?

Answer

Because different frames must relate time coordinates in a way that keeps light-speed measurements consistent. That requires t' to depend on both t and x (the vΔ x/c² term), which forces relativity of simultaneity.

Mind stretcher 2: Where does simultaneity come from in the equations?Extension

Time dilation can be seen from Δ x' = 0 even if you never think about simultaneity. So why does the Lorentz time equation still need the vΔ x/c² term?

Answer

Because the time transformation must handle events separated in space (Δ x ≠ 0). The vΔ x/c² term is exactly what makes simultaneity frame-dependent and is required for consistency with the invariance of c and the correct transformation of light-like intervals.

8. Optional/Enrichment: The One-Liner Summary

Special relativity is “space and time mix”. The Lorentz transformations encode that mixing; time dilation, length contraction, and relativity of simultaneity are all special cases of the same equations.

Next in the maintained sequence: the Special Relativity Hub revision/practice block.

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