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Time Dilation

Key idea: Learn time dilation using Δt = γΔτ, identify proper time, and solve exam-style problems including muons and moving clocks.

  • GCE A-Level H3 Physics 2027
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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.
A light clock with a vertical light path in its rest frame and a longer diagonal path in a frame where the clock moves
Both frames measure the same light speed, c. The longer diagonal path in the laboratory frame therefore takes a longer time.

Time dilation is the result that a moving clock ticks more slowly than a clock at rest in your frame. The key H3 formula is: Δ t = γ Δτ where Δτ is the proper time and γ is the Lorentz factor.

In the maintained path, pair this with simultaneity so you can justify why proper-time conditions are frame-specific.

1. Definitions (Must Know)

  • Lorentz factor: γ = 1/(square root of (1-v²/c²))
  • Proper time, Δτ: the time interval between two events measured in the frame where the two events occur at the same position (one clock can record both).
  • Dilated time, Δ t: the time interval measured in a frame where the same two events occur at different positions (the clock is moving relative to that frame).
  • Time dilation formula: Δ t = γ Δτ
  • Symbols used in this lesson: v relative speed (m s⁻¹), c speed of light (≈ 3.00 × 10⁸ m s⁻¹), γ (dimensionless), Δτ,Δ t (s).

Lorentz factor γ vs speed

The Lorentz factor γ increases slowly at low speeds but rises sharply as v approaches c.

Scroll across the graph to read all labels.

The Lorentz factor γ increases slowly at low speeds but rises sharply as v approaches c.The Lorentz factor γ increases slowly at low speeds but rises sharply as v approaches c.
Time dilation factor is γ: Δt = γΔτ.
Open full-size graph
View figure data
Values for Lorentz factor γ vs speed
Speed (v/c)γ
01
0.21.021
0.41.091
0.61.25
0.81.667
0.92.294
0.953.203
0.985.025
0.997.089

2. Key Ideas (What Earns Marks)

  • “Moving clocks run slow” means: for the same pair of events, Δ t ≥ Δτ with equality only when v = 0.
  • Proper time is measured in the clock’s rest frame (events at same place).
  • Time dilation is a consequence of the two postulates and the relativity of simultaneity.

Quick comparison:

QuantityMeaningHow to identify
Proper time ΔτOne clock measures both eventsEvents occur at same position in that frame (Δ x' = 0)
Dilated time Δ tOther inertial frame’s intervalSame events occur at different positions in that frame

3. Detailed Explanations

A. The light clock thought experiment (derivation)

In the clock’s rest frame, a light pulse travels up and down between two mirrors separated by distance d. One “tick” has proper time: Δτ = 2d/c

In a frame where the clock moves at speed v, the light travels a longer diagonal path, but still at speed c. For half a tick, geometry gives: ((cΔ t)/2)² = d² + ((vΔ t)/2)² Rearrange: Δ t = 2d/(c square root of (1-v²/c²))

Using Δτ = 2d/c gives

Δ t = γΔτ

B. What “proper” means (how to identify it)

If a single clock can stay at one place and record both events, that frame measures Δτ.

Example: “ticks of a wristwatch” are measured at the watch’s location in its own rest frame, so the watch measures its proper time.

4. Common Mistakes

  • Swapping Δ t and Δτ: proper time is the smaller interval.
  • Using time dilation for two events that occur at the same place in your frame (then your frame measures Δτ, not Δ t).
  • Treating γ as square root of (1-v²/c²) (it is the reciprocal).

5. Exam Tips

  • First decide which interval is proper: “Which frame sees both events at the same position?”
  • Write the formula in the safe direction: Δ t = γ Δτ where Δ t is the longer (“dilated”) time.
  • For v≪ c, you can sanity-check that γ ≈ 1 so the effect is small.

6. Worked Examples

Modelled example 1

Time dilation at 0.80c

Core

Problem

A spacecraft moves at 0.80c. A process takes Δτ = 1.0 μs in the spacecraft frame. Find its duration in the Earth frame.
Study the worked solution

  1. Identify proper time

    Method

    The spacecraft interval is proper.

    Reason

    The process occurs at one place in its own rest frame.

    Working

    Δτ = 1.0 μs

  2. Calculate gamma

    Method

    Use the speed ratio 0.80.

    Reason

    The Lorentz factor must exceed one.

    Working

    γ = 1/(square root of (1-0.80²)) = 1.67

  3. Dilate the interval

    Method

    Multiply the proper time by γ.

    Reason

    Earth sees the process move, so Earth measures the longer interval.

    Working

    Δ t = (1.67)(1.0 μs) = 1.7 μs

Guided practice 2

Finding speed from a time dilation factor

About 5 min

Problem

An observer measures Δ t = 2Δτ for a moving clock. Find v/c.

Try this before viewing the solution

Hints

Hint 1: extract gamma
Compare Δ t = γΔτ with the given factor.
Hint 2: square after isolating the root
From γ = 2, first write square root of (1-v²/c²) = 1/2.
View solution step by step

  1. Identify gamma

    Method

    The dilation factor equals the Lorentz factor.

    Reason

    Δ t/Δτ = γ.

    Working

    γ = 2

  2. Invert

    Method

    Solve the Lorentz-factor equation.

    Reason

    Squaring the isolated reciprocal root avoids a sign error.

    Working

    1-v²/c² = 1/4

  3. Calculate

    Method

    Select the positive speed magnitude.

    Reason

    v/c is non-negative here.

    Working

    v/c = square root of (3/4) = 0.866

Common misconception 3

Muon lifetime (application)

Find and correct the mistake

Learner claim

Muons have proper lifetime 2.2 μs and travel at 0.98c. A learner uses 2.2 μs as their Earth-frame lifetime. Explain the frame error and calculate the correct Earth-frame value.

Try this before viewing the solution

Frame measuring proper lifetime

View solution step by step

  1. Locate the events

    Method

    Creation and decay occur at the same position in the muon’s rest frame.

    Reason

    That frame therefore measures Δτ.

    Working

    Δτ = 2.2 μs

  2. Calculate gamma

    Method

    Evaluate at 0.98c.

    Reason

    The high speed produces a large factor.

    Working

    γ = 1/(square root of (1-0.98²)) = 5.03

  3. Find Earth time

    Method

    Earth measures the dilated interval.

    Reason

    The muon moves between the two events in Earth coordinates.

    Working

    Δ t = (5.03)(2.2 μs) = 11 μs

Examiner practice 4

How far can the muon travel? (Earth frame)

4 marks

Examination question

Using v = 0.98c and the unrounded Earth-frame lifetime 11.1 μs, estimate the distance a muon travels before decaying. [4 marks]

Try this before viewing the solution

View solution step by step

  1. Choose frame quantities

    1 mark

    Method

    Use speed and time measured in the Earth frame.

    Reason

    Distance d = vΔ t must use same-frame quantities.

    Working

    v = 0.98c, Δ t = 11.1 μs

  2. Convert time

    1 mark

    Method

    Write microseconds in seconds.

    Reason

    SI consistency is required with c.

    Working

    11.1 μs = 11.1 × 10⁻⁶ s

  3. Substitute

    1 mark

    Method

    Apply uniform-motion distance.

    Reason

    The Earth-frame muon speed is given.

    Working

    d = (0.98c)(11.1 × 10⁻⁶)

  4. Calculate

    1 mark

    Method

    Retain sensible significant figures.

    Reason

    The inputs are approximately two to three significant figures.

    Working

    d ≈ 3.3 × 10³ m

Challenge 5

Finding proper time from an observed interval

Minimal support

Independent transfer

A moving clock takes Δ t = 10.0 s between ticks in your frame and moves at 0.60c. Identify the proper-time frame and find Δτ.

Try this before viewing the solution

Hints

Hint 1: reverse the relation
Orient with Δ t = γΔτ and solve for the smaller interval.
View solution step by step

  1. Identify proper frame

    Method

    The moving clock’s rest frame measures proper time.

    Reason

    Both ticks occur at the same clock position there.

    Working

    Δτ belongs to the clock frame

  2. Calculate gamma

    Method

    At 0.60c, γ = 1.25.

    Reason

    1/square root of (1-0.60²) = 1/0.80.

    Working

    γ = 1.25

  3. Reverse the formula

    Method

    Divide the observed interval by γ.

    Reason

    The proper interval is the shorter one.

    Working

    Δτ = 10.0/1.25 = 8.0 s

7. Mind Stretchers

Mind stretcher 1: Why time dilation is not “just signal delay”Extension

Two observers correct for signal travel time in their own frames, yet still disagree on the time interval between the same two events. Why?

Answer

Because the disagreement is about the time coordinate assignment in different inertial frames (linked to how clocks are synchronized in each frame and the invariance of c), not about one observer “seeing late”. Even after correcting for light travel, the spacetime interval splits into different Δ t and Δ x in different frames.

Mind stretcher 2: “Each sees the other’s clock run slow” — so who is right?Extension

Two inertial observers move relative to each other. Each says the other’s moving clock runs slow. How can both statements be correct without contradiction?

Answer

Each statement is about how a moving clock’s tick interval relates to the observer’s own synchronized clocks and simultaneity convention in that observer’s inertial frame. There is no single “absolute” time interval for separated events; different frames assign different time coordinates consistently via Lorentz transformations.

8. Optional/Enrichment: The Low-Speed Approximation

For v≪ c, a useful approximation is: γ ≈ 1 + (1/2)v²/c² This shows why time dilation is negligible at everyday speeds.

Next in the maintained sequence: Length Contraction.

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