F3.3 Explain Einstein’s postulates and evidence for special relativity

Ontario Grade 12 Physics

Revolutions in Modern Physics: Quantum Mechanics and Special Relativity

How observations of light and moving particles support a new account of space and time

In SPH3U, you described motion using position, time, speed, and velocity. A reference frame is an observer’s coordinate system for describing where and when events happen. An inertial reference frame moves at constant velocity and does not rotate. The physical system in this lesson is the object or clock being observed. If it moves, choose its direction of travel as positive. Speed and elapsed time are scalars: they have magnitude but no direction. Velocity is a vector: it has magnitude and direction. Special relativity compares measurements made by inertial observers, especially when their relative speeds are very large.

What you will learn

State Einstein’s two postulates of special relativity.
Explain what an inertial reference frame is and why it matters.
Describe evidence for special relativity and distinguish measured results from what they support.
Use the Grade 12 time-dilation relationship in a calculation and check the result.

1. Einstein’s two postulates

A postulate is a starting statement used to build a model. Einstein’s first postulate says that the laws of physics have the same form in every inertial reference frame. No inertial frame is the absolute frame of the universe. For example, observers in a smoothly moving spacecraft and in a laboratory can use the same laws, even if they describe events from different frames.

Einstein’s second postulate says that every inertial observer measures the same speed for light in a vacuum. The speed is approximately

3.00 \times 10^8\ \mathrm{m/s}

. This result does not depend on how the light source or observer moves. The statement is specifically about light in a vacuum; light travels more slowly through materials such as glass.

In everyday motion, we often add the speed of a moving source to the speed of something it emits. That familiar rule does not predict the measured speed of light in a vacuum. In special relativity, inertial observers still measure the same value for that speed. The postulates therefore require a different account of how observers compare measurements of space and time.

c = 3.00 \times 10^8\ \mathrm{m/s}

An inertial frame moves at constant velocity and does not rotate.
The laws of physics have the same form in all inertial frames.
All inertial observers measure the same vacuum speed of light.

2. Time dilation as a consequence of the model

An event is something that happens at a particular place and time, such as a flash or a clock tick. A clock measures the elapsed time between two events. Observers in different inertial frames can measure different elapsed times between the same events. This effect is called time dilation.

Proper time is the elapsed time measured in the frame where the two events happen at the same location. For a clock, its own rest frame measures the time between ticks as proper time. An observer who sees that clock moving measures a longer interval. This does not mean that the clock’s owner notices their own clock behaving unusually. The statement compares measurements made in different frames.

At the Grade 12 algebra level, the relationship uses the proper time, the moving-frame time, and a factor that depends on relative speed. The symbol

v

is speed, so it has no direction in this equation. The ratio

v/c

has no units, and the factor is therefore unitless. The relationship applies to inertial frames and speeds below

c

. When

v

is much less than

c

, the factor is close to 1, so the difference is very small.

\Delta t = \gamma\Delta t_0,\qquad \gamma = \frac{1}{\sqrt{1-v^2/c^2}}

Proper time is measured in the frame where both events occur at one location.
A frame in which the clock moves measures a longer time interval.
For nonzero speed below $c$ , the time-dilation factor is greater than 1.

3. Evidence for special relativity

The Michelson–Morley experiment compared the travel of light along two perpendicular paths. The apparatus was turned, and the researchers looked for a shift in the interference pattern. An interference pattern is a set of bright and dark regions formed when light waves combine. Under an earlier expectation that Earth’s motion through a proposed light-carrying medium would change the travel times in the two directions, a shift was expected. The expected shift was not observed.

That result challenged the expectation of a detectable directional difference in light travel. It was consistent with the idea that the measured speed of light does not change in the expected way. It did not, by itself, prove every part of special relativity. It is important to state what was measured—the absence of the expected shift—separately from the interpretation.

Another line of evidence comes from muons. Muons are short-lived particles produced high in Earth’s atmosphere. Their speeds and the numbers reaching detectors near the ground have been measured. A calculation using the muons’ short proper lifetimes and everyday ideas about time would predict that very few travel far before decaying. Many are detected. The longer travel time calculated for the moving muons using time dilation agrees with this observation.

These are historical experimental findings, not results from a student experiment described here. Evidence supports a model when its predictions agree with measurements. The Michelson–Morley result and muon observations support important parts of special relativity; each should be described accurately without claiming that one result proves the whole theory.

Michelson–Morley found no expected interference-pattern shift.
Measured atmospheric muons are consistent with the longer time interval predicted by time dilation.
Separate the measured outcome from the explanation it supports.

4. A reliable calculation and evidence method

For a time-dilation problem, identify the system, frames, known values, and unknown first. The system may be a clock or a particle. State which frame measures proper time and which frame measures the moving clock’s interval. If motion direction is relevant to the description, choose a positive direction. The time-dilation equation uses speed, so a negative velocity is not substituted.

Calculate the factor using the speed as a fraction of the vacuum speed of light, then multiply by proper time. Keep units in substitutions. The speed ratio and the factor have no units, so if proper time is in seconds, the answer is also in seconds. Keep unrounded values during the calculation and round the final result to a sensible number of significant figures.

Finally, check the answer’s frame, units, and size. For a nonzero speed below

c

, the moving-frame time must be greater than proper time. For an evidence question, name the observation first, then state what it supports and what it does not establish.

\Delta t = \frac{\Delta t_0}{\sqrt{1-v^2/c^2}}

Assign each time interval to its frame before calculating.
Check that the speed is below $c$ and that the result has units of time.
In an evidence explanation, distinguish observation from interpretation.

Evidence and what it supports

Evidence	Measured result	What it supports
Michelson–Morley experiment	No expected interference-pattern shift	Challenges the expected directional change in light travel; consistent with the light-speed postulate
Atmospheric muons	Many muons reach detectors near the ground	Consistent with the longer Earth-frame time interval predicted by time dilation

Worked example

Comparing clock readings at high speed

A clock moves at

0.800c

relative to an inertial laboratory. Between two events at the clock’s location, it records

2.00\ \mathrm{s}

. Find the time between those events measured in the laboratory.

Set the system and frames
The system is the travelling clock. Its own frame measures proper time because both events occur at the clock’s location. The laboratory frame sees the clock move. Choose the direction of travel as positive; the equation needs the speed, not a signed velocity.
$\Delta t_0 = 2.00\ \mathrm{s},\qquad v = 0.800c$
Calculate the factor
Use the time-dilation relationship. The ratio of the speed to the speed of light is unitless.
$\gamma = \frac{1}{\sqrt{1-(0.800c)^2/c^2}} = 1.6667\ldots$
Find the laboratory interval
Multiply the proper time by the unrounded factor, then round the final result to three significant figures. The factor has no units, so the result remains in seconds.
$\Delta t = (1.6667\ldots)(2.00\ \mathrm{s}) = 3.33\ \mathrm{s}$

Answer: The laboratory measures an interval of

3.33\ \mathrm{s}

between the events.

Check: The result has units of seconds and is longer than the clock’s proper time of

2.00\ \mathrm{s}

. This is the expected direction of the time-dilation effect.

Worked example

Finding a particle’s speed from two lifetimes

A particle’s proper lifetime is

2.20\ \mu\mathrm{s}

. An inertial laboratory measures its lifetime as

5.50\ \mu\mathrm{s}

. Find its speed as a fraction of

c

and in SI units.

Identify the system and measurements
The system is the particle. Its rest frame measures the proper lifetime, and the laboratory measures the longer lifetime. The unknown is the particle’s speed relative to the laboratory.
$\Delta t_0 = 2.20\ \mu\mathrm{s},\qquad \Delta t = 5.50\ \mu\mathrm{s}$
Determine the factor
Divide the laboratory interval by proper time. The microsecond units cancel, leaving a unitless factor.
$\gamma = \frac{5.50\ \mu\mathrm{s}}{2.20\ \mu\mathrm{s}} = 2.50$
Solve for speed
Rearrange the factor relationship to find the positive speed ratio. Then multiply by the vacuum speed of light to express the result in metres per second.
$\frac{v}{c} = \sqrt{1-\frac{1}{(2.50)^2}} = 0.916,\qquad v = (0.916)(3.00 \times 10^8\ \mathrm{m/s}) = 2.75 \times 10^8\ \mathrm{m/s}$

Answer: The particle’s speed is

0.916c

, or

2.75 \times 10^8\ \mathrm{m/s}

to three significant figures.

Check: The speed has SI units when written in metres per second and is less than

c

. The laboratory lifetime exceeds proper lifetime, as required by time dilation.

Worked example

Evaluating a claim about Michelson–Morley

A student says, “The Michelson–Morley experiment found no expected fringe shift, so it directly measured a different speed of light for every observer.” Assess the claim.

State the experimental observation
The apparatus compared light travel along perpendicular paths and looked for an interference-pattern change as it was turned. The expected shift was not observed.
Separate evidence from interpretation
The observation was the absence of the expected shift. It challenged the expectation of a detectable directional difference in light travel. It was not a measurement showing different light speeds for different observers.
Connect the result to the postulate
The result is consistent with the postulate that inertial observers measure the same vacuum speed of light. A single result supports that account but does not establish every prediction of special relativity.

Answer: The claim is incorrect. The experiment found no expected fringe shift; it did not measure different light speeds for different observers.

Check: The conclusion accurately identifies the measured result and limits the claim to what that result supports.

Common mistakes and how to avoid them

Adding a light source’s speed to the vacuum speed of light.

Correction: Einstein’s second postulate says every inertial observer measures the same vacuum speed of light, regardless of source motion.

Calling the laboratory interval proper time when the clock or particle moves through the laboratory.

Correction: Proper time is measured in the frame where both events occur at the same location. For a particle’s lifetime, this is its rest frame.

Treating the Michelson–Morley result as proof of every prediction of special relativity.

Correction: State that no expected shift was observed, explain what expectation it challenged, and describe the result as evidence consistent with the postulates.

Using a negative velocity in the time-dilation factor because the object moves in the negative direction.

Correction: The factor uses speed, the magnitude of velocity. Use a non-negative value for

v

.

Lesson summary

The first postulate says the laws of physics have the same form in every inertial frame.
The second postulate says every inertial observer measures the same vacuum speed of light.
Time dilation relates proper time to a longer interval measured in a frame where the clock moves.
The Michelson–Morley result and measured atmospheric muons support important parts of special relativity.
Strong explanations distinguish a measured result from the conclusion it supports.

Check your understanding

Question 1

Two inertial observers move relative to one another. What does the second postulate say each measures for a light pulse in a vacuum?

The same speed, $c$
A speed equal to $c$ plus the source speed
A speed that depends on the observer’s direction of travel
A speed of zero if the observer travels with the source

Show answer and explanation

The same speed,

c

Every inertial observer measures the same vacuum speed of light, regardless of the motion of the source or observer.

Question 2

A clock’s proper time is

4.00\ \mathrm{s}

, and it moves at

0.600c

relative to an inertial observer. What interval does that observer measure?

$3.20\ \mathrm{s}$
$4.00\ \mathrm{s}$
$5.00\ \mathrm{s}$
$6.40\ \mathrm{s}$

Show answer and explanation

5.00\ \mathrm{s}

The factor is

1/\sqrt{1-0.600^2}=1.25

. The measured interval is

(1.25)(4.00\ \mathrm{s})=5.00\ \mathrm{s}

, which is longer than proper time.

Question 3

What key result did the Michelson–Morley experiment find, as described in this lesson?

It directly measured atmospheric muon lifetimes.
It found no expected interference-pattern shift as the apparatus was turned.
It showed that light in glass travels faster than light in a vacuum.
It proved that clocks in all frames have identical readings.

Show answer and explanation

It found no expected interference-pattern shift as the apparatus was turned.

The experiment found no expected fringe shift. This challenged an earlier expectation and was consistent with the special-relativity account of light speed.

Key terms

Reference frame: An observer’s coordinate system for describing positions and times.
Inertial frame: A reference frame that moves at constant velocity and does not rotate.
Postulate: A starting statement used to build a model.
Proper time: Elapsed time between two events measured in a frame where they occur at the same location.
Time dilation: A difference in elapsed-time measurements between inertial frames; a moving clock is measured to have a longer interval between ticks.
Interference pattern: A pattern of bright and dark regions formed when light waves combine.

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Published by DoAssignment. This AI-assisted lesson follows Ontario Grade 12 Physics (SPH4U), expectation F3.3. It is a study resource, not an official curriculum publication.

Before publication, the draft is checked for structure, mathematical or chemical notation, calculations, course boundaries, and readability, and then requires administrator approval. Errors can still occur, so corrections are welcomed.

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F3.3 · Explain Einstein’s postulates and evidence for special relativity

Ontario Grade 12 Physics

How observations of light and moving particles support a new account of space and time

What you will learn

1. Einstein’s two postulates

2. Time dilation as a consequence of the model

3. Evidence for special relativity

4. A reliable calculation and evidence method

Evidence and what it supports

Comparing clock readings at high speed

Finding a particle’s speed from two lifetimes

Evaluating a claim about Michelson–Morley

Common mistakes and how to avoid them

Lesson summary

Check your understanding

Question 1

Question 2

Question 3

Key terms

Continue through SPH4U

About this lesson and its review